New Perspectives on Kac–Moody Algebras Associated with Higher-Dimensional Manifolds

Abstract

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle ${\mathbb{S}}^1$, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, we present the construction of generalized current algebras $\mathfrak{g}\left(\mathcal{M}\right)$, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and we illustrate the framework through explicit realizations on $SU\left(2\right)$, $SU\left(2\right)/U\left(1\right)$, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications.

1. Introduction

1.1. From Finite- to Infinite-Dimensional Symmetry

For quite a long time, finite-dimensional simple Lie algebras have provided the backbone of symmetry in mathematics and physics. The classification by Cartan and the subsequent development of representation theory turned these structures into essential tools throughout geometry, number theory, and quantum theory. Beginning in the late 1960s, two independent lines of development—Moody’s and Kac’s—generalized this landscape to the infinite-dimensional realm, resulting in what are now called Kac–Moody (KM) algebras [1,2,3]. A particularly tractable and physically relevant subclass is formed by the affine algebras, realized as central extensions of loop algebras. Pressley and Segal’s monograph rigorously analyzed the functional-analytic and group-theoretic underpinnings in terms of loop groups and their projective unitary representations [4].

The importance of the affine KM symmetry in physics was then amplified by its tight relation to the Virasoro algebra, via the Sugawara construction and current algebra methods [5]. This connection underlies the solvability of two-dimensional conformal field theories (CFTs) and the Wess–Zumino–Witten (WZW) models, and it has been comprehensively discussed in the CFT literature [6,7]. At the same time, the broader spectrum of KM algebras beyond the affine case (e.g., indefinite and hyperbolic generalizations) has been invoked in diverse corners of high-energy theoretical physics, notably in hidden symmetries of supergravity, cosmology, and string/M-theory.

1.2. Beyond the Circle: Why?

The best-known laboratory for KM algebras is the loop algebra over a circle ${\mathbb{S}}^1$. Physically, the ubiquity of the circle form stems from the role of closed strings and from the reduction of two-dimensional field theories on a spatial circle. Mathematically, ${\mathbb{S}}^1$ is distinguished by its Fourier basis as well as by the resulting algebraic simplifications that enable a complete representation theory for affine algebras. However, many problems of current interest in mathematical physics naturally involve higher-dimensional manifolds:

• Compactifications in Kaluza–Klein (KK) theory and string/M-theory crucially involve group manifolds and coset spaces; examples include spheres, tori, and symmetric spaces, which are, e.g., discussed in [8,9,10,11,12,13].
• Non-compact, Riemannian target spaces (e.g., $\mathrm{SL}(2,\mathbb{R})$ or $\mathrm{SL}(2,\mathbb{R})/U\left(1\right)$) arise in (ungauged and gauged) supergravity theories in diverse space-time dimensions, as well as in the theory of black hole attractors (see, e.g., [14] and the references therein).
• Deformations of group manifolds (soft or non-homogeneous manifolds) appear in the effective descriptions of flux compactifications and in group-geometric approaches to supergravity [15].

In all such settings, the natural notion of “currents” is no longer related to loops on ${\mathbb{S}}^1$, but rather it is connected to functions (or sections) on a manifold $\mathcal{M}$, with values in a (generally finite-dimensional) Lie algebra $\mathfrak{g}$. The resulting current algebra $\mathfrak{g}\left(\mathcal{M}\right)$ inherits both the algebraic structure from $\mathfrak{g}$ and the geometric/analytic structure from $\mathcal{M}$. The central question we address in this review is, to what extent do the characteristic features of affine KM algebras—such as central extensions, semidirect actions by diffeomorphisms or isometries, and rich representation theory—survive when ${\mathbb{S}}^1$ is replaced by a higher-dimensional manifold $\mathcal{M}$?

1.3. A Unifying Viewpoint

Our viewpoint is twofold: geometric and representation theoretic. We begin by considering a Riemannian (or pseudo-Riemannian) manifold $\mathcal{M}$, equipped with a measure and a Hilbert space $L^2\left(\mathcal{M}\right)$ of square integrable functions. A key point is the existence of a Hilbert basis adapted to the symmetries of $\mathcal{M}$. On the one hand, for compact Lie groups $G_c$, the Peter–Weyl theorem provides such a basis in terms of matrix elements of unitary irreducible representations [16]. On the other hand, for non-compact Lie groups $G_{nc}$, the Plancherel theorem organizes $L^2\left(G_{nc}\right)$ functions as a ‘mixture’ of discrete and continuous series contributions [17,18]. This harmonic-analytic formalism allows us to expand $\mathfrak{g}$-valued fields on $\mathcal{M}$ and, thus, to define current algebras $\mathfrak{g}\left(\mathcal{M}\right),$ as well as their (semi)direct extensions by symmetry algebras generated by Killing vectors of $\mathcal{M}$ itself. In general, central extensions can be characterized cohomologically, in terms of closed currents on $\mathcal{M}$, and they can also be explicitly matched to local Schwinger terms in current commutators.

All this can be summarized into an algorithmic construction, which provides a coherent perspective encompassing spheres ${\mathbb{S}}^n$, compact groups, such as $SU\left(2\right)$ and their homogeneous spaces, non-compact groups, such as $\mathrm{SL}(2,\mathbb{R})$, and generalizations with a softened group structure. The steps of such an algorithmic construction are listed as follows:

• Choose $\mathcal{M}$ (compact or non-compact group manifold, a Riemannian or pseudo-Riemannian coset thereof, or soft deformations thereof, …) and its isometry algebra.
• Select an appropriate orthonormal basis on $L^2\left(\mathcal{M}\right)$ (as stated by the Peter–Weyl or Plancherel theorems), and then promote $\mathfrak{g}$-valued modes to generators.
• Determine the semidirect actions using isometries/diffeomorphisms, and describe/classify all compatible 2-cocycles that yield central extensions.
• Identify structural features (e.g., Witt/Virasoro analogs and area-preserving diffeomorphisms) related to the geometry of $\mathcal{M}$.

1.4. Infinite Dimensional Algebras in Physics

Historically, the amazingly fruitful cross-fertilization between infinite-dimensional algebras and physics has evolved along several lines of research. In two-dimensional CFT, the Virasoro and affine Kac–Moody algebras have provided the dynamical symmetry principles underlying the theory of exactly solvable models, modular covariance, and the classification of primary fields [4,5,6,7]. In string theory, worldsheet reparameterization and current algebra symmetries determine crucial consistency conditions (termed the anomaly cancellation condition, possibly exploiting BRST cohomology), while WZW models provide exact backgrounds with affine symmetry. The celebrated and surprising monstrous moonshine and generalized KM structures (Borcherds–Kac–Moody algebras) connect CFTs, sporadic groups, and automorphic forms, highlighting the depth of the algebraic structures that are emergent in quantum theories [19,20,21].

Another remarkable appearance of KM algebras pertains to the large hidden Lie symmetries arising in supergravity and in cosmological dynamics. For instance, cosmological billiards uncovered links between the asymptotic dynamics of gravity near space-like singularities and hyperbolic KM symmetries [22,23]. Moreover, extended supergravities and dimensional reductions have suggested hierarchies of very extended algebras, as well as infinite towers of dual potentials; see, for instance, [24,25,26,27,28]; for geometric and group-theoretic analyses, see [12,13,29]. Within this context, the KK theory has provided a wealth of examples in which harmonic analysis (on compact $\mathcal{M}$) describes the spectra of lower-dimensional fields [8,9,10,11]. Finally, current algebras naturally arise on $\mathcal{M}$, being defined in terms of Noether charges integrated over the internal spaces, and a possible subsequent mode expansion.

1.5. Scope and Plan of This Review

Expectedly, the generalization from ${\mathbb{S}}^1$ to higher-dimensional manifolds $\mathcal{M}$ generates both opportunities and challenges. The following lists a few of these:

• Central extensions. On ${\mathbb{S}}^1$, there is essentially a unique (up to normalization) central extension of the loop algebra, yielding an affine KM algebra. Typically, for higher-dimensional manifolds, the space of admissible 2-cocycles is infinite-dimensional. Cohomological constructions tied to closed $(n-1)$-currents on $\mathcal{M}$ and divergence-free vector fields can be exploited, directly relating to Schwinger terms in the current algebra.
• Symmetry by isometries and diffeomorphisms. The Lie algebra generated by the Killing vectors (or broader diffeomorphism algebras) acts naturally on $\mathfrak{g}\left(\mathcal{M}\right)$. In favorable cases, one can identify subalgebras, which are analogs of the Witt/Virasoro algebras (e.g., de Witt algebra on ${\mathbb{S}}^n$, or area-preserving diffeomorphisms on ${\mathbb{S}}^2$), yielding semidirect products that hint at affine–Virasoro interplay, which is familiar from two dimensions.
• Harmonic analysis and representation theory. On compact groups and the homogeneous spaces thereof, the Peter–Weyl theorem and Clebsch–Gordan machinery allow for expressing products of basis functions in terms of representation-theoretic data. On the other hand, for non-compact groups, such as $\mathrm{SL}(2,\mathbb{R})$, a ‘mixture’ of discrete and continuous series appears via the Plancherel theory (also implemented through Bargmann’s classification), and new unitary structures emerge.
• Applications. The above constructions provide an algebraic background for higher-dimensional current algebras in effective field theories, organize spectra in compactifications, and suggest new Virasoro-like structures that may control sectors of dynamics in higher-dimensional CFTs or holographic models.

This review aims to be pedagogical while remaining faithful to the breadth of the subject. We deliberately separate general principles from case studies:

• We present a construction based on manifolds that keeps track of metric, measure, and symmetry data, and we adopt a formalism based on the Hilbert space in order to emphasize unitarity and completeness of mode expansions.
• We use group- and representation-theoretic tools (such as the Peter–Weyl and Plancherel theorems or the Clebsch–Gordan coefficients) to highlight the algebraic structure and to allow explicit computations of structure constants in current algebras on $\mathcal{M}$.
• We discuss central extensions from two perspectives: the cohomology of current algebras on $\mathcal{M}$ and anomalies/Schwinger terms in local commutators.
• Throughout our treatment, we strive to highlight links to well-established areas in physics, such as two-dimensional CFT and string theory [4,5,6,7] or higher-dimensional supergravity and compactifications [8,9,10,11,12,13,24,25,26,27,28,29].

There exist several classic reference works on affine algebras and loop groups [4,5,6], on group manifolds and harmonic analysis [16,17], and on supergravity/sigma model applications [8,9,12,13,29]. In this respect, it is worth emphasizing that our focus is complementary: we aim to present a single construction, which applies to wide classes of manifolds, such as compact groups and cosets, non-compact groups and their homogeneous spaces, and (soft) deformations of group manifolds. As mentioned above, we also present a systematic analysis of the space of compatible central extensions. Various examples are discussed and treated in some detail in order to provide a glimpse of the wealth of the subject, rather than to exhaust all possibilities.

The plan of this review is as follows.

• Section 1: General framework on manifolds. We recall the geometric background on (pseudo-)Riemannian manifolds, measures, and Killing vectors; we introduce $L^2\left(\mathcal{M}\right)$ and orthonormal bases adapted to symmetries (exploiting the Peter–Weyl or Plancherel theorems). We then define the current algebra $\mathfrak{g}\left(\mathcal{M}\right)$, its semidirect extension by isometries, and set up the cohomological language for central extensions.
• Section 2: Compact group manifolds and cosets. Invoking the Peter–Weyl theorem, we construct $\mathfrak{g}\left(\mathcal{M}\right)$ on $G_c$ and on homogeneous spaces $G_c/H$. We explain how products of basis functions are controlled by Clebsch–Gordan coefficients and how central extensions are enumerated by closed currents. As examples, we highlight analogs of Witt/Virasoro subalgebras on spheres and their role in semidirect products.
• Section 3: Non-compact groups and Plancherel analysis. We treat $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{SL}(2,\mathbb{R})/U\left(1\right)$ as canonical examples, reviewing Bargmann’s discrete/continuous series and the Plancherel decomposition. We construct $\mathfrak{g}\left(\mathcal{M}\right)$ and compatible central extensions and discuss unitarity issues and constraints for the resulting representations.
• Section 4: Soft (deformed) group manifolds. We consider the so-called soft deformations of group manifolds motivated by supergravity and string compactifications, discussing various aspects in some detail.
• Section 5: Root systems and representation theory. We introduce a system of roots for the classes of Kac–Moody algebras introduced above. In the second part of this Section, we introduce some elements of representation theory, stressing important differences with respect to affine Lie algebras. For simplicity’s sake, we will here assume that $\mathfrak{g}$ is a compact Lie algebra.
• Section 6: Applications to physics. We discuss some applications of the above mathematical machinery to physics, most notably to (i) two-dimensional current algebra/CFT (WZW, Sugawara, Virasoro); (ii) higher-dimensional compactifications and spectra in Kaluza–Klein theory; (iii) structures emerging within the theory of cosmological billiards, and the underlying hidden symmetries of supergravity.
• Section 7: Outlook and open problems. We outline classification questions for central extensions on general $\mathcal{M}$, representation-theoretic challenges beyond compact groups, and possible applications to holography and integrability.

2. Kac–Moody (KM) Algebras on Higher-Dimension Manifolds: Some Generalities

Let $\mathfrak{g}$ be a semisimple (complex or real) Lie algebra. To the simplest (one-dimensional) manifold, namely the circle ${\mathbb{S}}^1$, one can associate two non-isomorphic infinite-dimensional Lie algebras. The first is the affine extension of the loop algebra of smooth maps from ${\mathbb{S}}^1$ to $\mathfrak{g}$, which allows for the construction of KM (or better, affine) Lie algebras [1,2,3,4,5]. A second possibility is given by the Virasoro algebra, which is a central extension of the de Witt algebra (i.e., the algebra of polynomial vector fields on ${\mathbb{S}}^1$), and is intimately connected to affine Kac–Moody algebras via the Sugawara construction (see, e.g., [7,30]).

A natural question concerns the possibility of defining infinite-dimensional KM algebras related to higher-dimensional manifolds, possibly maintaining some of the structural properties observed for the previous relevant cases. Over the years, a number of infinite-dimensional KM algebras have been constructed. The first examples are associated to specific manifolds, such as the two-sphere, three-sphere, or $n-$tori [13,31,32,33,34,35,36,37] cases. Castellani also applied infinite-dimensional KM algebras in his works on geometric field theories of (closed) strings [38,39]. Then, general studies for more general manifolds were undertaken in [40,41,42,43,44,45,46,47]. To this extent, various types of manifolds $\mathcal{M}$ were considered. This systematic study started with $\mathcal{M}$ being a compact $n-$dimensional manifold [40,41,42,43,44] (essentially, a compact Lie group $G_c$—not to be confused with G, the Lie group associated to $\mathfrak{g}$– or a coset space $G_c/H$ with H a closed subgroup of $G_c$). This construction is ultimately motivated by higher-dimensional physical theories in which harmonic expansions à la Kaluza–Klein play a crucial role (see, e.g., [8,9,10,11], with the latter reference being itself motivated by the supergravity context). The second series of extensions concerns non-compact manifolds, principally SL$(2,\mathbb{R})$ or the coset SL$(2,\mathbb{R})/$U$\left(1\right)$ [45,46]. These constructions were motivated by the fact that the non-compact manifold $SL(2,\mathbb{R})/U\left(1\right)$ appears as the target space of one complex scalar field in the bosonic sector of some Maxwell–Einstein supergravity theories in $D=3+1$ space-time dimensions. Finally, the last type of algebra [47] is associated to manifolds that are deformations of Lie groups, called a ‘soft’ group manifold (see, e.g., [15]).

In this section, before considering the KM algebras associated to the specific types of manifolds presented above, we briefly recall some general properties of real manifolds and explain the various steps to construct the corresponding Kac–Moody algebra.

2.1. Relevant Properties of Manifolds

Let $\mathcal{M}$ be an $n-$dimensional real Riemannian manifold. Let $\mathcal{U}\subset \mathcal{M}$ be an open set of $\mathcal{M}$. We assume that the manifold is such that $\mathcal{M}\setminus \mathcal{U}$ is of zero measure. Define a homomorphism from $\mathcal{U}$ to an open set of $\mathcal{O}\subset {\mathbb{R}}^n$ as

$$\begin{array}{c}\hfill \begin{array}{cccc}f:\hfill & \mathcal{U}& \to & \mathcal{O}\subset {\mathbb{R}}^n\hfill \\ & m& \mapsto & f\left(m\right)=(y^1,\dots ,y^n),\hfill \end{array}\end{array}$$

meaning that we can associate a system of coordinates for points m in $\mathcal{M}$ almost everywhere. Let $g_{IJ}$ be the metric tensor of $\mathcal{M}$ in the the system of coordinates $y^I=\left(y^1,\dots ,y^n\right)$:

$$\begin{array}{c}\hfill {\mathrm{d}}^{2}s=\mathrm{d}{y}^I\mathrm{d}{y}^J{g}_{IJ}\end{array}$$

and define $g=|det{g}_{IJ}|$. Consider now an infinitesimal diffeomorphism:

$$\begin{array}{c}\hfill y^I\to y^{\prime I}+ϵ{\xi }^I\left(y\right).\end{array}$$

This transformation is an isometry of $\mathcal{M}$, i.e., it preserves the tensor metric $g_{IJ}$, if the ${\xi }^I$ satisfies the Killing equation:

$$\begin{array}{c}\hfill {\nabla }_I{\xi }_J\left(y\right)+{\nabla }_J{\xi }_I\left(y\right)=0,\end{array}$$

(1)

where ${\xi }_I=g_{IJ}{\xi }^J$ and ${\nabla }_I$ is the covariant derivative [48]. Suppose that the Killing equation has $s-$independent solutions ${\xi }_A^I\left(y\right),A=1,\dots ,s$; then, the operators

$$\begin{array}{c}\hfill {\mathfrak{g}}_{\xi }=\left\{K_A=i{\xi }_A^I\left(y\right){\partial }_I,A=1,\dots ,s\right\}\end{array}$$

(2)

generate the Lie algebra with Lie brackets

$$\begin{array}{c}\hfill \left[K_A,K_B\right]=i{c}_{AB}{}^{C}K_C,\end{array}$$

(3)

since the composition of two isometries is an isometry, where

$$\begin{array}{c}\hfill -{\xi }_A^J{\partial }_J{\xi }_B^I+{\xi }_B^J{\partial }_J{\xi }_A^I=-c_{AB}{}^C\xi _C^I,\end{array}$$

because the Killing vectors are associated to independent infinitesimal isometries.

We endow the manifold $\mathcal{M}$ with a Hermitian scalar product:

$$\begin{array}{c}\hfill (f,g)={\int }_{\mathcal{O}}\sqrt{g}{\mathrm{d}}^{n}y\overline{f\left(y\right)}g\left(y\right)\end{array}$$

(4)

where $\mathcal{O}=f\left(\mathcal{U}\right)\subset {\mathbb{R}}^n$. The Hermiticity of the operators $K_A$ with respect to the scalar product (4) translates into two conditions:

$$\begin{array}{c}\hfill \begin{array}{cc}1:\hfill & {\partial }_I\left(\sqrt{g}{\xi }_A^I\right)=0,\hfill \\ 2:\hfill & {\xi }_A^I|=0,I=0,\dots ,n\hfill \end{array}\end{array}$$

(5)

where ${\xi }_A^I|=0$ means that the boundary term associated to all directions vanishes. In particular, note that if ${\xi }_A^I$ are periodic in a direction, condition 2 is automatically satisfied. Similarly, if a direction is unbounded, say $y^{I_0}\in \mathbb{R}$, condition 2 is also satisfied since ${\xi }_a^{I_0}\to 0$ whenever $y^{I_0}\to \pm \infty$. Furthermore, as ${\xi }_a^I$ is a Killing vector, we have a Levi-Civita connection satisfying ${\nabla }_I{g}_{JK}=0$; hence, the Killing condition (1) implies condition 1 above (recall that ${\nabla }_I{V}^I=1/\sqrt{g}{\partial }_I\left(\sqrt{g}{V}^I\right)$).

The last step of this general introduction is to consider the Hilbert space of square integrable functions on $\mathcal{M}$, namely $L^2\left(\mathcal{M}\right)$, or more precisely, $L^2\left(\mathcal{O}\right)$. It is known that any Hilbert space admits a Hilbert basis [49,50]. Let $\mathcal{B}=\{b_M,M\in \mathbb{N}\}$ be a Hilbert basis of $L^2\left(\mathcal{O}\right)$, i.e., a complete orthonormal set of vectors (for the scalar product (4)). In our case, the vectors $b_N$ become functions on $\mathcal{O}$. We also assume that for any generators $K_A$ of the Lie algebra generated by the Killing vectors (see (3)), and for any vector $b_M$ in $\mathcal{B}$, the function $K_A{b}_N$ is square integrable. Thus,

$$\begin{array}{c}\hfill K_A{b}_M\left(y\right)={{\left(M_A\right)}_M}^N{b}_N\left(y\right)\end{array}$$

(6)

where the symbol of summation is omitted, and ${{\left(M_A\right)}_M}^N$ is a matrix representing the action of ${\mathfrak{g}}_{\xi }$ on $b_M$. Indeed, this equation, in fact, means that the vectors $b_M$ transform according to a given representation of ${\mathfrak{g}}_{\xi }$. In general, this representation is fully reducible, but it may happen that a subset of vectors does not belong to an irreducible representation (see Section 4). Since the set of vectors $b_M$ is orthonormal and constitutes a Hilbert basis of $L^2\left(\mathcal{O}\right)$, these vectors also constitute a unitary representation of ${\mathfrak{g}}_{\xi }$; hence, the operators $K_A$ are Hermitian and, thus, satisfy (5) (in particular, the second condition).

In the following, two types of manifolds will be considered. The first type corresponds to group manifolds of compact Lie groups $G_c$, while the second corresponds to group manifolds of non-compact Lie groups $G_{nc}$. In both cases, the decomposition of square integrable functions (or the Hilbert basis $\mathcal{B}$) is organized within the representation theory of $G_c$ (resp., $G_{nc}$), but it should be observed that these two cases are structurally quite different. If the group is compact, its unitary representations are finite-dimensional, and it turns out that once correctly normalized, the set of all matrix elements constitutes an orthonormal Hilbert basis of $L^2\left(G_c\right)$. This corresponds to the Peter–Weyl Theorem [16]. If the group is non-compact, its unitary representations are infinite-dimensional, and square integrable functions decompose into a sum over the matrix elements of the discrete series and an integral over the matrix elements of the continuous series (see below). In this situation, the Plancherel theorem has to be used [51].

Returning to the Hilbert basis $\mathcal{B}$, we assume further that for any elements $b_N,b_M$ of $\mathcal{B}$, the product $b_N{b}_M$ is square integrable; thus, in particular, we have

$$\begin{array}{c}\hfill b_M\left(y\right)b_N\left(y\right)={C_{MN}}^P{b}_P\left(y\right)\end{array}$$

(7)

where the symbol of summation is omitted. In general, the coefficients ${C_{MN}}^P$ are difficult to compute explicitly, except (at least) for two types of manifolds, where (i) $\mathcal{M}$ is related to a compact Lie group $G_c$ or (ii) $\mathcal{M}$ is related to a non-compact Lie group $G_{nc}$. In both cases, the coefficients ${C_{MN}}^P$ can be expressed explicitly by means of the Clebsch–Gordan coefficients of $G_c$ (resp., $G_{nc}$).

Since $\mathcal{M}$ is a real manifold, one can choose the set of functions $b_M$ to be real functions. It is, however, useful for certain manifolds $\mathcal{M}$ to consider complex-valued functions. The reality of $\mathcal{M}$ imposes, then, that for any $b_M$ in $\mathcal{B}$, we have

$$\begin{array}{c}\hfill {\overline{b}}^N\left(y\right)={\eta }^{NM}{b}_M\left(y\right),\mathrm{or}{b}_M\left(y\right)={\eta }_{MN}{\overline{b}}^N\left(y\right),\end{array}$$

(8)

where ${\eta }_{MN}={\eta }^{MN}$, such that, for a given $M\in \mathbb{N}$, there exists a unique $N\in \mathbb{N}$ such that ${\eta }_{MN}$ is non-vanishing. One can, of course, choose ${\eta }_{MN}=1$, albeit for group manifolds, we could also have ${\eta }_{MN}=\pm 1$ (see Section 3). Finally, in this section, the index of the functions $b_M$ belongs to $\mathbb{N}$. In turn, other choices will be more relevant in the next section; in particular, negative integer values for M. In this case, condition (8) will be obvious, and ${\eta }_{MN}\ne 0$ if $N=-M$.

Since the basis $\mathcal{B}$ is complete, we have the completeness relation:

$$\begin{array}{c}\hfill \sum _{N\in \mathbb{N}}{\overline{b}}^N\left(y\right)b_N\left(y^{\prime }\right)={\displaystyle \frac{{\delta }^n(y-y^{\prime })}{\sqrt{g\left(y\right)}}}.\end{array}$$

(9)

2.2. A KM Algebra Associated to $\mathcal{M}$

Let $\mathfrak{g}=\{T_a,a=1,\dots ,d\}$ be a $d-$dimensional simple Lie algebra with Lie brackets

$$\begin{array}{c}\hfill \left[T_a,T_b\right]=i{f}_{ab}{}^{c}T_c.\end{array}$$

We assume, for simplicity, that $\mathfrak{g}$ is a compact Lie algebra with Killing form:

$$\begin{array}{c}\hfill {⟨T_a,T_b⟩}_0=k_{ab}=\mathrm{tr}\left(\mathrm{ad}\left(T_a\right)\mathrm{ad}\left(T_b\right)\right),\end{array}$$

where $\mathrm{ad}$ denotes the adjoint representation of $\mathfrak{g}$, and $k_{ab}$ is definite positive.

In the next step of our construction, we define the space of smooth mappings from $\mathcal{M}$ into $\mathfrak{g}$ as

$$\begin{array}{c}\hfill \mathfrak{g}\left(\mathcal{M}\right)=\left\{T_{aM}=T_a{b}_M\left(y\right),a=1,\dots ,d,M\in \mathbb{N}\right\},\end{array}$$

(10)

which inherits the structure of a Lie algebra

$$\begin{array}{c}\hfill \left[T_{aM},T_{bN}\right]={{if}_{ab}}^c{C_{MN}}^P{T}_{cP},\end{array}$$

(11)

where the coefficients ${C_{MN}}^P$ are defined in (7). The Killing form in $\mathfrak{g}\left(\mathcal{M}\right)$ is given by

$$\begin{array}{c}\hfill {⟨X,Y⟩}_1={\int }_{\mathcal{O}}\sqrt{g}\mathrm{d}{x}^n{⟨X,Y⟩}_0,\end{array}$$

for $X,Y\in \mathfrak{g}\left(\mathcal{M}\right)$. From (8), it follows that

$$\begin{array}{c}\hfill {⟨T_{aM},T_{bN}⟩}_1=k_{ab}{\eta }_{MN}.\end{array}$$

(12)

In fact, if we assume

$$\begin{array}{c}\hfill \left[T_a\left(y\right),T_b\left(y^{\prime }\right)\right]=i{f_{ab}}^c{T}_c\left(y\right){\displaystyle \frac{{\delta }^n(y-y^{\prime })}{\sqrt{g}}},\end{array}$$

(13)

where

$$\begin{array}{c}\hfill T_a\left(y\right)=\sum _{M\in \mathbb{N}}{T}_{aM}{\overline{b}}^M\left(y\right)\end{array}$$

(14)

then integrating both sides of the equation above by $\int \sqrt{g}{\mathrm{d}}^{n}y{b}_P\left(y\right)\int \sqrt{g}{\mathrm{d}}^n{y}^{\prime }{b}_q\left(y^{\prime }\right)$, (13), reproduces (11) using (9) and (7). Clearly, the algebra $\mathfrak{g}\left(\mathcal{M}\right)$ is the generalization of the loop algebra corresponding to the $n-$dimensional manifold $\mathcal{M}$ (the loop algebra is associated to $\mathcal{M}={\mathbb{S}}^1$, a circle).

Continuing the construction, we recall that $\mathcal{M}$ is such that the operators associated to the Killing vectors ${\mathfrak{g}}_{\xi }=\{K_A,A=1,\dots ,s\}$ are Hermitian with respect to the scalar product (4) (see (5)). Let $\mathfrak{g}\left(\mathcal{M}\right)⋊{\mathfrak{g}}_{\xi }$. This algebra admits a semidirect structure:

$$\begin{array}{ccc}\hfill \left[T_{aM},T_{bN}\right]& =& {{if}_{ab}}^c{C_{MN}}^P{T}_{cP},\hfill \\ \hfill \left[K_A,b_M\left(y\right)\right]& =& {{\left(M_A\right)}_M}^N{b}_N\left(y\right),\hfill \\ \hfill [K_A,K_B]& =& i{c_{AB}}^C{K}_C.\hfill \end{array}$$

(15)

It may be interesting to identify the maximal set of commuting operators (see (6)). Let r be the rank of ${\mathfrak{g}}_{\xi }$, and let $\{D_1,\dots ,D_r\}$ be a Cartan subalgebra. Without loss of generality, we can assume that the elements of the basis $\mathcal{B}$ are eigenvectors of the Ds:

$$\begin{array}{c}\hfill \left[D_i,b_M\left(y\right)\right]=M\left(i\right)b_M\left(y\right)\end{array}$$

(16)

with $M\left(i\right)\in \mathbb{R}$. It is important to emphasize that,

$$\begin{array}{c}\hfill \left[D_i,b_M\left(y\right)\right]=M\left(i\right)b_M\left(y\right)\iff \left[D_i,{\overline{b}}^M\left(y\right)\right]=-M\left(i\right){\overline{b}}^M\left(y\right).\end{array}$$

(17)

The algebra above can even be further extended. Introduce the Lie algebra of vector fields $\mathfrak{X}\left(\mathcal{M}\right)=\{V_{IM}=-i{b}_M\left(y\right){\partial }_I,I=1,\dots ,n,M\in \mathbb{N}\}$ on $\mathcal{M}$, where ${\partial }_I={\displaystyle \frac{\partial }{\partial y^I}}$. The algebra (15) extends to $\mathfrak{g}\left(\mathcal{M}\right)⋊\mathfrak{X}\left(\mathcal{M}\right)$:

$$\begin{array}{ccc}\hfill \left[T_{aM},T_{bN}\right]& =& {{i f}_{ab}}^c{C_{MN}}^P{T}_{cP},\hfill \\ \hfill \left[V_{IM},V_{JN}\right]& =& -i\left(\left({\partial }_I{b}_N\right)V_{JM}-\left({\partial }_J{b}_M\right)V_{IN}\right),\hfill \\ \hfill \left[V_{IM},T_{aN}\right]& =& -i{b}_M{\partial }_I{b}_N{T}_a\equiv -{{id}_{IM,N}}^P{T}_{aP},\hfill \end{array}$$

(18)

and also has a semidirect product structure. This algebra admits (15) as a subalgebra and also some analog of the de Witt algebras. Let

$$\begin{array}{c}\hfill {\mathrm{Witt}}_i\left(\mathcal{M}\right)=\left\{{\mathsf{\ell }}_{iM}=-b_M\left(y\right)D_i,M\in \mathbb{N}\right\},i=1,\dots ,s\end{array}$$

(19)

which are subalgebras of the algebra of vector fields on $\mathcal{M}$. The algebra (15) extends to $\mathfrak{g}\left(\mathcal{M}\right)⋊{\mathrm{Witt}}_i\left(\mathcal{M}\right)$

$$\begin{array}{ccc}\hfill \left[T_{aM},T_{bN}\right]& =& {{i f}_{ab}}^c{C_{MN}}^P{T}_{cP},\hfill \\ \hfill \left[{\mathsf{\ell }}_{iM},{\mathsf{\ell }}_{jN}\right]& =& (M\left(i\right)-N\left(j\right)){C_{MN}}^P{\mathsf{\ell }}_{iP}),\hfill \\ \hfill \left[{\mathsf{\ell }}_{iM},T_{aN}\right]& =& -M\left(i\right){C_{MN}}^P{T}_{aP}.\hfill \end{array}$$

(20)

The last step of the construction is to extend (centrally) the algebra $\mathfrak{g}\left(\mathcal{M}\right)$. Central extensions of $\mathfrak{g}\left(\mathcal{M}\right)$ were identified by Pressley and Segal in [4] (see Proposition 4.28 therein). Given a one-chain C (i.e., a closed one-dimensional, piecewise smooth curve), the central extension is given by the two-cocycle

$$\begin{array}{c}\hfill {\omega }_C(X,Y)={\oint }_C{⟨X,\mathrm{d}Y⟩}_0,\end{array}$$

(21)

where $\mathrm{d}Y={\partial }_{I}Y\mathrm{d}{y}^I$ is the exterior derivative of Y. The two-cocycle can be written in an alternative form. Indeed, we have [52]

$$\begin{array}{c}\hfill {\omega }_C(X,Y)={\int }_{\mathcal{O}}{⟨X,\mathrm{d}Y⟩}_0\wedge \gamma ,\end{array}$$

(22)

where $\gamma$ is a closed $(n-1)-$current (a distribution) associated to C. We now show that the cocycle (21) can also be associated to a vector field on $L\in \mathfrak{X}\left(\mathcal{M}\right)$. Let $L=f^I\left(y\right){\partial }_I$ be a vector field such that

$$\begin{array}{c}\hfill {\partial }_I\left(\sqrt{g}{f}^I\right)=0.\end{array}$$

(23)

To the operator L, one can associate the $1-$form:

$$\begin{array}{ccc}\hfill F\left(y\right)& =& f_I\left(y\right)\mathrm{d}{y}^I\hfill \end{array}$$

(24)

where $f_I=g_{IJ}{f}^J$, and the $(n-1)-$form:

$$\begin{array}{c}\hfill \gamma \left(y\right)=\sqrt{g}\sum _{I=1}^n{(-)}^{I-1}{f}^I\left(y\right)\mathrm{d}{y}^1\wedge \dots \mathrm{d}{y}^{I-1}\wedge \mathrm{d}{y}^{I+1}\wedge \dots \wedge \mathrm{d}{y}^n.\end{array}$$

(25)

Obviously, if $\mathcal{M}$ is orientable, $\gamma ={}^{\ast }F$ is the Hodge dual of F. Now, condition (23) implies that $\gamma$ is closed and, equivalently, that F is co-closed. Moreover, if the cohomology group $H^{n-1}\left(\mathcal{M}\right)=1$ is trivial, then F is co-exact or $\gamma$ is exact, and there exists a one-form G such that

$$\begin{array}{c}\hfill F={}^{\ast }\mathrm{d}^{\ast }G,\gamma ={(-1)}^{q+n-1}\mathrm{d}h,\end{array}$$

where $h={}^{\ast }G$, and the metric g has the signature $(p,q)$ with q minus signs. Since the $(n-1)-$form associated to the operator L is closed, the corresponding two-cocycle (22) reduces to

$$\begin{array}{c}\hfill \omega {(X,Y)}_f=\underset{\mathcal{O}}{\int }\sqrt{g}{\mathrm{d}}^{n}y{⟨X,LY⟩}_0=\underset{\mathcal{O}}{\int }{⟨X,\mathrm{d}Y⟩}_o\wedge \gamma .\end{array}$$

(26)

We conclude that there is an infinite number of central extensions associated to any operator L satisfying (23). In [53], Feigin showed that the number of central extensions is equal to the dimension of $H^2\left(\mathfrak{g}\left(\mathcal{M}\right)\right)$ and proved that when $dim\mathcal{M}>1$ and $\mathcal{M}$ is compact, we have an infinite number of central extensions. Central extensions were systematically computed in the case of the two- and three-sphere in [33,36,37].

The result above can also be obtained differently. Indeed, one can centrally extend the ‘loop’ algebra $\mathfrak{g}\left(\mathcal{M}\right)$ by using the current algebra and introducing an anomaly or a Schwinger term:

$$\begin{array}{c}\hfill \left[T_{a_1}\left(y_1\right),T_{a_2}\left(y_2\right)\right]=\left({{if}_{a_1{a}_2}}^b{T}_b\left(y_2\right)+k_{a_1{a}_2}{f}^I\left(y_2\right){\partial }_I^{\left(2\right)}\right){\displaystyle \frac{{\delta }^n(y_1-y_2)}{\sqrt{g\left(y_2\right)}}},\end{array}$$

(27)

where ${\partial }_I^{\left(2\right)}$ means that we take the derivative with respect to the second variable $y_2^I$. Thus,

$$\begin{array}{ccc}\hfill \left[\left[T_{a_1}\left(y_1\right),T_{a_2}\left(y_2\right)\right],T_{a_3}\left(y_3\right)\right]& =& \left[\left({{i f}_{a_1{a}_2}}^b{T}_b\left(y_2\right)+k_{a_1{a}_2}{f}^I\left(y_2\right){\partial }_I^{\left(2\right)}\right){\displaystyle \frac{{\delta }^n(y_1-y_2)}{\sqrt{g\left(y_2\right)}}},T_{a_3}\left(y_3\right)\right]\hfill \\ & =& \dots +i{f}_{a_1{a}_2{a}_3}{f}^I\left(y_3\right){\displaystyle \frac{{\delta }^n(y_1-y_2)}{\sqrt{g\left(y_2\right)}}}{\partial }_I^{\left(3\right)}\left({\displaystyle \frac{{\delta }^n(y_2-y_3)}{\sqrt{g\left(y_3\right)}}}\right)\hfill \end{array}$$

where … represents terms not involving the anomaly and $f_{a_1{a}_2{a}_3}={f_{a_1{a}_2}}^b{k}_{b{a}_3}$. Integration by parts, using the symmetry property of the Dirac $\delta$-distribution and the antisymmetry of the structure constants $f_{a_1{a}_2{a}_3}$, results in the computation $\left[\left[T_{a_1}\left(y_1\right),T_{a_2}\left(y_2\right)\right],T_{a_3}\left(y_3\right)\right]+$ cyclic terms, and the Jacobi identity leads to (23), which is the two-cocycle condition that we have obtained using the Pressley–Segal two-cocycle. Furthermore, performing integration $\int \sqrt{g\left(y\right)}{\mathrm{d}}^{n}y{b}_M\left(y\right)\int \sqrt{g\left(y^{\prime }\right)}{\mathrm{d}}^n{y}^{\prime }{b}_N\left(y^{\prime }\right)$ for the anomaly term in (27) leads (due to relation (23) above) to the two-cocycle (26). This means that the Pressley–Segal results on central extensions can equivalently be given in terms of an appropriate Schwinger term.

We now address the compatibility of central extensions and the differential operators of $\mathfrak{X}\left(\mathcal{M}\right)$. Assume that we have one central extension associated to the operator L above or the two-cocycle ${\omega }_L$. The new Lie brackets read

$$\begin{array}{c}\hfill {\left[X,Y\right]}_L{=[X,Y]}_0+{\omega }_L(X,Y),\end{array}$$

(28)

where ${[,]}_0$ are the Lie brackets without central extension, i.e., the Lie brackets of $\mathfrak{g}\left(\mathcal{M}\right)$. Let $L^{\prime }$ be an operator in $\mathfrak{X}\left(\mathcal{M}\right)$. The compatibility condition, i.e., the Jacobi identities, between the KM algebra with two-cocycle ${\omega }_L$ and $L^{\prime }$ turn out to be (see [33,34])

$$\begin{array}{c}\hfill {\omega }_L(L^{\prime }X,Y)+{\omega }_L(X,L^{\prime }Y)=0.\end{array}$$

Because of (23), after integration by parts, we see that $L^{\prime }=L$ is an obvious solution.

Now, let $(L,{\omega }_L)$ be a doublet differential operator/two-cocycle. On the one hand, the two-cocycle ${\omega }_L$ and the differential operator L are in duality (see (24) and (25)), and on the other hand, they are compatible. One may wonder if, for a given manifold $\mathcal{M}$, one can have more than one couple $({\omega }_L,L)$ in duality and if these can be compatible with each other. In fact, as will be seen in Section 3.1, in general, this is the case. In particular, if we consider the commuting Hermitian operators $D_i$ (see (16)) and the corresponding two-cocycle ${\omega }_i$,

$$\begin{array}{c}\hfill {\omega }_i(T_{aN},T_{bM})=M\left(i\right)k_{ab}{\eta }_{MN},\end{array}$$

(29)

the compatibility condition above between $D_j$ and ${\omega }_i$ reduces to

$$\begin{array}{c}\hfill {\omega }_i(D_j{T}_{aN},T_{bM})+{\omega }_i(T_{aN},D_j{T}_{bM})=k_{ab}M\left(i\right)(N\left(j\right)+M\left(j\right)){\eta }_{MN}=0.\end{array}$$

The condition of compatibility translates into

$$\begin{array}{c}\hfill (N\left(j\right)+M\left(j\right)){\eta }_{MN}=0,\forall M,N.\end{array}$$

(30)

Now, let ${\mathfrak{g}}_D=\left\{D_i,i=1,\dots ,r^{\prime }\le r:\left(30\right)\mathrm{is}\mathrm{satisfied}\right\}$. We thus define the KM algebra $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)=\{{\mathcal{T}}_{aM},a=1\dots ,d,M\in \mathbb{N},k^i,i=1,\dots r^{\prime }\}⋊\{D_i,i=1,\dots r^{\prime }\}$. From (11), (16), and (29), the non-vanishing Lie brackets are

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{aM},{\mathcal{T}}_{bN}\right]& =& i{f_{ab}}^c{C_{MN}}^P{\mathcal{T}}_{cP}+k_{ab}{\eta }_{MN}\sum _{i=1}^{r^{\prime }}{k}^{i}M\left(i\right),\hfill \\ \hfill \left[D_i,{\mathcal{T}}_{aM}\right]& =& M\left(i\right){\mathcal{T}}_{aM}.\hfill \end{array}$$

(31)

Note that $k^i,i=1,\dots r^{\prime }$ are the central charges associated to the two-cocycle ${\omega }_i$.

In the construction above, we determined the maximal set of compatible cocycles and differential operators $({\omega }_L,L)$. Here, we consider another possibility that will be relevant to the construction of the analog of the Virasoro algebra (see Section 3.4 for $\mathcal{M}=$ SU$\left(2\right)$). Let ${\omega }_L$ be a cocycle associated to the differential operator L, and let T be any function in $L^2\left(\mathcal{M}\right)$. The compatibility condition of the cocycle ${\omega }_L$ with the operator L reads

$$\begin{array}{c}\hfill {\omega }_L\left(T\left(m\right)LX,Y\right)+{\omega }_L\left(X,T\left(m\right)LY\right)=0,\end{array}$$

(32)

whose second term on the LHS is

$$\begin{array}{c}\hfill {\omega }_L\left(X,T\left(m\right)LY\right)=\underset{\mathcal{M}}{\int }{⟨X,L\left(\left(T\right(m)LY\right)⟩}_0\sqrt{g}{\mathrm{d}}^{n}m,\end{array}$$

while the first one, after an integration by parts, reduces to

$$\begin{array}{ccc}\hfill {\omega }_L\left(T\left(m\right)LX,Y\right)& =& \underset{\mathcal{M}}{\int }{⟨T\left(m\right)LX,LY⟩}_0\sqrt{g}{\mathrm{d}}^{n}m\hfill \\ & =& -\underset{\mathcal{M}}{\int }{\left({⟨X,L\left(\left(T\right(m)LY\right)⟩}_0\sqrt{g}+\left.⟨X,\left(T\right(m)LY\right)\right.⟩}_0{\partial }_m\left({\xi }^m\sqrt{g}\right)){\mathrm{d}}^{n}m\hfill \\ & & +\underset{\mathcal{M}}{\int }{\partial }_m{\left({\xi }^m\left.⟨X,\left(T\right(m)LY\right)\right.⟩}_0\sqrt{g}){\mathrm{d}}^{n}m,\hfill \end{array}$$

where we have used $L={\xi }^m{\partial }_m$. The second term in the second equality vanishes because of (23). On the other hand, the boundary term vanishes along any compact direction of $\mathcal{M}$, but also vanishes along non-compact directions of $\mathcal{M}$, since X and Y go to zero at infinity. Finally, Equation (32) is satisfied, and the set of differential operators

$$\begin{array}{c}\hfill D_L=\left\{L^T:=T\left(m\right)L,T\left(m\right)\in L^2\left(\mathcal{M}\right)\right\}\end{array}$$

(33)

is compatible with the cocycle ${\omega }_L$. Furthermore, we have the following commutation relations:

$$\begin{array}{c}\hfill \left[L^{T_1},L^{T_2}\right]=L^{T_{12}},T_{12}\left(m\right)=T_1\left(m\right)\left(L{T}_2\left(m\right)\right)-T_2\left(m\right)\left(L{T}_1\left(m\right)\right).\end{array}$$

Thus, $D_L\subset \mathfrak{X}\left(\mathcal{M}\right)$ is a subalgebra of the algebra of vector fields on $\mathcal{M}$, which should be regarded as the analog of the de Witt algebra in the case of $\mathcal{M}=$ U$\left(1\right)$. This algebra was found in [33] for $\mathcal{M}=$ SU$\left(2\right)$. We shall see in Section 3.4 that, in the case of SU$\left(2\right)$, a convenient choice of L enables us to introduce an analog of the Virasoro algebra on SU$\left(2\right)$, i.e., to centrally extend the algebra $D_L$. This construction certainly extends to other manifolds, as, for instance, any compact group manifold $\mathcal{M}=G_c$. In particular, for a given cocycle associated to L, further compatible differential operators can be added. For instance, if $L=D_i$, the set $D_{D_i}$ is the de Witt algebra (see (19) and the second equation of (20)). Next, the algebra $D_{D_i}$ can certainly be centrally extended along the lines of Section 3.4, yielding an analog of the semidirect product of the Virasoro algebra with the Kac–Moody algebra.

Before ending this section, it should be noted that the loop algebra $\mathfrak{g}\left(\mathcal{M}\right)$ also admits other central extensions. See, e.g., [54,55] for the three-dimensional case and [56] (and references therein) for higher-dimensional cases.

3. KM Algebras on Compact Group Manifolds

In the previous section, we gave a general strategy to construct a generalized KM algebra associated to a manifold $\mathcal{M}$. We have assumed this throughout (6) and (7). Let $G_c$ be a compact Lie group manifold, and let $H\subset G_c$ be a closed subgroup of $G_c$. In this section, we consider the case where $\mathcal{M}$ is either the compact group manifold $\mathcal{M}=G_c$ or the coset space $\mathcal{M}=G_c/H$. In both cases, conditions (6) and (7) (together with (5) (2)) are natural.

3.1. Compact Group Manifolds

Let $G_c$ be an $n-$dimensional compact Lie group. Let $m^A$ with $i=A,\dots ,n$ be a parameterization of $G_c$. Then, an element of $G_c$ connected to the identity takes the form

$$\begin{array}{c}\hfill g\left(m\right)=e^{i{m}^A{J}_A},\end{array}$$

(34)

where $J_A,A=1,\dots ,n$ are the generators of ${\mathfrak{g}}_c$, the Lie algebra of $G_c$ (not to be confused with the generators of $\mathfrak{g}$). Then, the coordinates of a group element (in a local coordinate chart) are

$$\begin{array}{c}\hfill g{\left(m\right)}^M\equiv m^M,M=1,\dots ,n.\end{array}$$

(35)

The indices $A,B,\dots$ are tangent space indices, i.e., flat indices, while the indices $M,N,\dots$ are world indices, i.e., curved indices. It should be mentionned that for specific parameterizations, the variables $m^A$ decompose into $m^A=({\phi }^i,{\theta }^r),i=1,\dots ,p,r=1,\dots ,q,p+q=n$, where the matrix elements of $m^M$ are periodic in $\phi$ and non-periodic in $\theta$ (see Section 3).

The Vielbein one-form associated to the parameterization (34) is, thus,

$$\begin{array}{c}\hfill e\left(m\right)=g{\left(m\right)}^{-1}\mathrm{d}g\left(m\right),\end{array}$$

(36)

where $\mathrm{d}g\left(m\right)$ is the exterior derivative of g. The Vielbein satisfies the Maurer–Cartan equation

$$\begin{array}{c}\hfill \mathrm{d}e+e\wedge e=0.\end{array}$$

(37)

Expanding the Vielbein in the basis $J_A$: $e=i{e}^A{J}_A$, we have for the one-forms $e^A$

$$\begin{array}{c}\hfill e^A\left(m\right)={e_M}^A\left(m\right)\mathrm{d}{m}^M.\end{array}$$

The one-forms are left-invariant by construction, i.e., invariant by the left multiplication $L_h\left(g\right)=hg$ with $h\in G_c$. Note that by defining the alternative Vielbein $e^{\prime }\left(m\right)=\mathrm{d}g\left(m\right)g{\left(m\right)}^{-1}$, we obtain a right-invariant one-form, i.e., invariant by the right multiplication $R_h\left(g\right)=gh$. The metric tensor on $G_c$ can be defined by the left- or right-invariant one-forms

$$\begin{array}{c}\hfill g_{MN}={e_M}^A\left(m\right){e_N}^B\left(m\right){\delta }_{AB}={e_M^{\prime }}^A\left(m\right){e_N^{\prime }}^B\left(m\right){\delta }_{AB}.\end{array}$$

(38)

We endow the manifold $G_c$ with the scalar product

$$\begin{array}{c}\hfill (f,g)={\displaystyle \frac{1}{V}}\underset{G_c}{\int }\sqrt{g}\mathrm{d}{\phi }^p\mathrm{d}{\theta }^q\overline{f(\phi ,\theta )}g(\phi ,\theta ),\end{array}$$

(39)

where V is the volume of $G_c$.

To identify a Hilbert basis of $L^2\left(G_c\right)$, we first recall that since $G_c$ is a compact Lie group, its unitary irreducible representations are finite-dimensional. Let $\widehat{\mathcal{R}}=\{{\mathcal{R}}_k,k\in {\widehat{G}}_c\}$ be the set of all irreducible unitary representations of $G_c$, and let ${\widehat{G}}_c$ be the set of labels of such representations (see below). Let $d_k$ be the dimension of the representation ${\mathcal{R}}_k$, and let ${{D_{\left(k\right)}}^i}_j\left(g\right)$ be its matrix elements. With these notations, we have the following theorem:

Theorem 1 

(Peter–Weyl [16]). Let $\widehat{\mathcal{R}}=\{{\mathcal{R}}_k,k\in {\widehat{G}}_c\}$ be the set of all unitary irreducible representations of $G_c$, and let $D_{\left(k\right)}\left(g\right)\in {\mathcal{R}}_k$ for $g\in G_c$. Then, the set of functions on $G_c$,

$$\begin{array}{c}\hfill {{{\psi }_{\left(k\right)}}^i}_j\left(g\right)=\sqrt{d_k}{{D_{\left(k\right)}}^i}_j\left(g\right),k\in {\widehat{G}}_c,i,j=1,\dots ,d_k,g\in G_c,\end{array}$$

(40)

forms a complete Hilbert basis of $L^2\left(G_c\right)$ with inner product

$$\begin{array}{c}\hfill {\int }_{G_c}\sqrt{g}\mathrm{d}{\phi }^p\mathrm{d}{\theta }^q{{{\overline{\psi }}^{\left(k\right)}}_i}^j\left(g\right){{{\psi }_{\left(k^{\prime }\right)}}^{i^{\prime }}}_{j^{\prime }}\left(g\right)={\delta }_{k^{\prime }}^k{\delta }_i^{i^{\prime }}{\delta }_{j^{\prime }}^j.\end{array}$$

Since the notations of Theorem 1 are not appropriate for our purpose, we introduce more convenient notations. Assume that the Lie algebra ${\mathfrak{g}}_c$ is of rank ℓ. Then, ${\mathfrak{g}}_c$ admits an ℓ independent Casimir operator $\{C_1,\dots ,C_{\mathsf{\ell }}\}$. Let $\mathcal{R}$ be a unitary representation of $G_c$, and $\mathcal{R}$ is uniquely specified by the eigenvalues of the ℓ Casimir operators (alternatively, the representation can also be specified by the highest weight with respect to a given Cartan subalgebra $\mathfrak{h}\in {\mathfrak{g}}_c$). Denote $Q=(c_1,\dots ,c_{\mathsf{\ell }})$ the eigenvalues $\{C_1,\dots ,C_{\mathsf{\ell }}\}$. Racah showed in [57] that a number of $1/2(dimg-\mathsf{\ell })$ internal labels are required to separate the states within the multiplet $\mathcal{R}$. Given a Cartan subalgebra, its eigenvalues constitute an appropriate set of ℓ labels. The choice of the remaining additional internal labels is far from being unique and usually depends on a specific chain of proper subalgebras

$${\mathfrak{g}}_1\subset {\mathfrak{g}}_2\subset \dots \subset {\mathfrak{g}}_c$$

such that, in each step, the Casimir operators of the subalgebra are used to separate states [58,59]. Let $D_{\left(k\right)}$ and $D_{\left(k\right)}^{\prime }$ be two matrix elements of the representation ${\mathcal{R}}_k$. Since the product $D_{\left(k\right)}{D}_{\left(k\right)}^{\prime }$ is still a matrix element of ${\mathcal{R}}_k$, the lines (resp., the column) of $D_{\left(k\right)}$ are in the representation ${\mathcal{R}}_k$ (the lines are in the left representations, and the columns are in the right representations). Consequently, we now denote any matrix element appearing in Theorem 1 by ${\Psi }_{LQR}$, where Q denotes the ℓ eigenvalues of the Casimir operators that identify the representation, and L (resp., R) represents the $1/2(dimg-\mathsf{\ell })$ labels to separate the states within the left (resp., right) action of $G_c$. Let $\mathcal{I}$ be the range for the variables $L,Q$ and R; with these notations, the Hilbert basis of $L^2\left(G_c\right)$ associated to the Peter–Weyl theorem is given by

$$\begin{array}{c}\hfill {\mathcal{B}}_{G_c}=\left\{{\Psi }_{LQR},(L,Q,R)\in \mathcal{I}\right\},\end{array}$$

(41)

and the matrix elements are normalized:

$$\begin{array}{c}\hfill ({\Psi }_{LQR},{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }})={\delta }_{Q^{\prime }}^Q{\delta }_{L^{\prime }}^L{\delta }_{R^{\prime }}^R.\end{array}$$

The parameterization $m^A=({\phi }^i,{\theta }^r)$ leads naturally to a differential realization of the Lie algebra ${\mathfrak{g}}_c$ for the generators of the left and right actions. Indeed, since the manifold $G_c$ has a natural left and right action, Killing Equation (1) automatically admits solutions, namely the generators of the left (resp., right) action and ${\mathfrak{g}}_{\xi }={\left({\mathfrak{g}}_c\right)}_L\oplus {\left({\mathfrak{g}}_c\right)}_R$. Let $L_A$ (resp., $R_A$) $A=1,\dots n$, be the generators of the left (resp., right) action. We have

$$\begin{array}{c}\hfill \left[L_A,L_B\right]=i{c_{AB}}^C{L}_C,\left[R_A,R_B\right]=i{c_{AB}}^C{R}_C,\left[L_A,R_B\right]=0,\end{array}$$

(42)

where ${c_{AB}}^C$ are the structure constants of ${\mathfrak{g}}_c$. Furthermore, the operators $L_A$ and $R_A$ are Hermitian with respect to the scalar product (39) and act naturally on the matrix elements ${\Psi }_{LQR}$:

$$\begin{array}{ccc}\hfill L_A{\Psi }_{LQR}(\phi ,\theta )& =& {{\left(M_A^Q\right)}_L}^{L^{\prime }}{\Psi }_{L^{\prime }QR}(\phi ,\theta )\hfill \\ \hfill R_A{\Psi }_{LQR}(\phi ,\theta )& =& {{\left(M_A^Q\right)}_R}^{R^{\prime }}{\Psi }_{LQ{R}^{\prime }}(\phi ,\theta )\hfill \end{array}$$

where ${{\left(M_A^Q\right)}_L}^{L^{\prime }},{{\left(M_A^Q\right)}_R}^{R^{\prime }}$ are the matrix elements of the left or right action for the representation specified by Q.

Let ${\mathcal{D}}_Q,{\mathcal{D}}_{Q^{\prime }}$ be two representations of $G_c$, specified by the eigenvalues of the Casimir operators. Consider the tensor product:

$$\begin{array}{c}\hfill {\mathcal{D}}_Q\otimes {\mathcal{D}}_{Q^{\prime }}=\underset{Q^{″}}{⨁}{\mathcal{D}}_{Q^{″}}.\end{array}$$

Introducing the Clebsch–Gordan coefficient $\left(\begin{array}{ccc}Q& Q^{\prime }& Q^{″}\\ L& L^{\prime }& L^{″}\end{array}\right)$, we have

$$\begin{array}{c}\hfill |Q^{″},L^{″}⟩=\sum \left(\begin{array}{ccc}Q& Q^{\prime }& Q^{″}\\ L& L^{\prime }& L^{″}\end{array}\right)|Q,L⟩\otimes |Q^{\prime },L^{\prime }⟩\end{array}$$

(43)

Thus, observing that

$$\begin{array}{c}\hfill {\Psi }_{LQR}\left(0\right)=\sqrt{d}{\delta }_{LR},\end{array}$$

where d is the dimension of the representation ${\mathcal{D}}_Q$, we have

$$\begin{array}{c}\hfill {\Psi }_{LQR}\left(m\right){\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}\left(m\right)=C_{LQR;L^{\prime }{Q}^{\prime }{R}^{\prime }}^{L^{″}{Q}^{″}{R}^{″}}{\Psi }_{L^{″}{Q}^{″}{R}^{″}}\left(m\right),\end{array}$$

(44)

where the summation is implicit, and with

$$\begin{array}{c}\hfill C_{LQR;L^{\prime }{Q}^{\prime }{R}^{\prime }}^{L^{″}{Q}^{″}{R}^{″}}=\sqrt{{\displaystyle \frac{d{d}^{\prime }}{d^{″}}}}\left(\begin{array}{ccc}Q& Q^{\prime }& Q^{″}\\ L& L^{\prime }& L^{″}\end{array}\right)\overline{\left(\begin{array}{ccc}Q& Q^{\prime }& Q^{″}\\ R& R^{\prime }& R^{″}\end{array}\right)},\end{array}$$

(45)

where d (resp., $d^{\prime },d^{″}$) are the dimensions of ${\mathcal{D}}_Q$ (resp., ${\mathcal{D}}_{Q^{\prime }},{\mathcal{D}}_{Q^{″}}$).

3.2. Coset Spaces of Compact Group Manifolds

Now, let H be a closed subgroup of $G_c$, let $\pi :G_c\to G_c/H$ be the quotient map, and let $n=dim{G}_c,s=dimH$. Then, we can always find a chart $\left(U,\phi =(y^1,\dots ,y^n)\right)$ around the identity in $G_c$ such that the following applies:

• $\phi \left(U\right)=\left\{({\xi }^1,\dots ,{\xi }^n)\left|\right|{\xi }^p|<\epsilon ,p=1,\dots ,n\right\}$ for some $\epsilon >0$.
• Each slice with $y^{s+1}={\xi }^{s+1},\dots ,y^n={\xi }^n$ is a relatively open set in some coset $gH$, and these cosets are all distinct.
• The restriction of $\pi$ to the slice $y^q=0,q=1,\dots s$ is an open homeomorphism and, hence, determines a chart around the identity element in $G_c/H$.

Using these charts, the coset $G_c/H$ can be endowed with the structure of a differential manifold such that the $\pi$ and the action of $G_c$ are differentiable. If, in addition, H is a normal subgroup, then $G_c/H$ inherits the structure of a Lie group, and the differential of the $G_c\to G_c/H$ corresponds to the quotient map ${\mathfrak{g}}_c\to {\mathfrak{g}}_c/\mathfrak{h}$, where $\mathfrak{h}$ is the Lie algebra of H.

In the general case (i.e., there is some $g\in G_c$ with $g^{-1}Hg\ne H$), the coset space $G_c/H$ is not a Lie group, and hence, the factor space ${\mathfrak{g}}_c/\mathfrak{h}$ is not a Lie algebra. We write the generators of ${\mathfrak{g}}_c$, namely $T_A$ (with $A=1,\dots ,dim{\mathfrak{g}}_c$), as follows: $U_i$ with $i=1,\dots ,dim\mathfrak{h}$, and $V_p$ with $p=1,\dots ,dim{\mathfrak{g}}_c-dim\mathfrak{h}$, where $V_p$ denotes the complementary space of $\mathfrak{h}$ in ${\mathfrak{g}}_c$. The commutation relations take the generic form

$$\begin{array}{c}\hfill \begin{array}{cc}\left(a\right)\hfill & [U_j,U_k]=i{g_{jk}}^{\mathsf{\ell }}{U}_{\mathsf{\ell }},\hfill \\ \left(b\right)\hfill & [U_j,V_p]=i{N_{jp}}^{\mathsf{\ell }}{U}_{\mathsf{\ell }}+i{{\left(R_j\right)}_p}^q{V}_q,\hfill \\ \left(c\right)\hfill & [V_p,V_q]=i{g_{pq}}^j{U}_j+i{g_{pq}}^r{V}_r.\hfill \end{array}\end{array}$$

The relations (a) are trivially satisfied, as $\mathfrak{h}$ is a Lie subalgebra of ${\mathfrak{g}}_c$, whereas the relations (b) imply that ${\mathfrak{g}}_c/\mathfrak{h}$ is a representation of ${\mathfrak{g}}_c$, whenever the condition ${N_{jp}}^{\mathsf{\ell }}=0$ is satisfied. In particular, the coset space $G_c/H$ is called reductive if there exists an $Ad\left(H\right)$-invariant subspace $\mathfrak{m}$ of ${\mathfrak{g}}_c$ that is complementary to $\mathfrak{h}$ in ${\mathfrak{g}}_c$ (the space $\mathfrak{m}$ is sometimes called the Lie subspace for $G_c/H$). It follows from the invariance that $\left[\mathfrak{h},\mathfrak{m}\right]\subset \mathfrak{m}$. A sufficient condition for the coset space to be reductive is that either H or $\mathrm{Ad}\left(H\right)$ is compact [17]. If, in addition, ${g_{pq}}^r=0$ holds in (c), then the manifold $G_c/H$ is related to a specially relevant class of homogeneous spaces (endowed with a Riemannian structure), the so-called symmetric spaces [60].

In the following, we shortly summarize the construction steps of the previous section, adapted to (generic) homogeneous spaces. A generic element of $G_c/H$ is of the form

$$\begin{array}{c}\hfill L=e^{i{m}^p{V}_p},\end{array}$$

and the Vielbein one-form is given by

$$\begin{array}{c}\hfill e=L^{-1}\mathrm{d}L,\end{array}$$

(46)

which decomposes to

$$\begin{array}{c}\hfill e\left(m\right)=i{e}^p{U}_p+i{e}^i{V}_i.\end{array}$$

(47)

Denoting

$$\begin{array}{c}\hfill e^p\left(m\right)={e^p}_{\tilde{p}}\left(m\right)\mathrm{d}{m}^{\tilde{p}},\end{array}$$

where $p,q,\dots$ represent the indices in the flat tangent space, and $\tilde{p},\tilde{q},\dots$ represent the curved indices of the manifold $G_c/H$; the metric tensor on the coset space is given by

$$\begin{array}{c}\hfill g_{\tilde{p}\tilde{q}}={e^p}_{\tilde{p}}\left(m\right){e^q}_{\tilde{q}}\left(m\right){\delta }_{pq}.\end{array}$$

(48)

We identify the Killing vectors and the corresponding symmetry algebra of the manifold $G_c/H$

$$\begin{array}{c}\hfill {\mathfrak{g}}_{\xi }=\left\{K_A=i{\xi }_A^p{\partial }_p:{\xi }_A^p\mathrm{Killing}\mathrm{vector}\mathrm{of}{g}_{\tilde{p}\tilde{q}}\right\}\subset {\left({\mathfrak{g}}_c\right)}_L\oplus {\left({\mathfrak{g}}_c\right)}_R.\end{array}$$

(49)

We endow the manifold $G_c/H$ with the scalar product

$$\begin{array}{c}\hfill (f,g)={\displaystyle \frac{1}{V^{\prime }}}\underset{G_c/H}{\int }\sqrt{g^{\prime }}\mathrm{d}{m}^N\overline{f\left(m\right)}g\left(m\right),\end{array}$$

(50)

where $V^{\prime }$ is the volume of $G_c/H$, which is the restriction of (39) to $G_c/H$ and $N=dim{\mathfrak{g}}_c-dim\mathfrak{h}$.

We now identify a Hilbert basis of $G_c/H$. Given a representation ${\mathcal{R}}_k$ of $G_c$, ${\mathcal{R}}_k$ admits a scalar representation with respect to the embedding $H\subset G_c$. Let ${\mathcal{R}}_k{|}_H$ be the set of representations with this property. The set of matrix elements ${\Psi }_{LQR}$ of a representation ${\mathcal{R}}_k{|}_H$ decomposes into

$$\begin{array}{c}\hfill \left\{{\Psi }_{LQR},\left(LQR\right)\in \mathcal{I}\right\}=\left\{{\Psi }_{LQ{R}_0},\left(LQ{R}_0\right)\in \mathcal{I}:{\Psi }_{LQ{R}_0}\mathrm{trival}\mathrm{under}\mathrm{the}\mathrm{right}\mathrm{action}\mathrm{of}H\right\}\oplus \dots \end{array}$$

Stated differently, $R_0$ corresponds to the set of indices associated to the trivial representation of the right $H-$action. The matrix elements ${\Psi }_{LQ{R}_0}$ contribute to the harmonic analysis in $G_c/H$. Using the notation of Section 3.1, we denote the corresponding normalized matrix elements

$$\begin{array}{c}\hfill {\varphi }_{LQ}={\psi }_{LQ{R}_0},\end{array}$$

(51)

where the index $R_0$ means that ${\varphi }_{LQ{R}_0}$ transform trivially under the right action of H [8,40]. The number of internal labels is ℓ (the rank of ${\mathfrak{g}}_c$) to separate the representations, and $1/2(dim{\mathfrak{g}}_c-\mathsf{\ell })$ internal labels are used to distinguish states within representations [40]. Solving the Killing equation for the coset leads to the generators of the left action $G_c$ and possibly additional generators associated to the right action, which survive the coset process. Finally, by computing the product ${\varphi }_{LQ}{\varphi }_{L^{\prime }{Q}^{\prime }}$, we obtain coefficients that can be deduced from (45) (see Section 3.3).

3.3. KM Algebras Associated to Compact Lie Groups

First, consider $\mathcal{M}=G_c$. The first step to associate a KM algebra to $G_c$ is to define the Lie algebra

$$\begin{array}{c}\hfill \mathfrak{g}\left(G_c\right)=\left\{T_{aLQR}\left(m\right)=T_a{\psi }_{LQR}\left(m\right),A=1,\dots ,d,(L,Q,R)\in \mathcal{I}\right\}\end{array}$$

(see (41)) with Lie brackets

$$\begin{array}{c}\hfill \left[T_{aLQR}\left(m\right),T_{a^{\prime }{L}^{\prime }{Q}^{\prime }{R}^{\prime }}\left(m\right)\right]=i{f_{a{a}^{\prime }}}^{a^{″}}{C}_{LQR;L^{\prime }{Q}^{\prime }{R}^{\prime }}^{L^{″}{Q}^{″}{R}^{″}}{T}_{a^{″}{L}^{″}{Q}^{″}{R}^{″}}\left(m\right).\end{array}$$

(52)

This algebra can be obtained in a different way. Let

$$\begin{array}{c}\hfill T_a\left(m\right)=\sum _{(L,Q,R)\in \mathcal{I}}{T}_{aLQR}\overline{{\psi }^{LQR}}\left(m\right),\end{array}$$

then,

$$\begin{array}{c}\hfill \left[T_a\left(m\right),T_b\left(m^{\prime }\right)\right]=i {f_{ab}}^c{T}_c\left(m^{\prime }\right){\displaystyle \frac{{\delta }^n(m-m^{\prime })}{\sqrt{g}}}\end{array}$$

(53)

leads to (52) upon integration by $\int {\mathrm{d}}^{n}m\sqrt{g}{\Psi }_{LQR}\left(m\right)\int {\mathrm{d}}^n{m}^{\prime }\sqrt{g}{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}\left(m^{\prime }\right)$.

This algebra can, indeed, be obtained easily in the context of Kaluza–Klein theories. Consider the space-time $K={\mathbb{R}}^{1,D-1}\times G_c$, i.e., a $D-$dimensional Minkowski space-time with a compact manifold $G_c$ of dimension $dim{\mathfrak{g}}_c=n$ as internal space. Denote $X^I=(x^{\mu },m^M)$ the components in K where $\mu =0,\dots ,D-1$ and $M=1,\dots ,n$. Assume further the metric on K (in this simple analysis, we do not endow ${\mathbb{R}}^{1,d-1}$ with a Riemannian structure; thus, the metric is Minkowskian)

$$\begin{array}{c}\hfill \mathrm{d}{s}^2=\mathrm{d}{x}^{\mu }\mathrm{d}{x}^{\nu }{\eta }_{\mu \nu }-\mathrm{d}{m}^M\mathrm{d}{m}^N{g}_{MN}:=g_{IJ}\mathrm{d}{X}^I\mathrm{d}{X}^J,\end{array}$$

where ${\eta }_{\mu \nu }=$ diag$(1,-1,\dots ,-1)$ is the Minkowski metric on ${\mathbb{R}}^{1,d-1}$, and $g_{MN}$ is the metric (38) on $G_c$.

Consider a massless, free complex scalar field, with action

$$\begin{array}{c}\hfill \mathcal{S}=\underset{{\mathbb{R}}^{1,D-1}}{\int }\mathrm{d}{x}^D\underset{G_c}{\int }\mathrm{d}{m}^n\sqrt{g}{g}^{IJ}{\partial }_I{\Phi }^{†}(x,m){\partial }_J\Phi (x,m).\end{array}$$

The equations of the motion read

$$\begin{array}{c}\hfill (\square -{\nabla }^2)\Phi (x,m)=0,\end{array}$$

where $\square ={\partial }_{\mu }{\partial }_{\nu }{\eta }^{\mu \nu }$ and ${\nabla }^2={\displaystyle \frac{1}{\sqrt{g}}}{\partial }_M\left(g^{MN}{\partial }_N\right)$ are, respectively, the $D-$dimensional d’Alembertian and the Laplace–Beltrami operator of $G_c$. Next, decomposing the scalar field in the compact manifold $G_c$ as

$$\begin{array}{c}\hfill \Phi (x,m)=\sum _{(L,Q,R)\in \mathcal{I}}{\varphi }_{LQR}\left(x\right){\overline{\psi }}^{LQR}\left(m\right),\end{array}$$

and observing that the action of the Laplace–Beltrami operator on the scalar functions is related to the quadratic Casimir operator [17,61],

$$\begin{array}{c}\hfill {\nabla }^2=-k{C}_2,k>0,\end{array}$$

(where $C_2$ is the quadratic Casimir operator of ${\mathfrak{g}}_c$ with eigenvalue $c_2$), we obtain

$$\begin{array}{c}\hfill {\nabla }^2{\overline{\psi }}^{LQR}\left(m\right)=-k{c}_2{\overline{\psi }}^{LQR}\left(m\right),c_2\ge 0,\end{array}$$

with the mass of the field ${\varphi }_{LQR}$ given by $\sqrt{k{c}_2}$. Thus, the solution of the relativistic wave equation takes the following form:

$$\begin{array}{c}\hfill \Phi (x,m)=\sum _{(L,Q,R)\in \mathcal{I}}\int {\displaystyle \frac{{\mathrm{d}}^{D-1}x}{\sqrt{2E_{c_2}}}}\left(a_{\overrightarrow{p},QLR}{e}^{-i\mathbf{p}·\mathbf{x}}+b_{\overrightarrow{p},QLR}^{†}{e}^{i\mathbf{p}·\mathbf{x}}\right){\overline{\psi }}^{LQR}\left(m\right),\end{array}$$

where $\mathbf{p}·\mathbf{x}$ is the scalar product in the Minkowski space-time, and $\mathbf{p}·\mathbf{p}=E^2-\overrightarrow{p}·\overrightarrow{p}=k{c}_2$ or $E_{c_2}\equiv \sqrt{\overrightarrow{p}·\overrightarrow{p}+k{c}_2}$.

Let $\Pi (x,m)={\partial }_0{\Phi }^{†}(x,m)$ be the conjugate momentum of $\Phi$. If we now quantize the scalar field by the usual equal-time commutation relations, an elementary computation leads to

$$\begin{array}{ccc}& & \left[\Phi (t,\overrightarrow{x},m),\Pi (t,{\overrightarrow{x}}^{\prime },m^{\prime })\right]=i{\delta }^{D-1}(\overrightarrow{x}-{\overrightarrow{x}}^{\prime }){\displaystyle \frac{{\delta }^n(m-m^{\prime })}{\sqrt{g}}}\hfill \\ & \iff & \left\{\begin{array}{ccc}\left[a_{\overrightarrow{p},LQR},a_{{\overrightarrow{p}}^{\prime },L^{\prime }{Q}^{\prime }{R}^{\prime }}^{†}\right]\hfill & =\hfill & {\left(2\pi \right)}^{D-1}{\delta }^{D-1}(\overrightarrow{p}-{\overrightarrow{p}}^{\prime }){\delta }_{L{L}^{\prime }}{\delta }_{Q{Q}^{\prime }}{\delta }_{R{R}^{\prime }}\hfill \\ \left[b_{\overrightarrow{p},LQR},b_{{\overrightarrow{p}}^{\prime },L^{\prime }{Q}^{\prime }{R}^{\prime }}^{†}\right]\hfill & =\hfill & {\left(2\pi \right)}^{D-1}{\delta }^{D-1}(\overrightarrow{p}-{\overrightarrow{p}}^{\prime }){\delta }_{L{L}^{\prime }}{\delta }_{Q{Q}^{\prime }}{\delta }_{R{R}^{\prime }}.\hfill \end{array}\right.\hfill \end{array}$$

Now, consider that the scalar field is in a representation $\mathcal{R}$ of the compact Lie algebra $\mathfrak{g}$. Let $M_a$ be the corresponding matrix representation. Then, the Nœther theorem leads to the conserved current

$$\begin{array}{c}\hfill j_a^{\mu }(t,\overrightarrow{x},m)=i\left(\Pi (t,\overrightarrow{x},m)M_a\Phi (t,\overrightarrow{x},m)-{\Phi }^{†}(t,\overrightarrow{x},m)M_a{\Pi }^{†}(t,\overrightarrow{x},m)\right)\end{array}$$

and the equal-time commutation relations imply

$$\begin{array}{c}\hfill \left[j_a^0(t,\overrightarrow{x},m),j_b^0(t,{\overrightarrow{x}}^{\prime },m^{\prime })\right]=i {f_{ab}}^c{j}_c^0(t,\overrightarrow{x},m){\delta }^{D-1}(\overrightarrow{x}-{\overrightarrow{x}}^{\prime }){\displaystyle \frac{{\delta }^n(m-m^{\prime })}{\sqrt{g}}}.\end{array}$$

Upon space integration $\int \mathrm{d}{x}^{D-1}$, we reproduce (53). A second integration $\int \sqrt{g}\mathrm{d}{m}^d$ leads to (52). Thus, the conserved charges $Q_a=\int \mathrm{d}{x}^{D-1}\int \sqrt{g}\mathrm{d}{m}^d{j}_a^0(t,\overrightarrow{x},m)$ generate the Lie algebra (52), which appears naturally upon compactification. This fact was already observed in [62] for the case $G_c=U\left(1\right)$.

The construction of the KM algebras associated to $G_c$ follows naturally from the description given in Section 2.2. We now introduce the generators of the Cartan subalgebra of ${\mathfrak{g}}_c$ for the left (resp., right action) $D_i^L,i=1,\dots ,{\mathsf{\ell }}^{\prime }\le \mathsf{\ell }$ (resp., $D_i^R,i=1\dots ,{\mathsf{\ell }}^{\prime }$) satisfying (30), together with their associated two-cocycles

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\omega }_i^L(T_{aLQR},T_{a^{\prime }{L}^{\prime }{Q}^{\prime }{R}^{\prime }})& =\hfill & \int \mathrm{d}{m}^d\sqrt{g}{⟨T_{aLQR},D_i^L{T}_{a^{\prime }{L}^{\prime }{Q}^{\prime }{R}^{\prime }}⟩}_0=k_{a{a}^{\prime }}{L}^{\prime }\left(i\right){\eta }_{LQR,L^{\prime }{Q}^{\prime }{R}^{\prime }},\\ \hfill {\omega }_i^R(T_{aLQR},T_{a^{\prime }{L}^{\prime }{Q}^{\prime }{R}^{\prime }})& =\hfill & \int \mathrm{d}{m}^d\sqrt{g}{⟨T_{aLQR},D_i^R{T}_{a^{\prime }{L}^{\prime }{Q}^{\prime }{R}^{\prime }}⟩}_0=k_{a{a}^{\prime }}{R}^{\prime }\left(i\right){\eta }_{LQR,L^{\prime }{Q}^{\prime }{R}^{\prime }},\end{array}\end{array}$$

(54)

(see (25)). As the detailed steps are given in Section 2.2, we merely indicate the results without further details. The KM algebra associated to $G_c$ can be defined (with the notations of Section 2.2) by $\tilde{\mathfrak{g}}\left({\mathrm{G}}_c\right)=\{{\mathcal{T}}_{aLQR},a=1\dots ,d,\left(LQR\right)\in \mathcal{I},D_i^L,D_i^R,k_L^i,k_R^j,i=1,\dots {\mathsf{\ell }}^{\prime }\}$, with non-vanishing Lie brackets:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left[{\mathcal{T}}_{aLQR},{\mathcal{T}}_{a^{\prime }{L}^{\prime }{Q}^{\prime }{R}^{\prime }}\right]& =i{f_{a{a}^{\prime }}}^{a^{″}}{C}_{LQR;L^{\prime }{Q}^{\prime }{R}^{\prime }}^{L^{″}{Q}^{″}{R}^{″}}{\mathcal{T}}_{a^{″}{L}^{″}{Q}^{″}{R}^{″}}\hfill \\ & +k_{a{a}^{\prime }}{\eta }_{LQR,L^{\prime }{Q}^{\prime }{R}^{\prime }}\sum _{i=1}^{{\mathsf{\ell }}^{\prime }}\left(k_L^{i}L\left(i\right)+k_R^{i}R\left(i\right)\right),\hfill \\ \hfill \left[D_i^L,{\mathcal{T}}_{aLRQ}\right]& =L\left(i\right){\mathcal{T}}_{aLQM},\hfill \\ \hfill \left[D_i^R,{\mathcal{T}}_{aLRQ}\right]& =R\left(i\right){\mathcal{T}}_{aLQP}.\hfill \end{array}\end{array}$$

Note that, here, $L\left(i\right),R\left(i\right)$ are the components of the weight vectors of the corresponding representation.

Now, consider briefly the construction of a KM algebra associated to a coset $G_c/H$, where $H\subset G_c$ is a closed subgroup of $G_c$. The construction of a KM algebra associated to the manifold $G_c/H$ is entirely analogous to the construction of the KM algebra associated to $G_c$, mutatis mutandis, the following replacements:

• The metric on $G/H$ is given by (48).
• The scalar product reduces to (50).
• The Hilbert basis of $L^2($${\mathrm{G}}_c/$H) is given by the set of $H-$invariant vectors of the right-action (51).
• The Killing vectors and the corresponding symmetry algebra of $G/H$ are given in (49). Denote $D_i,i=1,\dots ,{\mathsf{\ell }}^{\prime }\le \mathsf{\ell }$ the generators of the Cartan subalgebra satisfying (30).
• The two-cocycles are obtained through (29) (with the commuting Hermitian operators satisfying (30) $D_i,i=1,\dots ,{\mathsf{\ell }}^{\prime }\le \mathsf{\ell }$).

The Lie brackets of the algebra $\tilde{\mathfrak{g}}($${\mathrm{G}}_c/$H) are given by the Lie brackets (55) modulo the substitutions above.

3.4. KM Algebra Associated to SU$\left(2\right)$ and SU$\left(2\right)/$U$\left(1\right)$

In this section, we give explicit examples of KM algebras associated to compact group manifolds or cosets. A detailed account of their construction was given in [40]; for this reason, we merely present two examples here.

We first consider $G_c=$ SU$\left(2\right)$. Here, we give the principal steps, the details of which can also be found in [40,47]. We have

$$\begin{array}{c}\hfill \mathrm{SU}\left(2\right)=\left\{z_1,z_2\in {\mathbb{C}}^2:|z_1{|}^2+{\left|z_2\right|}^2\right\}\cong {\mathbb{S}}^3,\end{array}$$

where ${\mathbb{S}}^3$ is the three-sphere. A parameterization of ${\mathbb{S}}^3$ is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill z_1=cos{\displaystyle \frac{\theta }{2}}{e}^{i{\displaystyle \frac{\phi +\psi }{2}}},z_2=sin{\displaystyle \frac{\theta }{2}}{e}^{i{\displaystyle \frac{\phi -\psi }{2}}},0\le \theta \le \pi ,0\le \phi <2\pi ,0\le \psi <4\pi .\end{array}\end{array}$$

(55)

From

$$\begin{array}{c}\hfill m=\left(\begin{array}{cc}{z}_1& -{\overline{z}}_2\\ z_2& \phantom{-}{\overline{z}}_1\end{array}\right)\in \mathrm{SU}\left(2\right)\end{array}$$

(56)

(36) leads to the left-invariant one-forms

$$\begin{array}{ccc}\hfill {\lambda }_1& =& sin\phi \mathrm{d}\theta -cos\phi sin\theta \mathrm{d}\psi \hfill \\ \hfill {\lambda }_2& =& cos\phi \mathrm{d}\theta +sin\phi sin\theta \mathrm{d}\psi \hfill \\ \hfill {\lambda }_3& =& cos\theta \mathrm{d}\psi +\mathrm{d}\phi \hfill \end{array}$$

(57)

and to the metric tensor

$$\begin{array}{c}\hfill \mathrm{d}{s}^2={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2=\mathrm{d}{\theta }^2+\mathrm{d}{\phi }^2+\mathrm{d}{\psi }^2+2cos\theta \mathrm{d}\phi \mathrm{d}\psi .\end{array}$$

(58)

The left/right invariant vector fields obtained by solving Killing Equation (1) are

$$\begin{array}{c}\hfill \begin{array}{cccccc}{L}_{\pm }\hfill & =\hfill & e^{\pm i\psi }\left(-{\displaystyle \frac{i}{sin\theta }}{\partial }_{\phi }+icot\theta {\partial }_{\psi }\pm {\partial }_{\theta }\right),\hfill & L_0\hfill & =\hfill & -i{\partial }_{\psi }\hfill \\ R_{\pm }\hfill & =\hfill & e^{\pm i\phi }\left(\phantom{-}{\displaystyle \frac{i}{sin\theta }}{\partial }_{\psi }-icot\theta {\partial }_{\phi }\mp {\partial }_{\theta }\right),\hfill & R_0\hfill & =\hfill & -i{\partial }_{\phi }\hfill \end{array}\end{array}$$

and satisfy the commutation relations

$$\begin{array}{c}\hfill \begin{array}{cccccc}\left[L_0,L_{\pm }\right]\hfill & =\hfill & \pm L_{\pm },\hfill & \left[L_{+},L_{-}\right]\hfill & =\hfill & 2L_0,\hfill \\ \left[R_0,R_{\pm }\right]\hfill & =\hfill & \pm R_{\pm },\hfill & \left[R_{+},R_{-}\right]\hfill & =\hfill & 2R_0,\hfill \end{array}\left[L_a,R_b\right]=0.\end{array}$$

The quadratic Casimir operator reduces to

$$\begin{array}{c}\hfill C_2=-{\partial }_{\theta }^2-cot\theta {\partial }_{\theta }-{\displaystyle \frac{1}{{sin}^2\theta }}\left({\partial }_{\phi }^2+{\partial }_{\psi }^2\right)+2{\displaystyle \frac{cos\theta }{{sin}^2\theta }}{\partial }_{\phi }{\partial }_{\psi }.\end{array}$$

Finally, the SU$\left(2\right)-$scalar product is given by

$$\begin{array}{c}\hfill (f,g)={\displaystyle \frac{1}{16{\pi }^2}}\underset{{\mathbb{S}}^3}{\int }sin\theta \mathrm{d}\theta \mathrm{d}\psi \mathrm{d}\varphi \overline{f(\theta ,\phi ,\psi )}g(\theta ,\phi ,\psi ).\end{array}$$

We now identify the matrix elements. Since $dim\mathfrak{su}\left(2\right)=3$ and $\mathrm{rk}\mathfrak{su}\left(2\right)=1$, we need two labels to identify the matrix elements of $\mathfrak{su}\left(2\right)$, one external corresponding to the eigenvalue of the quadratic Casimir operator, and one internal label determined by the Cartan subalgebra eigenvalues of the left/right action of $\mathfrak{su}\left(2\right)$. Thus, we obtain [47] $(s\in {\displaystyle \frac{1}{2}}\mathbb{N},-s\le n,m\le s$):

$$\begin{array}{c}\hfill {\psi }_{nsm}(\theta ,\phi ,\psi )=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^{m-n}\sqrt{(2s+1)}}{(n-m)!}}\sqrt{{\displaystyle \frac{(s+n)!}{(s-n)!}}{\displaystyle \frac{(s-m)!}{(s+m)!}}}{e}^{im\phi +in\psi }{cos}^{-n-m}{\displaystyle \frac{\theta }{2}}{sin}^{n-m}{\displaystyle \frac{\theta }{2}}\hfill & n\ge m\\ {}_{2}F_1(-m-s,-m+s+1;1-m+n;{sin}^2{\displaystyle \frac{\theta }{2}})\hfill & -n-m\ge 0\\ {\displaystyle \frac{\sqrt{(2s+1)}}{(m-n)!}}\sqrt{{\displaystyle \frac{(s+m)!}{(s-m)!}}{\displaystyle \frac{(s-n)!}{(s+n)!}}}{e}^{im\phi +in\psi }{cos}^{-n-m}{\displaystyle \frac{\theta }{2}}{sin}^{m-n}{\displaystyle \frac{\theta }{2}}\hfill & m\ge n\\ {}_{2}F_1(-n-s,-n+s+1;1-n+m;{sin}^2{\displaystyle \frac{\theta }{2}})\hfill & -n-m\ge 0\\ {\displaystyle \frac{{(-1)}^{m-n}\sqrt{2s+1}}{(n-m)!}}\sqrt{{\displaystyle \frac{(s+n)!}{(s-n)!}}{\displaystyle \frac{(s-m)!}{(s+m)!}}}{e}^{im\phi +in\psi }{cos}^{n+m}{\displaystyle \frac{\theta }{2}}{sin}^{n-m}{\displaystyle \frac{\theta }{2}}\hfill & n\ge m\\ {}_{2}F_1(n-s,n+s+1;1+n-m;{sin}^2{\displaystyle \frac{\theta }{2}})\hfill & n+m\ge 0\\ {\displaystyle \frac{\sqrt{2s+1}}{(m-n)!}}\sqrt{{\displaystyle \frac{(s+m)!}{(s-m)!}}{\displaystyle \frac{(s-n)!}{(s+n)!}}}{e}^{im\phi +in\psi }{cos}^{n+m}{\displaystyle \frac{\theta }{2}}{sin}^{m-n}{\displaystyle \frac{\theta }{2}}\hfill & m\ge n\\ {}_{2}F_1(m-s,m+s+1;1+m-n;{sin}^2{\displaystyle \frac{\theta }{2}})\hfill & n+m\ge 0\end{array}\right.\end{array}$$

(59)

where ${}_{2}F_1$ denotes the Euler hypergeometric polynomial (see, e.g., [45] for definitions). A direct inspection gives rise to

$$\begin{array}{c}\hfill \overline{{\psi }^{nsm}}(\theta ,\phi ,\psi )={(-1)}^{n-m}{\psi }_{-ns-m}(\theta ,\phi ,\psi ),\end{array}$$

leading to ${\eta }_{nm,n^{\prime }{m}^{\prime }}={(-1)}^{n-m}{\delta }_{n,-n^{\prime }}{\delta }_{m,-m^{\prime }}$ (see Section 2.1). These matrix elements were obtained by solving the differential equations

$$\begin{array}{ccc}\hfill L_0{\psi }_{nsm}(\theta ,\phi ,\psi )& =& n{\psi }_{nsm}(\theta ,\phi ,\psi ),\hfill \\ \hfill R_0{\psi }_{nsm}(\theta ,\phi ,\psi )& =& m{\psi }_{nsm}(\theta ,\phi ,\psi ),\hfill \\ \hfill C_2{\psi }_{nsm}(\theta ,\phi ,\psi )& =& s(s+1){\psi }_{nsm}(\theta ,\phi ,\psi )\hfill \end{array}$$

where $C_2$ is the quadratic Casimir operator [47], and we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L_{\pm }{\psi }_{nsm}(\theta ,\phi ,\psi )=& \sqrt{(s\mp n)(s\pm n+1)}{\psi }_{n\pm 1sm},\hfill \\ \hfill L_0{\psi }_{nsm}(\theta ,\phi ,\psi )=& n{\psi }_{nsm}(\theta ,\phi ,\psi )\hfill \\ \hfill R_{\pm }{\psi }_{nsm}(\theta ,\phi ,\psi )=& \sqrt{(s\mp m)(s\pm m+1)}{\psi }_{nsm\pm 1},\hfill \\ \hfill R_0{\psi }_{nsm}(\theta ,\phi ,\psi )=& m{\psi }_{nsm}(\theta ,\phi ,\psi ).\hfill \end{array}\end{array}$$

The set

$$\begin{array}{c}\hfill \mathcal{B}=\left\{{\psi }_{nsm},s\in {\displaystyle \frac{1}{2}}\mathbb{N},-s\le n,m\le s\right\}\end{array}$$

(60)

constitutes a Hilbert basis of $L^2\left(\mathrm{SU}\left(2\right)\right)$, and the following is satisfied:

$$\begin{array}{c}\hfill ({\psi }_{nsm},{\psi }_{n^{\prime },s^{\prime },m^{\prime }})={\delta }_{s{s}^{\prime }}{\delta }_{n{n}^{\prime }}{\delta }_{m{m}^{\prime }}.\end{array}$$

(61)

We now have all the ingredients to define the KM algebra associated to SU$\left(2\right)$. First, observe that the differential operators $L_0,R_0$ satisfy both (30). Let ${\omega }_L,{\omega }_R$ be the corresponding two-cocycles. One may now wonder whether there exists additional differential operators compatible with the two cocycles ${\omega }_L,{\omega }_R$. This will be studied in the next section. Thus, in turn, we are able to define the possible extensions of the algebra $\mathfrak{g}\left(\mathcal{M}\right)=\left\{T_a{\psi }_{nsm}(\theta ,\phi ,\psi ),a=1,\dots ,d,\mathsf{\ell }\in {\displaystyle \frac{1}{2}}\mathbb{N},-\mathsf{\ell }\le n,m\le \mathsf{\ell }\right\}$ (see (52)) by

$$\begin{array}{c}\hfill \tilde{\mathfrak{g}}\left(\mathrm{SU}\left(2\right)\right)⋊\{L_0,R_0\}=\left\{{\mathcal{T}}_{ansm},k_L,k_R,a=1,\dots ,d,\mathsf{\ell }\in {\displaystyle \frac{1}{2}}\mathbb{N},-\mathsf{\ell }\le n,m\le \mathsf{\ell }\right\}⋊\left\{L_0,R_0\right\}\end{array}$$

with Lie brackets (see [40])

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{ansm},{\mathcal{T}}_{a^{\prime }{n}^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& i {f_{a{a}^{\prime }}}^{a^{″}}{C}_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{\mathcal{T}}_{a^{″}n+n^{\prime }{s}^{″}m+m^{\prime }}+k_{ab}{(-1)}^{m-n}{\delta }_{s{s}^{\prime }}{\delta }_{n,-n^{\prime }}{\delta }_{m,-m^{\prime }}(k_L{n}^{\prime }+k_R{m}^{\prime })\hfill \\ \hfill \left[L_0,{\mathcal{T}}_{ansm}\right]& =& n{\mathcal{T}}_{ansm}\hfill \\ \hfill \left[R_0,{\mathcal{T}}_{ansm}\right]& =& m{\mathcal{T}}_{ansm}.\hfill \end{array}$$

(62)

Here, we have

$$\begin{array}{c}\hfill {C_{s_1{s}_2}^S}_{n_1{n}_2{m}_1{m}_2}=\sqrt{{\displaystyle \frac{(2s_1+1)(2s_2+1)}{2S+1}}}\left(\begin{array}{ccc}{s}_1& s_2& S\\ n_1& n_2& n_1+n_2\end{array}\right)\overline{\left(\begin{array}{ccc}{s}_1& s_2& S\\ m_1& m_2& m_1+m_2\end{array}\right)}\end{array}$$

(63)

with $\left(\begin{array}{ccc}{s}_1& s_2& S\\ n_1& n_2& n_1+n_2\end{array}\right)$ the Clebsch–Gordan coefficients. Note that we recover the algebra obtained in [40,47]. There exists a second algebra associated to $\mathfrak{g}\left(\mathrm{SU}\right(2\left)\right)$, which will be studied in the next section.

The KM algebra associated to SU$\left(2\right)/$U$\left(1\right)\cong {\mathbb{S}}^2$ follows directly from the construction above. A point in the coset is parameterized by

$$\begin{array}{c}\hfill m=\left(\begin{array}{cc}{e}^{i\phi }cos{\displaystyle \frac{\theta }{2}}& -sin{\displaystyle \frac{\theta }{2}}\\ sin{\displaystyle \frac{\theta }{2}}& e^{-i\phi }cos{\displaystyle \frac{\theta }{2}}\end{array}\right),\end{array}$$

obtained by the substitution $\psi =-\phi$ in (55). From (47), we obtain the left-invariant vector fields

$$\begin{array}{c}\hfill {\omega }_1=sin\theta sin\phi \mathrm{d}\phi -sin\phi \mathrm{d}\theta ,{\omega }_2=sin\theta cos\phi \mathrm{d}\phi +cos\phi \mathrm{d}\theta ,\end{array}$$

such that the metric on the two-sphere reduces to

$$\begin{array}{c}\hfill \mathrm{d}{s}^2=\mathrm{d}{\theta }^2+{sin}^2\theta \mathrm{d}{\phi }^2.\end{array}$$

The invariant vector fields obtained by solving Killing Equation (1) are

$$\begin{array}{c}\hfill L_{\pm }=e^{\pm i\phi }\left(icot\theta {\partial }_{\phi }\pm {\partial }_{\theta }\right),L_0=-i{\partial }_{\phi },\end{array}$$

and generate the $\mathfrak{so}\left(3\right)-$Lie algebra. We finally define the scalar product

$$\begin{array}{c}\hfill (f,g)={\displaystyle \frac{1}{4\pi }}\underset{{\mathbb{S}}^2}{\int }sin\theta \mathrm{d}\theta \mathrm{d}\phi \overline{f(\theta ,\phi )}g(\theta ,\phi ).\end{array}$$

Following Section 3.2, the only representations to be considered for harmonic analysis on the two-sphere are those having matrix elements with eigenvalues $R_0$ equal to zero (see (51)). This is possible if $s\in \mathbb{N}$ and for $m=0$. By substituting $m=0$ in (59), we obtain the following $(\mathsf{\ell }\in \mathbb{N}$):

$$\begin{array}{cc}& Y_{\mathsf{\ell }n}(\theta ,\phi )={\displaystyle \frac{{(-1)}^{\mathsf{\ell }+\frac{\left|n\right|+n}{2}}\sqrt{2\mathsf{\ell }+1}}{\left|n\right|!}}\sqrt{{\displaystyle \frac{(\mathsf{\ell }+|n\left|\right)!}{(\mathsf{\ell }-|n\left|\right)!}}}{e}^{in\phi }{cos}^{\left|n\right|}{\displaystyle \frac{\theta }{2}}{sin}^{\left|n\right|}{\displaystyle \frac{\theta }{2}}\\ & {}_{2}F_1\left(\right|n|-\mathsf{\ell },\left|n\right|+\mathsf{\ell }+1;1+\left|n\right|;{sin}^2{\displaystyle \frac{\theta }{2}}),\end{array}$$

where

$$\begin{array}{c}\hfill \overline{Y^{\mathsf{\ell }n}}(\theta ,\phi )={(-1)}^n{Y}^{\mathsf{\ell }-n}(\theta ,\phi ).\end{array}$$

The KM algebras associated to ${\mathbb{S}}^2$ follow immediately:

$$\begin{array}{c}\hfill \tilde{\mathfrak{g}}(\mathrm{SU}\left(2\right)/\mathrm{U}\left(1\right))=\left\{{\mathcal{T}}_{a\mathsf{\ell }m},k,a=1,\dots ,d,\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}⋊\left\{L_0\right\}\end{array}$$

with Lie brackets (see [40])

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{a\mathsf{\ell }m},{\mathcal{T}}_{a^{\prime }{\mathsf{\ell }}^{\prime }{m}^{\prime }}\right]& =& i{f_{a{a}^{\prime }}}^{a^{″}}{C}_{\mathsf{\ell }{\mathsf{\ell }}^{\prime }m{m}^{\prime }}^{{\mathsf{\ell }}^{″}}{\mathcal{T}}_{a^{″}m+m^{\prime }{\mathsf{\ell }}^{″}}+k{k}_{ab}{(-1)}^m{\delta }_{\mathsf{\ell }{\mathsf{\ell }}^{\prime }}{\delta }_{m,-m^{\prime }}\hfill \\ \hfill \left[L_0,{\mathcal{T}}_{a\mathsf{\ell }m}\right]& =& m{\mathcal{T}}_{a\mathsf{\ell }m}\hfill \end{array}$$

Since in the coset SU$\left(2\right)/$U$\left(1\right)$, the matrix elements of the left action are obtained from the matrix elements of SU$\left(2\right)$, ${\psi }_{n\mathsf{\ell }m}$ for $m=0$, the coefficients (63) reduce to

$$\begin{array}{c}\hfill {C_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2}^L}_{n_1{n}_2}=\sqrt{{\displaystyle \frac{(2{\mathsf{\ell }}_1+1)(2{\mathsf{\ell }}_2+1)}{2L+1}}}\left(\begin{array}{ccc}{\mathsf{\ell }}_1& {\mathsf{\ell }}_2& L\\ n_1& n_2& n_1+n_2\end{array}\right)\overline{\left(\begin{array}{ccc}{\mathsf{\ell }}_1& {\mathsf{\ell }}_2& L\\ 0& 0& 0\end{array}\right)}\end{array}$$

This algebra was obtained for the first time in [31]; see also [40,47].

3.5. Virasoro Algebra Associated to the Two- and Three-Sphere

In this paragraph, we show that, for two- and three-spheres, an analog of the Virasoro algebra can be constructed.

First, consider the case of ${\mathbb{S}}^2$. Let L be a vector field on ${\mathbb{S}}^2$:

$$\begin{array}{c}\hfill L=A_{\phi }(\theta ,\phi ){\partial }_{\phi }+A_{\theta }(\theta ,\phi ){\partial }_{\theta }\end{array}$$

As $H^1\left({\mathbb{S}}^2\right)=1$, every closed one-form on the sphere is exact. Let $\gamma =\mathrm{d}h$ be a closed one-form associated to the cocycle

$$\begin{array}{c}\hfill {\omega }_h(X,Y)={\displaystyle \frac{1}{4\pi }}\underset{{\mathbb{S}}^2}{\int }⟨X,\mathrm{d}Y⟩\wedge \mathrm{d}h.\end{array}$$

The compatibility condition between the KM algebra with a two-cocycle ${\omega }_h$ is

$$\begin{array}{c}\hfill {\left[X,Y\right]}_h{=[X,Y]}_0+{\omega }_h(X,Y),\end{array}$$

where ${[,]}_0$ are the Lie brackets without central extension, i.e., the Lie brackets of $\mathfrak{g}\left({\mathbb{S}}^2\right)$, which turn out to be

$$\begin{array}{c}\hfill Lh=c,\end{array}$$

with c a constant whose solutions are given by (see [33])

$$\begin{array}{ccc}\hfill L_h^T(\theta ,\phi )& =& T(\theta ,\phi )({\partial }_{\theta }h{\partial }_{\phi }-{\partial }_{\phi }h{\partial }_{\theta }),\hfill \\ \hfill \Lambda (\theta ,\phi )& =& {\displaystyle \frac{{\partial }_{\theta }h{\partial }_{\phi }+{\partial }_{\phi }h{\partial }_{\theta }}{{\partial }_{\theta }h{\partial }_{\phi }h}}.\hfill \end{array}$$

(64)

The generators $L_h^T$ generate the area-preserving diffeomorphism algebra [63,64]. The second solution is possible if the denominator does not vanish. In particular, if we define the cocycle associated to $L_0$, the generators of the de Witt algebra

$$\begin{array}{c}\hfill {\mathsf{\ell }}_{\mathsf{\ell }m}=-Y_{\mathsf{\ell }m}(\theta ,\phi )L_0\end{array}$$

(65)

are compatible with the two-cocycle ${\omega }_0$. Thus, we are able to define the algebra

$$\begin{array}{cc}& {\tilde{\mathfrak{g}}}^{\prime }(\mathrm{SU}\left(2\right)/\mathrm{U}\left(1\right))⋊\mathrm{Witt}\\ & =\left\{{\mathcal{T}}_{a\mathsf{\ell }m},ka=1,\dots ,d,\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}⋊\left\{{\mathsf{\ell }}_{\mathsf{\ell }m},\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}\end{array}$$

with the following Lie brackets:

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{a\mathsf{\ell }m},{\mathcal{T}}_{a^{\prime }{\mathsf{\ell }}^{\prime }{m}^{\prime }}\right]& =& i {f_{a{a}^{\prime }}}^{a^{″}}{C}_{\mathsf{\ell }{\mathsf{\ell }}^{\prime }m{m}^{\prime }}^{{\mathsf{\ell }}^{″}}{\mathcal{T}}_{a^{″}{\mathsf{\ell }}^{″}m+m^{\prime }}+k{m}^{\prime }{k}_{ab}{(-1)}^m{\delta }_{\mathsf{\ell }{\mathsf{\ell }}^{\prime }}{\delta }_{m,-m^{\prime }}\hfill \\ \hfill \left[L_0,{\mathcal{T}}_{a\mathsf{\ell }m}\right]& =& m{\mathcal{T}}_{a\mathsf{\ell }m}\hfill \end{array}$$

(66)

A similar analysis holds for ${\mathbb{S}}^3$. As $H^2\left({\mathbb{S}}^3\right)=1$, a closed one-form is exact. Let $(u_1,u_2,u_3)=(\phi ,\psi ,cos\theta )$, and let $h\left(u\right)=h_i\left(u\right)\mathrm{d}{u}^i$. The corresponding two-cocycle associated to the closed form $\mathrm{d}h$ is given by

$$\begin{array}{c}\hfill {\omega }_h(X,Y)=\underset{{\mathbb{S}}^3}{\int }⟨X,\mathrm{d}Y⟩\wedge \mathrm{d}h.\end{array}$$

From $\mathrm{d}(X\mathrm{d}Y\wedge h)=\mathrm{d}X\wedge \mathrm{d}Y\wedge h-X\mathrm{d}Y\wedge \mathrm{d}h$, and since ${\mathbb{S}}^3$ has no boundary, we have

$$\begin{array}{ccc}\hfill {\omega }_h(X,Y)& =& \underset{{\mathbb{S}}^3}{\int }⟨X\wedge \mathrm{d}Y⟩\wedge \mathrm{d}h=\underset{{\mathbb{S}}^3}{\int }⟨\mathrm{d}X\wedge \mathrm{d}Y⟩\wedge h\hfill \\ & =& \underset{{\mathbb{S}}^3}{\int }\left(h_1{\{X,Y\}}_{23}+h_2{\{X,Y\}}_{31}+h_3{\{X,Y\}}_{12}\right)\mathrm{d}{u}_1\mathrm{d}{u}_2\mathrm{d}{u}_3,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\{X,Y\}}_{ij}={\partial }_{i}X{\partial }_{j}Y-{\partial }_{j}X{\partial }_{i}Y.\end{array}$$

It can be shown that the vector field (see also (33))

$$\begin{array}{c}\hfill L_h=A(u_1,u_2,u_3)\left(\left({\partial }_1{h}_2{\partial }_3-{\partial }_3{h}_2{\partial }_1\right)+\left({\partial }_2{h}_3{\partial }_1-{\partial }_1{h}_3{\partial }_3\right)+\left({\partial }_3{h}_1{\partial }_2-{\partial }_2{h}_1{\partial }_3\right)\right)\end{array}$$

is compatible with the two-cocycle ${\omega }_h$ [33]. The algebra generated by the operators of the form $L_h$ above is called area-preserving diffeomorphisms in [63,64]. Thus, we are able to define (pay attention to the fact that, in this case, we only have one central charge, say $k_L=k$)

$$\begin{array}{cc}& \widetilde{{\mathfrak{g}}^{\prime }}\left(\mathrm{SU}\left(2\right)\right)⋊{\mathrm{Witt}}_{L_0}=\\ & \left\{{\mathcal{T}}_{ansm},k,a=1,\dots ,d,s\in {\displaystyle \frac{1}{2}}\mathbb{N},-s\le n,m\le s\right\}⋊\left\{{\mathsf{\ell }}_{msn},s\in {\displaystyle \frac{1}{2}}\mathbb{N},-s\le n,m\le s\right\}\end{array}$$

where

$$\begin{array}{c}\hfill {\mathsf{\ell }}_{msn}=-{\psi }_{msn}(\phi ,\psi ,\theta )L_0\end{array}$$

and with Lie brackets:

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{ansm},{\mathcal{T}}_{a^{\prime }{n}^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& i {f_{a{a}^{\prime }}}^{a^{″}}{C}_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{\mathcal{T}}_{a^{″}n+n^{\prime }{s}^{″}m+m^{\prime }}+n^{\prime }k{k}_{a{a}^{\prime }}{(-1)}^{m+n}{\delta }_{s{s}^{\prime }}{\delta }_{n,-n^{\prime }}{\delta }_{m,-m^{\prime }}\hfill \\ \hfill \left[{\mathsf{\ell }}_{nsm},{\mathcal{T}}_{a^{\prime }{n}^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& -n^{\prime }{C}_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{\mathcal{T}}_{an+n^{\prime }{s}^{″}m+m^{\prime }}\hfill \\ \hfill \left[{\mathsf{\ell }}_{nsm},{\mathsf{\ell }}_{n^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& (n-n^{\prime })C_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{\mathsf{\ell }}_{n+n^{\prime }{s}^{″}m+m^{\prime }}\hfill \end{array}$$

(67)

We now show that the de Witt algebra of the two- and three-sphere can be centrally extended. Before analysing possible central extensions, let us make several observations. A basis of the algebra of vector fields $\mathfrak{X}\left({\mathbb{S}}^2\right)$ is given by $\{X_{\mathsf{\ell }m}^{\phi }=Y_{\mathsf{\ell }m}{\partial }_{\phi },X_{\mathsf{\ell }m}^{\theta }=Y_{\mathsf{\ell }m}{\partial }_{\theta },\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\}$. Several subalgebras of $\mathfrak{X}\left({\mathbb{S}}^2\right)$ with different properties may be considered. For instance, Floratos and Iliopoulos studied the area-preserving diffeomorphism algebra that appears in the analysis of bosonic membranes. If the membrane has the topology of a two-sphere, this algebra is generated by the vector fields of the form [63] $L_h={\partial }_{cos\theta }h(\theta ,\phi ){\partial }_{\phi }-{\partial }_{\phi }h(\theta ,\phi ){\partial }_{cos\theta }$ and satisfies

$$\begin{array}{c}\hfill \left[V_{h_1},V_{h_2}\right]=V_{\{h_2,h_1\}},\end{array}$$

(68)

where $\{h_1,h_2\}={\partial }_{cos\theta }{h}_1(\theta ,\phi ){\partial }_{\phi }{h}_2(\theta ,\phi )-{\partial }_{cos\theta }{h}_2(\theta ,\phi ){\partial }_{\phi }{h}_1(\theta ,\phi )$ is the Poisson bracket. In particular, for $V_{\mathsf{\ell }m}={\partial }_{cos\theta }{Y}_{\mathsf{\ell }m}(\theta ,\phi ){\partial }_{\phi }-{\partial }_{\phi }{Y}_{\mathsf{\ell }m}(\theta ,\phi ){\partial }_{cos\theta }$, we have

$$\begin{array}{c}\hfill \left[V_{{\mathsf{\ell }}_1{m}_1},V_{{\mathsf{\ell }}_2{m}_2}\right]=-g_{{\mathsf{\ell }}_1{m}_1{\mathsf{\ell }}_2{m}_2}^{{\mathsf{\ell }}_3{m}_3}{V}_{{\mathsf{\ell }}_3{m}_3},\end{array}$$

(69)

where the structure constants

$$\begin{array}{c}\hfill \{Y_{{\mathsf{\ell }}_1{m}_1},Y_{{\mathsf{\ell }}_2{m}_2}\}=g_{{\mathsf{\ell }}_1{m}_1{\mathsf{\ell }}_2{m}_2}^{{\mathsf{\ell }}_3{m}_3}{Y}_{{\mathsf{\ell }}_3{m}_3},\end{array}$$

are given in [65]. The second relevant algebra is that associated with the compatibility of the cocycle ${\omega }_h$ [33]. This algebra is generated by the vector fields of the form $L_h^T=T(\theta ,\phi )\left({\partial }_{cos\theta }h(\theta ,\phi ){\partial }_{\phi }-{\partial }_{\phi }h(\theta ,\phi ){\partial }_{cos\theta }\right)$:

$$\begin{array}{c}\hfill \left[L_h^{T_1},L_h^{T_2}\right]=L_h^{T_3},T_3=T_1\{T_2,h\}-T_2\{T_1,h\}.\end{array}$$

(70)

The difference between the algebras given in (68) and (70) is that, for the former, the function h in $V_h$ varies, whereas in the latter, the function h in $V_h^T$ is fixed. It has been shown in [64] that the algebra (68), or the area-preserving diffeomorphism algebra, does not admit central extensions. In contrast, we now show that the algebra (70) for $h(\theta ,\phi )=Y_{\mathsf{\ell }m}(\theta ,\phi )$ admits a central extension. Introduce the spherical harmonics in the form

$$\begin{array}{c}\hfill Y_{\mathsf{\ell }m}(\theta ,\phi )=\sqrt{{\displaystyle \frac{(\mathsf{\ell }-m)!}{(\mathsf{\ell }+m)!}}}{P}_{\mathsf{\ell }m}\left(\cos \theta \right)e^{im\phi }=Q_{\mathsf{\ell }m}\left(\cos \theta \right)e^{im\phi },\end{array}$$

(71)

where $P_{\mathsf{\ell }m}$ are the associated Legendre functions, satisfying the orthonormality property for ${\mathsf{\ell }}_1,{\mathsf{\ell }}_2\ge \left|m\right|$:

$$\begin{array}{c}\hfill (Q_{{\mathsf{\ell }}_{1}m},Q_{{\mathsf{\ell }}_{2}m})={\displaystyle \frac{1}{2}}\underset{0}{\overset{\pi }{\int }}\sin \theta \mathrm{d}\theta Q_{{\mathsf{\ell }}_{2}m}\left(\cos \theta \right)Q_{{\mathsf{\ell }}_{1}m}\left(\cos \theta \right)={\delta }_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2}.\end{array}$$

(72)

We also have

$$\begin{array}{c}\hfill Q_{{\mathsf{\ell }}_1{m}_1}\left(\cos \theta \right)Q_{{\mathsf{\ell }}_2{m}_2}\left(\cos \theta \right)=C_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2{m}_1{m}_2}^{\mathsf{\ell }}{Q}_{\mathsf{\ell }{m}_1+m_2}\left(\cos \theta \right),\end{array}$$

(73)

and

$$\begin{array}{c}\hfill Q_{\mathsf{\ell }-m}\left(\cos \theta \right)={(-1)}^m{Q}_{\mathsf{\ell }m}\left(\cos \theta \right),\end{array}$$

(74)

which are a direct consequence of analogous relations for $Y_{\mathsf{\ell }m}$ and from the definition of $Q_{\mathsf{\ell }m}$ (see (71)). Let

$$\begin{array}{ccc}\hfill L(\theta ,\phi )& =& \sum _{m=-\infty }^{+\infty }\left(\sum _{\mathsf{\ell }=\left|m\right|}^{+\infty }{L}_{\mathsf{\ell }m}{Q}_{\mathsf{\ell }m}\left(\cos \theta \right)\right)e^{im\phi }=\sum _{m=-\infty }^{+\infty }{L}_m\left(\theta \right)e^{im\phi }.\hfill \end{array}$$

We further assume that the $L_m\left(\theta \right)$ satisfies the relations

$$\begin{array}{ccc}\hfill \left[L_m\left(\theta \right),L_n\left({\theta }^{\prime }\right)\right]& =& \left((m-n)L_{m+n}+{\displaystyle \frac{c}{12}}m(m^2-1){\delta }_{m,-n}\right)\delta (\cos \theta -\cos {\theta }^{\prime })\hfill \end{array}$$

(75)

Now, using the orthonormality relations (72), (73), and (74), integration by

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\int \mathrm{d}u{Q}_{\mathsf{\ell }m}\left(u\right){\displaystyle \frac{1}{2}}\int \mathrm{d}{u}^{\prime }{Q}_{{\mathsf{\ell }}^{\prime }{m}^{\prime }}\left(u^{\prime }\right),u=\cos \theta ,u^{\prime }=\cos {\theta }^{\prime }\end{array}$$

leads to

$$\begin{array}{ccc}\hfill \left[L_{{\mathsf{\ell }}_1{m}_1},L_{{\mathsf{\ell }}_2{m}_2}\right]& =& (m-n)C_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2{m}_1{m}_2}^{{\mathsf{\ell }}_3}{L}_{{\mathsf{\ell }}_3{m}_1+m_2}+{(-1)}^{m_1}{\displaystyle \frac{c}{12}}{m}_1(m_1^2-1){\delta }_{m_1,-m_2}{\delta }_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2}\hfill \end{array}$$

We now check that the Jacobi identities are satisfied using (75). Indeed,

$$\begin{array}{cc}& [[L_{m_1}\left({\theta }_1\right),L_{m_2}\left({\theta }_2\right)],L_{m_3}\left({\theta }_3\right)]\\ & +[[L_{m_2}\left({\theta }_2\right),L_{m_3}\left({\theta }_3\right)],L_{m_1}\left({\theta }_1\right)]\\ & +[[L_{m_3}\left({\theta }_3\right),L_{m_1}\left({\theta }_1\right)],L_{m_2}\left({\theta }_2\right)]\\ & =-{\displaystyle \frac{c}{12}}(m_2-m_3)(m_1-m_3)(m_1-m_2)(m_1+m_3+m_2)\\ & {\delta }_{m_1+m_2+m_3,0}\delta (\cos {\theta }_1-\cos {\theta }_2)\delta (\cos {\theta }_2-\cos {\theta }_3)=0\end{array}$$

where we have used $(m_1-m_2)(m_3^3-m_3)+(m_2-m_3)(m_1^3-m_1)+(m_3-m_1)(m_2^3-m_2)=(m_2-m_3)(m_1-m_3)(m_1-m_2)(m_1+m_3+m_2)$.

Thus, altogether, the algebra

$$\begin{array}{cc}& {\tilde{\mathfrak{g}}}^{\prime }\left({\mathbb{S}}^2\right)⋊\mathrm{Vir}\left({\mathbb{S}}^2\right)=\\ & \left\{{\mathcal{T}}_{a\mathsf{\ell }m},k,a=1,\dots ,d,\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}⋊\left\{L_{\mathsf{\ell }m},\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}\end{array}$$

has Lie brackets:

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{a\mathsf{\ell }m},{\mathcal{T}}_{a^{\prime }{\mathsf{\ell }}^{\prime }{m}^{\prime }}\right]& =& i {f_{a{a}^{\prime }}}^{a^{″}}{C}_{\mathsf{\ell }{\mathsf{\ell }}^{\prime }m{m}^{\prime }}^{{\mathsf{\ell }}^{″}}{\mathcal{T}}_{a^{″}{\mathsf{\ell }}^{″}m+m^{\prime }}+k{m}^{\prime }{k}_{ab}{(-1)}^m{\delta }_{\mathsf{\ell }{\mathsf{\ell }}^{\prime }}{\delta }_{m,-m^{\prime }}\hfill \\ \hfill \left[L_{{\mathsf{\ell }}_1{m}_1},L_{{\mathsf{\ell }}_2{m}_2}\right]& =& (m-n)C_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2{m}_1{m}_2}^{{\mathsf{\ell }}_3}{L}_{{\mathsf{\ell }}_3{m}_1+m_2}+{(-1)}^{m_1}{\displaystyle \frac{c}{12}}{m}_1(m_1^2-1){\delta }_{m_1+m_2}{\delta }_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2}\hfill \\ \hfill \left[L_{\mathsf{\ell }m},{\mathcal{T}}_{a_1{\mathsf{\ell }}_1{m}_1}\right]& =& -m_1{C}_{\mathsf{\ell }{\mathsf{\ell }}_{1}m{m}_1}^{{\mathsf{\ell }}^{\prime }}{\mathcal{T}}_{a_1{\mathsf{\ell }}^{\prime }m+m_1},\hfill \end{array}$$

(76)

where now $L_0=-L_{00}$.

It remains to be shown that the algebra above is the only centrally extended Lie algebra obtained from $\mathfrak{g}\left({\mathbb{S}}^2\right)$ and $\mathfrak{X}\left({\mathbb{S}}^2\right)$ that one can consistently define. Recall that $\mathfrak{g}\left({\mathbb{S}}^2\right)⋊\mathfrak{X}\left({\mathbb{S}}^2\right)=\left\{T_{a\mathsf{\ell }m},a=1,\dots ,dim\mathfrak{g},\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}⋊\left\{X_{\mathsf{\ell }m}^{\theta }=Y_{\mathsf{\ell }m}(\theta ,\phi ){\partial }_{\theta },X_{\mathsf{\ell }m}^{\phi }=Y_{\mathsf{\ell }m}(\theta ,\phi ){\partial }_{\phi },\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}$, which contains as a subalgebra $\mathfrak{g}\left({\mathbb{S}}^2\right)⋊\{L_0,L_{+},L_{-}\}$ (where $\{L_0,L_{+},L_{-}\}$ are the algebra associated to the isometry of ${\mathbb{S}}^2$) and has Lie brackets:

$$\begin{array}{ccc}\hfill \left[T_{a_1{\mathsf{\ell }}_1{m}_1},T_{a_2{\mathsf{\ell }}_2{m}_2}\right]& =& i {f_{a_1{a}_2}}^{a_3}{C}_{{\mathsf{\ell }}_1{\mathsf{\ell }}_2{m}_1{m}_2}^{{\mathsf{\ell }}_3}{T}_{a_3{\mathsf{\ell }}_3{m}_1+m_2}\hfill \\ \hfill [L_0,T_{a_1{\mathsf{\ell }}_1{m}_1}]& =& m_1{T}_{a_1{\mathsf{\ell }}_1{m}_1}\hfill \\ \hfill [L_{\pm },T_{a_1{\mathsf{\ell }}_1{m}_1}]& =& \sqrt{({\mathsf{\ell }}_1\mp m_1)({\mathsf{\ell }}_1\pm m_1+1)}{T}_{a_1{\mathsf{\ell }}_1{m}_1\pm 1}\hfill \end{array}$$

The algebra $\mathfrak{g}\left({\mathbb{S}}^2\right)$ admits an infinite number of central extensions [4]. Explicit expressions can be found in [33,36,37]. Now, the compatibility of the algebra which centrally extends $\mathfrak{g}\left({\mathbb{S}}^2\right)$ and $\mathfrak{X}\left({\mathbb{S}}^2\right)$ is two-fold: on the one hand, only one cocycle can be considered, say ${\omega }_h$. On the other hand, only the vector fields (64) are compatible with ${\omega }_h$. If we now centrally extend $\mathfrak{g}\left({\mathbb{S}}^2\right)$ by introducing the two-cocyle ${\omega }_0$, the only generators of the first algebra that are compatible with ${\omega }_0$ are the operators ${\mathsf{\ell }}_{\mathsf{\ell }m}$ (see (65)), and the only operator of the second algebra that is compatible with ${\omega }_0$ is $L_0$ (strictly speaking, the operator $R_0$ is also compatible with ${\omega }_0$):

$$\begin{array}{ccc}& & \left\{{\mathcal{T}}_{a\mathsf{\ell }m},k,a=1,\dots ,dim\mathfrak{g},\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}⋊\left\{L_0\right\}\hfill \\ & \subset & \left\{{\mathcal{T}}_{a\mathsf{\ell }m},k,a=1,\dots ,dim\mathfrak{g},\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}⋊\left\{{\mathsf{\ell }}_{\mathsf{\ell }m},\mathsf{\ell }\in \mathbb{N},-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}.\hfill \end{array}$$

Finally, if we centrally extend the de Witt algebra $\left\{{\mathsf{\ell }}_{\mathsf{\ell }m},\mathsf{\ell }\in N,-\mathsf{\ell }\le m\le \mathsf{\ell }\right\}$, we obtain the Virasoro algebra of the two-sphere. Thus, in conclusion, the algebra ${\tilde{\mathfrak{g}}}^{\prime }\left({\mathbb{S}}^2\right)⋊\mathrm{Vir}\left({\mathbb{S}}^2\right)$ with brackets given in (76) is the only centrally extended algebra that one can define along these lines.

The analysis for the three-sphere is similar, starting from

$$\begin{array}{ccc}\hfill L(\theta ,\phi ,\psi )& =& \sum _{ϵ=0,1/2}\sum _{\mathsf{\ell }\in \mathbb{N}+ϵ}\sum _{m,n=-\mathsf{\ell }}^{\mathsf{\ell }}{L}_{n\mathsf{\ell }m}{\psi }_{n\mathsf{\ell }m}(\theta ,\phi ,\psi )\hfill \\ & =& \sum _{ϵ=0,1/2}\sum _{n\in \mathbb{Z}+ϵ}\left(\sum _{\mathsf{\ell }\ge \left|n\right|}\sum _{m=-\mathsf{\ell }}^{\mathsf{\ell }}{L}_{n\mathsf{\ell }m}{F}_{\mathsf{\ell }m}(\theta ,\phi )\right)e^{in\psi }\hfill \\ & \equiv & \sum _{ϵ=0,1/2}\sum _{n\in \mathbb{Z}+ϵ}{L}_n(\theta ,\phi )e^{in\psi }\hfill \end{array}$$

We have introduced the functions $F_{n\mathsf{\ell }m}$ defined by ${\psi }_{nsm}(\theta ,\phi ,\psi )=F_{nsm}(\theta ,\phi )e^{in\psi }$. Thus, if we assume

$$\begin{array}{ccc}\hfill \left[L_n(\theta ,\phi ),L_{n^{\prime }}({\theta }^{\prime },{\phi }^{\prime })\right]& =& \left((n-n^{\prime })L_{n+n^{\prime }}(\theta ,\phi )+{\displaystyle \frac{c}{12}}(n^3-n){\delta }_{n,-n^{\prime }}\right){\displaystyle \frac{\delta (\theta -{\theta }^{\prime })\delta (\phi -{\phi }^{\prime })}{sin{\theta }^{\prime }}}\hfill \end{array}$$

as for the two-sphere, we obtain the algebra

$$\begin{array}{cc}& {\tilde{g}}^{\prime }\left(\mathrm{SU}\left(2\right)\right)⋊\mathrm{Vir}\left(\mathrm{SU}\left(2\right)\right)=\\ & \left\{{\mathcal{T}}_{ansm},k,a=1,\dots ,d,s\in {\displaystyle \frac{1}{2}}\mathbb{N},-s\le n,m\le s\right\}⋊\left\{L_{msn},s\in {\displaystyle \frac{1}{2}}\mathbb{N},-s\le n,m\le s\right\}\end{array}$$

where

$$\begin{array}{c}\hfill L_{msn}=-{\psi }_{msn}(\phi ,\psi ,\theta )L_0\end{array}$$

and with Lie brackets,

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{ansm},{\mathcal{T}}_{a^{\prime }{n}^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& i {f_{a{a}^{\prime }}}^{a^{″}}{C}_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{\mathcal{T}}_{a^{″}n+n^{\prime }{s}^{″}m+m^{\prime }}+n^{\prime }k{k}_{a{a}^{\prime }}{(-1)}^{m+n}{\delta }_{s{s}^{\prime }}{\delta }_{n,-n^{\prime }}{\delta }_{m,-m^{\prime }}\hfill \\ \hfill \left[L_{nsm},{\mathcal{T}}_{a^{\prime }{n}^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& -n^{\prime }{C}_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{\mathcal{T}}_{an+n^{\prime }{s}^{″}m+m^{\prime }}\hfill \\ \hfill \left[L_{nsm},L_{n^{\prime }{s}^{\prime }{m}^{\prime }}\right]& =& (n-n^{\prime })C_{s{s}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{s^{″}}{L}_{n+n^{\prime }{s}^{″}m+m^{\prime }}+{\displaystyle \frac{c}{12}}(n^3-n){\delta }_{s{s}^{\prime }}{\delta }_{n,-n^{\prime }}{\delta }_{m,-m^{\prime }}.\hfill \end{array}$$

We now have $L_0=-L_{000}$. This algebra is defined in [44].

4. KM Algebras on Non-Compact Lie Groups

The purpose of this section is to extend the results of the previous section to the case of non-compact manifolds.

4.1. Some Generalities

Let $G_{nc}$ be a non-compact Lie group, and let ${\mathfrak{g}}_{nc}=\{J_A,A=1,\dots ,n\}$ be its corresponding Lie algebra. In contrast to compact Lie algebras, where the Killing form is positive definite (we use the physicists’ notation, in which the Lie brackets have an additional $i-$factor. Thus, for unitary representations, the generators of the Lie algebra $\mathfrak{g}$ are Hermitian, and the Killing form is definite positive for a compact semisimple Lie algebra). For non-compact Lie algebras ${\mathfrak{g}}_{nc}$, the Killing form has the signature $(n_{+},n_{-})$ with $n=n_{+}+n_{-}$. Despite this difference, the metric $g_{MN}$ on $G_{nc}$ is obtained in a similar way to the metric on $G_c$ (see Equations (34)–(38)). Since, in this case, the manifold $G_{nc}$ is non-compact, its volume is infinite (the volume is finite for a compact Lie group $G_c$). Thus, we define the scalar product for $f,g\in L^2\left(G_{nc}\right)$ by

$$\begin{array}{c}\hfill (f,g)=\underset{G_{nc}}{\int }\sqrt{g}\mathrm{d}{m}^n\overline{f\left(m\right)}g\left(m\right)\end{array}$$

(77)

for the variables m, with the same notations as in Section 3.1. As it holds for compact manifolds, when solving the Killing equations, the generators of the left (resp., right) action, and ${\mathfrak{g}}_{\xi }={\left({\mathfrak{g}}_{nc}\right)}_L\oplus {\left(g_{nc}\right)}_R$ corresponding to the left/right action of $G_{nc}$, automatically appear as solutions. For the corresponding generators, here, we adopt the same notation used for $G_c$ (see (42)).

The next step in the construction of a KM algebra associated to $G_{nc}$ is to decompose square integrable functions on $L^2\left(G_{nc}\right)$. However, the situation for non-compact Lie groups is very different than the situation for compact Lie groups. The first difference resides in the unitary representations of $G_{nc}$. Due to non-compactness, unitary representations are infinite-dimensional (recall that for a compact Lie group, all unitary representations are finite-dimensional). Next, there exist two types of unitary representations: the discrete series and the continuous series [66]. A non-compact Lie group always admits a continuous series as a unitary representation, but the former exists iff the rank of the non-compact group is equal to the rank of its maximal compact subgroup. The discrete series are characterized by discrete eigenvalues of the Casimir operators, whereas the continuous series have continuous eigenvalues of the Casimir operators. This, in particular, means that if we realize the Lie algebra on the manifold $G_{nc}$, the continuous series are non-normalizable, while the discrete series are normalizable. Hence, the harmonic analysis on $L^2\left(G_{nc}\right)$ is more involved in this case and is summarized in the Plancherel theorem [51]. This theorem basically states that any square integrable function on $G_{nc}$ decomposes as a sum over the matrix elements of the discrete series and an integral over the matrix elements of the continuous series. However, since we are considering Hermitian operators with continuous spectra, their eigenfunctions are not normalizable. This means that some care must be taken. Thus, we consider a Gel’fand triple of a Hilbert space $\mathcal{H}$ (see, for instance, [67]). Let $\mathcal{S}$ be a dense subspace of smooth functions of $\mathcal{H}$ and its dual ${\mathcal{S}}^{\prime }$

$$\begin{array}{c}\hfill \mathcal{S}\stackrel{J}{\subset }\mathcal{H}\stackrel{K}{\subset }{\mathcal{S}}^{\prime },\end{array}$$

(78)

where $J:\mathcal{S}\to \mathcal{H}$ is an injective bounded operator with a dense image, and K is the composition of the canonical isomorphism $\mathcal{H}\simeq {\mathcal{H}}^{\prime }$ determined by the inner product (given by the Riesz theorem) and the dual $J^{\prime }:{\mathcal{H}}^{\prime }\to {\mathcal{S}}^{\prime }$ of J. The set $\mathcal{S}$ we consider here is the space of rapidly decreasing functions in any non-compact directions of $G_{nc}$ or, for short, Schwartz functions. Schwartz functions (which generalize the well-known set of Schwartz functions for $\mathbb{R}$) were defined for any semisimple Lie group by Harish–Chandra (see also [68] and the references therein). With these definitions, it turns out that the matrix elements of the continuous series belong to ${\mathcal{S}}^{\prime }$, and a closed expression of the Plancherel theorem can be given (see, for instance, [67], Theorem 1, p. 426, and (34)–(37) p. 429 or [68] Chapter 8).

We also consider a second way to expand any square integrable functions on $G_{nc}$ by identifying a Hilbert basis of $L^2\left(G_{nc}\right)$. This is always possible, since any Hilbert space admits a Hilbert basis (i.e., a countable, complete set of orthonormal vectors) [49].

Therefore, following the steps of Section 3.3, one can naturally associate to $G_{nc}$ a corresponding KM algebra. In this review, we do not consider a generic non-compact manifold $G_{nc}$; we only focus on a specific $G_{nc}$, namely $\mathrm{SL}(2,\mathbb{R})$. There are two isomorphic presentations of the KM algebra associated to $\mathrm{SL}(2,\mathbb{R})$. The first one is based on the Plancherel theorem and involves integrals and Dirac $\delta -$distributions in its Lie brackets. The second is based on the identification of a Hilbert basis of $L^2\left(G_{nc}\right)$ and involves only sums and Kronecker symbols in its Lie brackets. Since we are only considering Schwartz functions throughout, both presentations are isomorphic.

4.2. A KM Algebra Associated to SL$(2,\mathbb{R})$

In this section, we give some details about the construction of the KM algebra associated to $\mathrm{SL}(2,\mathbb{R})$. At the end of the section, we briefly comment on the construction associated to $\mathrm{SL}(2,\mathbb{R})/\mathrm{U}\left(1\right)$.

4.2.1. The Group SL$(2,\mathbb{R})$

The group $\mathrm{SL}(2,\mathbb{R})\cong \mathrm{SU}(1,1)$ is defined by the set of $2\times 2$ complex matrices

$$\begin{array}{ccc}\hfill \mathrm{SL}(2,\mathbb{R})& =& \left\{U=\left(\begin{array}{cc}{z}_1& {\overline{z}}_2\\ z_2& {\overline{z}}_1\end{array}\right),z_1,z_2\in \mathbb{C}:|z_1{|}^2-{\left|z_2\right|}^2=1\right\}\hfill \\ & \cong & \left\{z_1,z_2\in {\mathbb{C}}^2:|z_1{|}^2-{\left|z_2\right|}^2=1\right\}\equiv {\mathbb{H}}_{2,2},\hfill \end{array}$$

(79)

where ${\mathbb{H}}_{2,2}$ is the hyperboloid, which can be parameterized as follows:

$$z_1=cosh\rho e^{i{\phi }_1},z_2=sinh\rho e^{i{\phi }_2},\rho \ge 0,0\le {\phi }_1,{\phi }_2<2\pi .$$

From this parameterization, we obtain the left-invariant one-forms (see (36))

$$\begin{array}{ccc}\hfill {\lambda }_0& =& {cosh}^2\rho \mathrm{d}{\phi }_1-{sinh}^2\rho \mathrm{d}{\phi }_2\hfill \\ \hfill {\lambda }_1& =& sin({\phi }_1+{\phi }_2)\mathrm{d}\rho -sinh\rho cosh\rho cos({\phi }_1+{\phi }_2)(\mathrm{d}{\phi }_1-\mathrm{d}{\phi }_2)\hfill \\ \hfill {\lambda }_2& =& -cos({\phi }_1+{\phi }_2)\mathrm{d}\rho -sinh\rho cosh\rho sin({\phi }_1+{\phi }_2)(\mathrm{d}{\phi }_1-\mathrm{d}{\phi }_2).\hfill \end{array}$$

Thus, the metric tensor is

$$\begin{array}{c}\hfill \mathrm{d}{s}^2={\lambda }_0^2-{\lambda }_2^2-{\lambda }_3^2=-\mathrm{d}{\rho }^2+{cosh}^2\rho \mathrm{d}{\phi }_1^2-{sinh}^2\rho \mathrm{d}{\phi }^2.\end{array}$$

The generators of the left/right actions of $\mathfrak{sl}(2,\mathbb{R})$, obtained by solving Killing Equation (1), are

$$\begin{array}{c}\hfill \begin{array}{ccccccc}{L}_{\pm }\hfill & =\hfill & {\displaystyle \frac{1}{2}}{e}^{i({\phi }_1\mp {\phi }_2)}& \left[itanh\rho {\partial }_1\pm {\partial }_{\rho }-icoth\rho {\partial }_2\right],& L_0\hfill & =\hfill & {\displaystyle \frac{i}{2}}({\partial }_2-{\partial }_1),\hfill \\ R_{\pm }\hfill & =\hfill & {\displaystyle \frac{1}{2}}{e}^{\pm i({\phi }_1+{\phi }_2)}& \left[-itanh\rho {\partial }_1\mp {\partial }_{\rho }-icoth\rho {\partial }_2\right],& R_0\hfill & =\hfill & -{\displaystyle \frac{i}{2}}({\partial }_2+{\partial }_1),\hfill \end{array}\end{array}$$

and satisfy the commutation relations

$$\begin{array}{c}\hfill \begin{array}{cccccc}\left[L_0,L_{\pm }\right]\hfill & =\hfill & \pm L_{\pm },\hfill & \left[L_{+},L_{-}\right]\hfill & =\hfill & -2L_0,\hfill \\ \left[R_0,R_{\pm }\right]\hfill & =\hfill & \pm R_{\pm },\hfill & \left[R_{+},R_{-}\right]\hfill & =\hfill & -2R_0,\hfill \end{array}\left[L_a,R_b\right]=0.\end{array}$$

The Casimir operator is given by

$$\begin{array}{ccc}\hfill C_2& =& {\displaystyle \frac{1}{2}}coth\left(2\rho \right){\partial }_{\rho }+{\displaystyle \frac{1}{4}}{\partial }_{\rho }^2-{\displaystyle \frac{1-{tanh}^2\rho }{4}}{\partial }_1^2+{\displaystyle \frac{{coth}^2\rho -1}{4}}{\partial }_2^2.\hfill \end{array}$$

and the scalar product on ${\mathbb{H}}_{2,2}$ reduces to

$$(f,g)={\displaystyle \frac{1}{4{\pi }^2}}\underset{{\mathbb{H}}_{2,2}}{\int }cosh\rho sinh\rho \mathrm{d}\rho \mathrm{d}{\phi }_1\mathrm{d}{\phi }_2\overline{f(\rho ,{\phi }_1,{\phi }_2)}g(\rho ,{\phi }_1,{\phi }_2).$$

(80)

We recall that unitary representations ${\mathcal{D}}_{\Lambda }=\left\{|\Lambda ,n\right.⟩,n\in I_{\Lambda }\}$ of $\mathfrak{sl}(2,\mathbb{R})$ were classified by Bargmann [18] (see also [45]):

$$\begin{array}{ccc}\hfill L_0|\Lambda ,n>& =& n|\Lambda ,n>,\hfill \\ \hfill L_{+}|\Lambda ,n>& =& \vartheta \sqrt{(n+\Lambda )(n-\Lambda +1)}|\Lambda ,n+1>,\hfill \\ \hfill L_{-}|\Lambda ,n>& =& \vartheta \sqrt{(n-\Lambda )(n+\Lambda -1)}|\Lambda ,n-1>,\hfill \\ \hfill C_2|\Lambda ,n>& =& \Lambda (\Lambda +1)|L,n>,\hfill \end{array}$$

(81)

with the sign $\vartheta$ being conveniently chosen for each representation (see Proposition 1). Unitarity restricts $\Lambda$ and $I_{\Lambda }$:

Proposition 1. 

Unitary representations are as follows:

1. 

${\mathcal{D}}_{\lambda }^{+}$: Discrete series bounded from below $\Lambda =\lambda ,\lambda \in \mathbb{N}\setminus \left\{0\right\}$ or ${\displaystyle \frac{1}{2}}+\mathbb{N}$ and $I_{\lambda }=\{n\ge \lambda \}$;

2. 

${\mathcal{D}}_{\lambda }^{-}$: Discrete series bounded from above $\Lambda =\lambda ,\lambda \in \mathbb{N}\setminus \left\{0\right\}$ or ${\displaystyle \frac{1}{2}}+\mathbb{N}$ and $I_{\lambda }=\{n\le -\lambda \}$;

3. 

${\mathcal{C}}_{i\sigma }^{ϵ}$: Principal continuous series $\Lambda ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{2}}\sigma ,\sigma >0$ and $I_{i\sigma }=\mathbb{Z}$ or ${\displaystyle \frac{1}{2}}+\mathbb{Z}$ ($n\in {\displaystyle \frac{1}{2}}+\mathbb{Z}$ or $n\in \mathbb{Z}$);

4. 

${\mathcal{C}}_{\sigma }$: Supplementary continuous series $\Lambda ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{\sigma }{2}},0<{\sigma }^2<1$ and $I_{\sigma }=\mathbb{Z}$ ($n\in \mathbb{Z}$).

The sign ϑ can be taken to equal 1; however, conveniently, we take $\vartheta =1$ for the discrete series bounded from below, as well as for the (principal and supplementary) continuous series, and we take $\vartheta =-1$ for the discrete series bounded from above.

The discrete series ${\mathcal{D}}_{\lambda }^{+}$ is bounded from below, whereas the discrete series ${\mathcal{D}}_{\lambda }^{-}$ is bounded from above. The continuous series ${\mathcal{C}}_{i\sigma }^{ϵ}$ and ${\mathcal{C}}_{\sigma }$ are unbounded. The eigenvalue of $C_2$ is discrete for the two discrete series, and it is continuous for the two continuous series. Note that $\mathfrak{sl}(2,\mathbb{R})$ admits discrete series because $\mathfrak{u}\left(1\right)\subset \mathfrak{sl}(2,\mathbb{R})$ and rk $\mathfrak{u}\left(1\right)=$ rk $\mathfrak{sl}(2,\mathbb{R})$.

4.2.2. Matrix Elements of $\mathfrak{sl}(2,\mathbb{R})$ and the Plancherel Theorem

The Plancherel theorem only involves the two discrete series and the principal continuous series. Since the supplementary continuous series plays no role (see [51,69,70]), we hereafter only consider the discrete and continuous principal series. The corresponding matrix elements are denoted as ${\psi }_{m\overline{\Lambda }n}$ or, more specifically, for the discrete series, ${\psi }_{m\lambda n}^{\eta }$, where $\overline{\Lambda }=(\lambda ,\eta )$, $\eta =\pm ,\lambda >1/2,\eta m,\eta n\ge \lambda$ for the discrete series bounded from below ($\eta =1$) and above ($\eta =-1$) (the unitarity of $\mathfrak{sl}(2,\mathbb{R})-$representations implies $\lambda >0$, but in order to have normalizable matrix elements, we now have $\lambda >1/2$ [18,45]), and for the continuous principal series, ${\psi }_{mi\sigma n}^{ϵ}$, where $\overline{\Lambda }=({\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{2}}\sigma ,ϵ)$, $\sigma >0,ϵ=0,1/2,n,m\in \mathbb{Z}+ϵ$, for the continuous principal bosonic (resp., fermionic) $ϵ=0$ (resp. $ϵ=1/2$). These matrix elements are obtained by solving the differential equations

$$\begin{array}{ccc}\hfill L_0{\psi }_{m\Lambda n}(\rho ,{\phi }_1,{\phi }_2)& =& m{\psi }_{m\Lambda n}(\rho ,{\phi }_1,{\phi }_2)\hfill \\ \hfill R_0{\psi }_{m\Lambda n}(\rho ,{\phi }_1,{\phi }_2)& =& n{\psi }_{m\Lambda n}(\rho ,{\phi }_1,{\phi }_2)\hfill \\ \hfill C_2{\psi }_{m\Lambda n}(\rho ,{\phi }_1,{\phi }_2)& =& \Lambda (\Lambda +1){\psi }_{m\Lambda n}(\rho ,{\phi }_1,{\phi }_2).\hfill \end{array}$$

We can unify all matrix elements in the following form (for $m\ge n$):

$$\begin{array}{ccc}\hfill {\psi }_{n\overline{\Lambda }m}(\rho ,{\phi }_1,{\phi }_2)& =& {\displaystyle \frac{\mathcal{N}}{(m-n)!}}\sqrt{{\displaystyle \frac{\Gamma (\vartheta m+1-\Lambda )\Gamma (\vartheta m+\Lambda )}{\Gamma (\vartheta n+1-\Lambda )\Gamma (\vartheta n+\Lambda )}}}{e}^{i(m+n){\phi }_1+i(m-n){\phi }_2}\hfill \\ & & {cosh}^{-\vartheta (m+n)}\rho {sinh}^{\vartheta (m-n)}\rho {}_{2}F_1(-\vartheta n+\Lambda ,-\vartheta n-\lambda +1;1+\vartheta (m-n);-{sinh}^2\rho )\hfill \end{array}$$

where $\overline{\Lambda }=(\Lambda =\lambda ,+),(\Lambda =\lambda ,-)$ with $\lambda >1/2$ for the discrete series, and $\overline{\Lambda }=(\Lambda ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{2}}\sigma ,0),(\Lambda ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{2}}\sigma ,{\displaystyle \frac{1}{2}})$ with $\sigma >0$ for the continuous series. We have defined $\vartheta =1$ for the discrete series bounded from below and the continuous series, as well as $\vartheta =-1$ for the discrete series bounded from above, and $\mathcal{N}=\sqrt{2(2\lambda -1)}$ (resp., $\mathcal{N}=1$) for the discrete version (resp. continuous series). The matrix elements for $n\ge m$ are obtained by using the substitution $m\leftrightarrow n$ everywhere, except in the exponential factor, which is unaffected, and are multiplied by an overall factor ${(-1)}^{n-m}$. Recall that ${}_{2}F_1$ is a hypergeometric function (see [45] for a precise definition). The matrix elements of discrete series are, thus, expressed in terms of hypergeometric polynomials, whereas the matrix elements of continuous series are expressed in terms of hypergeometric functions [45]. They are normalized such that

$$\begin{array}{c}\hfill \begin{array}{cccc}\hfill ({\psi }_{m_1{\lambda }_1{n}_1}^{{\eta }_1},{\psi }_{m_2{\lambda }_2{n}_2}^{{\eta }_2})& =\hfill & {\delta }^{{\eta }_1{\eta }_2}{\delta }_{{\lambda }_1{\lambda }_2}{\delta }_{m_1{m}_2}{\delta }_{n_1{n}_2}\hfill & \mathrm{discrete}\mathrm{series};\hfill \\ \hfill {\psi }_{mi\sigma n}^{ϵ}\left(0\right)& =\hfill & {\delta }_{mn}\hfill & \mathrm{principal}\mathrm{continuous}\mathrm{series},\hfill \end{array}\end{array}$$

(82)

and satisfy (81) for the left and right actions. As stated previously, the matrix elements of the discrete series are normalizable (see (82)) but satisfy

$$\begin{array}{c}\hfill {\psi }_{m\lambda n}^{\eta }\left(0\right)=\sqrt{2(2\lambda -1)}{\delta }_{mn},\end{array}$$

(83)

whereas the matrix elements of the continuous principal series are not normalizable and satisfy

$$\begin{array}{c}\hfill ({\psi }_{ni\sigma m}^{ϵ},{\psi }_{n^{\prime }i{\sigma }^{\prime }{m}^{\prime }}^{{ϵ}^{\prime }})={\displaystyle \frac{1}{\sigma tanh\pi (\sigma +iϵ)}}{\delta }_{ϵ{ϵ}^{\prime }}{\delta }_{m{m}^{\prime }}{\delta }_{n{n}^{\prime }}\delta (\sigma -{\sigma }^{\prime }),\end{array}$$

(84)

(see [45]). Finally, we have

$$\begin{array}{ccc}\hfill \overline{{\psi }_{\eta }^{m\lambda n}}(\rho ,{\phi }_1,{\phi }_2)& =& {\psi }_{-n\lambda -m}^{-\eta }(\rho ,{\phi }_1,{\phi }_2),\hfill \\ \hfill \overline{{\psi }_{ϵ}^{mi\sigma n}}(\rho ,{\phi }_1,{\phi }_2)& =& {\psi }_{-mi\sigma -n}^{ϵ}(\rho ,{\phi }_1,{\phi }_2).\hfill \end{array}$$

(85)

Let $\mathcal{S}$ be the set of Schwartz (or rapidly decreasing) functions in the $\rho -$direction, and let ${\mathcal{S}}^{\prime }$ be its dual (see (78)). The asymptotic (when $\rho \to +\infty$) expansion of the matrix elements of the discrete series was studied in [18,45], and it turns out that the matrix elements ${\psi }_{m\lambda n}^{\eta }$ belong to $\mathcal{S}$. Moreover, the matrix elements of the continuous principal series ${\psi }_{mi\sigma n}^{ϵ}$ belong to ${\mathcal{S}}^{\prime }$; see, for instance, [67], Theorem 1, p. 426, and (34)–(37) p. 429 or [68] Chapter 8. Then, considering a function $f\in \mathcal{S}$, the Plancherel theorem enables us to expand f as follows (see [69,70], pp. 336–337):

$$\begin{array}{ccc}\hfill f(\rho ,{\phi }_1,{\phi }_2)& =& \sum _{\lambda >{\displaystyle \frac{1}{2}}}\sum _{m,n\ge \lambda }{f}_{+}^{n\lambda m}{\psi }_{n\lambda m}^{+}(\rho ,{\phi }_1,{\phi }_2)+\sum _{\lambda >{\displaystyle \frac{1}{2}}}\sum _{m,n\le -\lambda }{f}_{-}^{n\lambda m}{\psi }_{n\lambda m}^{-}(\rho ,{\phi }_1,{\phi }_2)\hfill \\ & & +\underset{0}{\overset{+\infty }{\int }}\mathrm{d}\sigma \sigma tanh\pi \sigma \sum _{m,n\in \mathbb{Z}}{f}_0^{nm}\left(\sigma \right){\psi }_{ni\sigma m}^0(\rho ,{\phi }_1,{\phi }_2)\hfill \\ & & +\underset{0}{\overset{+\infty }{\int }}\mathrm{d}\sigma \sigma coth\pi \sigma \sum _{m,n\in \mathbb{Z}+{\displaystyle \frac{1}{2}}}{f}_{{\displaystyle \frac{1}{2}}}^{nm}\left(\sigma \right){\psi }_{ni\sigma m}^{{\displaystyle \frac{1}{2}}}(\rho ,{\phi }_1,{\phi }_2)\hfill \end{array}$$

(86)

i.e., as a sum over the matrix elements of the discrete series and an integral over the matrix elements of the principal continuous series. This is the Plancherel theorem for $\mathrm{SL}(2,\mathbb{R})$. The components of f are given by the scalar products (80)

$$\begin{array}{ccc}\hfill f_{\pm }^{n\lambda m}& =& ({\psi }_{n\lambda m}^{\pm },f)\hfill \\ \hfill f_{nm}^{ϵ}\left(\sigma \right)& =& ({\psi }_{ni\sigma m}^{ϵ},f).\hfill \end{array}$$

(87)

Introducing the symbol $\sum \int$ to denote the summation over all discrete and continuous series, we can rewrite (86) as

$$\begin{array}{c}\hfill f(\rho ,{\phi }_1,{\phi }_2)=\sum _{\Lambda ,m,n}\int f^{n\Lambda m}{\psi }_{n\Lambda ,m}(\rho ,{\phi }_1,{\phi }_2)\end{array}$$

(88)

with $\Lambda =(+,\lambda ),(-,\lambda ),(0,i\sigma ),(1/2,i\sigma )$.

4.2.3. Hilbert Basis of $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$

The Plancherel theorem gives rise to an expansion of Schwartz functions as a sum over the matrix elements of the discrete series and an integral over the matrix elements of the continuous series (see (86)). As any Hilbert space is known to admit a Hilbert basis [49], we would now like to identify a Hilbert basis of $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$. Let $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)=L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}\oplus L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$, where $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}$ constitutes the set of square integrable functions expanded within the discrete series (first line of (86)). By definition, the set of matrix elements of a discrete series is a Hilbert basis of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}$. In other words, since the matrix elements of the continuous series are not normalizable, they do not constitute a Hilbert basis of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$. V. Losert identified a Hilbert basis for $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$ (see [45]). Let $W_{mn}$ be the eigenspaces of the operators $L_0$ and $R_0$

$$\begin{array}{c}\hfill W_{nm}=\left\{F\in L^2\left(\mathrm{SL}(2,\mathbb{R})\right),F(\rho ,{\phi }_1,{\phi }_2)=e^{i(m+n){\phi }_1+i(m-n){\phi }_2}f\left(\rho \right)\right\}.\end{array}$$

For $F\in W_{nm}$, we get

$$\begin{array}{ccc}\hfill L_{0}F(\rho ,{\phi }_1,{\phi }_2)& =& nF(\rho ,{\phi }_1,{\phi }_2),\hfill \\ \hfill R_{0}F(\rho ,{\phi }_1,{\phi }_2)& =& mF(\rho ,{\phi }_1,{\phi }_2).\hfill \end{array}$$

(89)

The main idea of V. Losert is to identify a Hilbert basis of $W_{nm}$ for each n and m. Let

$$\begin{array}{c}\hfill {\mathcal{B}}_{nm}=\left\{{\Phi }_{nnk}(\rho ,{\phi }_1,{\phi }_2)=e^{i(m+n){\phi }_1+i(m-n){\phi }_2}{e}_{nmk}\left(\cosh 2\rho \right),k\in \mathbb{N}\right\}\end{array}$$

be a Hilbert basis of $W_{nm}$. Observing that for the discrete series bounded from below (resp., above), we have $nm>0,n,m\ge \lambda >1/2$ (resp., $mn>0,n,m\le -\lambda <-1/2$) (see Proposition 1), three cases have to be considered for $W_{nm}$ [45]:

• $mn>0,m,n>1/2$: ${\mathcal{B}}_{nm}$ contains matrix elements of the discrete series bounded from below, together with elements of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$.
• $mn>0,m,n<-1/2$: ${\mathcal{B}}_{nm}$ contains matrix elements of the discrete series bounded from above, together with elements of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$.
• $mn<0$ or $m=0,1/2$ or $n=0,1/2$: ${\mathcal{B}}_{nm}$ contains only elements of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$.

Let $ϵ=0$ if $m,n\in \mathbb{Z}$ and $ϵ=1/2$ if $m,n\in \mathbb{Z}+{\displaystyle \frac{1}{2}}$, and define the condition

$$\begin{array}{c}\hfill \mathcal{C}:mn>0,\left|m\right|,\left|n\right|>1/2\end{array}$$

(90)

which ensures that $W_{mn}\cap L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}\ne \varnothing$. Then, set

$$\begin{array}{c}\hfill k_{\min }=\left\{\begin{array}{cccc}\min \left(\right|m|,|n\left|\right)-ϵ& \mathrm{if}\hfill & \mathcal{C}\hfill & \mathrm{is}\mathrm{satisfied}\hfill \\ 0& \mathrm{if}\hfill & \mathcal{C}\hfill & \mathrm{is}\mathrm{not}\mathrm{satisfied}\hfill \end{array}\right.\end{array}$$

(91)

such that if

$$\begin{array}{ccc}\hfill k<k_{\min }& \Rightarrow & {\Phi }_{nmk}\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}\hfill \\ \hfill k\ge k_{\min }& \Rightarrow & {\Phi }_{nmk}\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }},\hfill \end{array}$$

(92)

this shows that $W_{mn}\cap L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}=\varnothing$ when condition (90) does not hold. Considering $x=cosh2\rho$, for $n\ge m$ [45], we have

$$\begin{array}{ccc}\hfill e_{nmk}\left(x\right)& =& \sqrt{2^{2m-1}{\displaystyle \frac{(2m-2k-1)k!(m+n-k-1)!}{(2m-k-1)!(n-m+k)!}}}\times \hfill \\ & & {(x-1)}^{{\displaystyle \frac{n-m}{2}}}{(x+1)}^{-{\displaystyle \frac{m+n}{2}}}{P}_k^{(n-m,-n-m)}\left(x\right)\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}},\hfill \\ & & 0\le k<k_{\min },\hfill \\ \hfill e_{nmk}\left(x\right)& =& \sqrt{2^{2k+2ϵ+1}{\displaystyle \frac{(2k+n-m+2ϵ+1)(k+n-m+2ϵ)!k!}{(k+2ϵ)!(n-m+k)!}}}\times \hfill \\ & & {(x-1)}^{{\displaystyle \frac{n-m}{2}}}{(x+1)}^{{\displaystyle \frac{m-n}{2}}-k-ϵ-1}{P}_k^{(n-m,m-n-2k-2ϵ-1)}\left(x\right)\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }},\hfill \\ & & k\ge k_{\min },\hfill \end{array}$$

(93)

where $P_k^{(a,b)}$ are Jacobi polynomials (the matrix elements of the discrete series are expressed in terms of hypergeometric polynomials. However, one may show that these hypergeometric polynomials can be easily related to Jacobi polynomials [45]). Recall that $deg{P}_k^{(a,b)}=k$. Thus, $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}$ involves Jacobi polynomials of degree $<k_{\min }$, and $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$ involves Jacobi polynomials of degree $\ge k_{\min }$ [45]. We have similar expressions for $n\ge m$ and the conjugation relations:

$$\begin{array}{c}\hfill \overline{{\Phi }^{nmk}}(\rho ,{\phi }_1,{\phi }_2)={\Phi }_{-n-mk}(\rho ,{\phi }_1,{\phi }_2).\end{array}$$

(94)

The set ${\cup }_{n,m\in \mathbb{Z}}{\mathcal{B}}_{nm}{\cup }_{n,m\in \mathbb{Z}+1/2}{\mathcal{B}}_{nm}$ is a complete orthonormal Hilbert basis of $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$ called the Losert basis:

$$\begin{array}{c}\hfill ({\Phi }_{nmk},{\Phi }_{n^{\prime }{m}^{\prime }{k}^{\prime }})={\delta }_{n{n}^{\prime }}{\delta }_{m{m}^{\prime }}{\delta }_{k{k}^{\prime }}.\end{array}$$

(95)

Therefore, if $f\in L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$, we have

$$\begin{array}{ccc}\hfill f(\rho ,{\phi }_1,{\phi }_2)& =& \sum _{ϵ=0,1/2}\sum _{n,m\in \mathbb{Z}+ϵ}\sum _{k=0}^{+\infty }{f}^{nmk}{\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2),\hfill \\ \hfill f_{nmk}& =& ({\Phi }_{mnk},f).\hfill \end{array}$$

(96)

Since the $\mathfrak{sl}(2,\mathbb{R})$ generators act on $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$, and since $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\mathrm{d}}\subset L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$ is a sub-representation, i.e., an invariant subspace of $L^2\left(\mathrm{SL}(2,\mathbb{R})\right)$, it follows that $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$ is also a representation of $\mathrm{SL}(2,\mathbb{R})$. The action of $L_{\pm },R_{\pm }$ on ${\Phi }_{nmk}\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cccc}{L}_{+}\left({\Phi }_{nmk}\right)\hfill & =\hfill & {\alpha }_{nmk}^{+L}{\Phi }_{n+1mk+1}+{\beta }_{nmk}^{+L}{\Phi }_{n+1mk}+{\gamma }_{nmk}^{+L}{\Phi }_{n+1nk-1}\hfill & \left\{\begin{array}{cc}{\gamma }_{nmk}^{+L}=0\hfill & m>n\hfill \\ {\alpha }_{nmk}^{+L}=0\hfill & n\ge m\hfill \end{array}\right.\hfill \\ L_{-}\left({\Phi }_{nmk}\right)\hfill & =\hfill & {\alpha }_{nmk}^{-L}{\Phi }_{n-1mk+1}+{\beta }_{nmk}^{-L}{\Phi }_{n-1mk}+{\gamma }_{nmk}^{-L}{\Phi }_{n-1mk-1}\hfill & \left\{\begin{array}{cc}{\alpha }_{nmk}^{-L}=0\hfill & m\ge n\hfill \\ {\gamma }_{nmk}^{+L}=0\hfill & n>m\hfill \end{array}\right.\hfill \\ R_{+}\left({\Phi }_{nmk}\right)\hfill & =\hfill & {\alpha }_{nmk}^{+R}{\Phi }_{nm+1k+1}+{\beta }_{nmk}^{+R}{\Phi }_{nm+1k}+{\gamma }_{nmk}^{+R}{\Phi }_{nm+1k-1}\hfill & \left\{\begin{array}{cc}{\alpha }_{nmk}^{+L}=0\hfill & m\ge n\hfill \\ {\gamma }_{nmk}^{+L}=0\hfill & n>m\hfill \end{array}\right.\hfill \\ R_{-}\left({\Phi }_{nmk}\right)\hfill & =\hfill & {\alpha }_{nmk}^{-R}{\Phi }_{nm-1k+1}+{\beta }_{nmk}^{-R}{\Phi }_{nm-1k}+{\gamma }_{nmk}^{-R}{\Phi }_{nm-1k-1}\hfill & \left\{\begin{array}{cc}{\gamma }_{nmk}^{-L}=0\hfill & m>n\hfill \\ {\alpha }_{nmk}^{+L}=0\hfill & n\ge m\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(97)

The action of the Casimir operator reduces to

$$\begin{array}{c}\hfill C_2{\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2)=a_{nmk}{\Phi }_{nmk-1}(\rho ,{\phi }_1,{\phi }_2)+b_{nmk}{\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2)+c_{nmk}{\Phi }_{nmk+1}(\rho ,{\phi }_1,{\phi }_2).\end{array}$$

This action clearly shows that this representation is unitary but is not irreducible. Explicit expressions of the coefficients $\alpha ,\beta$ and $a,b,c$ are given in [45].

The asymptotic behavior of the functions $e_{nmk}\in W_{nm}$ was studied in [45], and it turns out that the functions ${\Phi }_{nmk}$ are actually Schwartz functions. Thus, for ${\Phi }_{nmk}\in W_{nm}\cap L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$, one can use the Plancherel theorem and write (86)

$$\begin{array}{c}\hfill {\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2)=\underset{0}{\overset{+\infty }{\int }}\mathrm{d}\sigma \sigma tanh\pi (\sigma +iϵ)f_{nmk}\left(\sigma \right){\psi }_{ni\sigma ϵm}\left(\rho \right),\end{array}$$

(98)

where $ϵ=0$ if $n,m$ are integers, and $ϵ=1/2$ if $n,m$ are half-integers. It should be observed that there is no summation on n and m. Furthermore, using (87), we have (with the scalar product (80))

$$\begin{array}{c}\hfill f^{nmk}\left(\sigma \right)=({\psi }_{ni\sigma ϵm},e_{nmk}).\end{array}$$

This implies that relation (98) can be inverted

$$\begin{array}{c}\hfill {\psi }_{ni\sigma ϵm}(\rho ,{\phi }_1,{\phi }_2)=\sum _{k\ge k_{\min }}\overline{f^{mnk}}\left(\sigma \right){\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2),\end{array}$$

(99)

with $k_{\min }$ defined in (92). Thus, we are able to express the matrix elements of the (principal) continuous series in terms of the Losert basis. This observation is important for the construction of the algebra below.

4.2.4. Clebsch–Gordan Coefficients

The next step in the construction of a KM algebra associated to SL$(2,\mathbb{R})$ is the computation of the Clebsch–Gordan coefficients corresponding to the decomposition ${\mathcal{D}}_{\Lambda }\otimes {\mathcal{D}}_{{\Lambda }^{\prime }}$. This was studied in [71,72]. The coupling of two discrete series was studied by means of a bosonic realization of the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$; for the case in which at least one continuous series is involved, the result was obtained by a cumbersome analytic continuation. A heuristic decomposition can be deduced from a back-and-forth process. More precisely, let ${\mathcal{D}}_{\Lambda },{\mathcal{D}}_{{\Lambda }^{\prime }}$ be two representations of $\mathrm{SL}(2,\mathbb{R})$ in the Plancherel basis; expand the matrix elements of ${\mathcal{D}}_{\Lambda }$ and ${\mathcal{D}}_{{\Lambda }^{\prime }}$ in the Losert basis using (99); subsequently, decompose the product ${\Psi }_{m\Lambda n}{\Psi }_{m^{\prime }{\Lambda }^{\prime }{n}^{\prime }}$ in the Losert basis, and then express the results back in the Plancherel basis using (98). For instance, consider two discrete series bounded from above: ${\mathcal{D}}_{\lambda }^{+}\otimes {\mathcal{D}}_{{\lambda }^{\prime }}^{+}$. The product of two matrix elements ${\psi }_{n\lambda m}^{+}{\psi }_{n^{\prime }{\lambda }^{\prime }{m}^{\prime }}^{+}$ is such that $nm>0,n^{\prime }{m}^{\prime }>0,n,m,n^{\prime },m^{\prime }>1/2$ (see condition (90)). So we have $(n+n^{\prime })(m+m^{\prime })>0,(n+n^{\prime }),(m+m^{\prime })>1/2$. Thus, the product ${\mathcal{D}}_{\lambda }^{+}\otimes {\mathcal{D}}_{{\lambda }^{\prime }}^{+}$ decomposes only in the discrete series bounded from above. The situation is very different for the product ${\mathcal{D}}_{{\lambda }_{+}}^{+}\otimes {\mathcal{D}}_{{\lambda }_{-}}^{-}$, which involves both matrix elements of the discrete series and continuous series. Indeed, the product of two matrix elements ${\psi }_{m_{+}{\lambda }_{+}{n}_{+}}^{+}{\psi }_{m_{-}{\lambda }_{-}{n}_{-}}^{-}$ is such that $m_{\pm }{n}_{\pm }>0$ and $\pm m_{\pm },\pm n_{\pm }\ge {\lambda }_{\pm }$. So, it may happen that $(m_{+}+m_{-})(n_{+}+n_{-})<0$ (depending on the respective value of $m_{\pm }$ and $n_{\pm }$), and thus, ${\psi }_{m_{+}{\lambda }_{+}{n}_{+}}^{+}{\psi }_{m_{-}{\lambda }_{-}{n}_{-}}^{-}$ belongs to $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$. So, in this case, according to (98), ${\psi }_{m_{+}{\lambda }_{+}{n}_{+}}^{+}{\psi }_{m_{-}{\lambda }_{-}{n}_{-}}^{-}$ decomposes as an integral over the matrix elements of continuous representations.

Thus, we generically write

$$\begin{array}{ccc}\hfill {\psi }_{m_1{\overline{\Lambda }}_1{m}_1^{\prime }}(\rho ,{\phi }_1,{\phi }_2){\psi }_{m_2{\overline{\Lambda }}_2{m}_2^{\prime }}(\rho ,{\phi }_1,{\phi }_2)& =& \sum _{\overline{\Lambda }}\int {C_{{\overline{\Lambda }}_1,{\overline{\Lambda }}_2}^{\overline{\Lambda }}}_{m_1,m_2,m_1^{\prime },m_2^{\prime }}{\psi }_{m_1+m_2\overline{\Lambda }{m}_1^{\prime }+m_2^{\prime }}(\rho ,{\phi }_1,{\phi }_2)\hfill \end{array}$$

(100)

with the notations of (88), and where ${\overline{\Lambda }}_1,{\overline{\Lambda }}_2=(\lambda ,+),(\lambda ,-),(i\sigma ,0)$ or $(i\sigma ,1/2)$ and $\overline{\Lambda }$ takes one of the allowed values occurring in the tensor product decomposition (see [45,71,72] for a case-by-case treatment with the same notations). The coefficients are given by the corresponding product of Clebsch–Gordan coefficients (see [45,46]).

Similarly, if we proceed with the Losert basis, as the product $e_{nmk}\left(x\right)e_{n^{\prime }{m}^{\prime }{k}^{\prime }}\left(x\right)$ is square integrable (even better, it is a Schwartz function) [45], we have

$$\begin{array}{c}\hfill {\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2){\Phi }_{n^{\prime }{m}^{\prime }{k}^{\prime }}(\rho ,{\phi }_1,{\phi }_2)=C_{k{k}^{\prime }n{n}^{\prime }m{m}^{\prime }}^{k^{″}}{\Phi }_{n+n^{\prime }m+m^{\prime }{k}^{″}}(\rho ,{\phi }_1,{\phi }_2).\end{array}$$

(101)

From now on, we call the Plancherel basis the set of matrix elements of the discrete and principal continuous series, i.e., the $\psi$s and the Losert basis of the $\Phi$s.

4.2.5. A KM Algebra Associated to $\mathfrak{sl}(2,\mathbb{R})$

The KM algebra associated to SL$(2,\mathbb{R})$ is obtained in an analogous way to the construction of the KM algebra associated to a compact Lie group $G_c$. It is a central extension of the algebra $\mathfrak{g}\left(\mathrm{SL}\right(2,\mathbb{R}\left)\right)$. In the Plancherel basis (PB), we have

$$\begin{array}{ccc}\hfill \mathfrak{g}\left(\mathrm{SL}\right(2,\mathbb{R}\left)\right)& =& \{T^a{\psi }_{n\lambda m}^{+}(\rho ,{\phi }_1,{\phi }_2),T^a{\psi }_{n\lambda m}^{-}(\rho ,{\phi }_1,{\phi }_2),\lambda >{\displaystyle \frac{1}{2}},mn>0,\left|m\right|,\left|n\right|>\lambda ,\hfill \\ & & T^a{\psi }_{ni\sigma m}^{ϵ}(\rho ,{\phi }_1,{\phi }_2),ϵ=0,{\displaystyle \frac{1}{2}},\sigma >0,m,n\in \mathbb{Z}+ϵ,a=1,\dots ,dim\mathfrak{g}\}\hfill \\ & =& \left\{T_{m\overline{\Lambda }n}^a=T^a{\psi }_{m\overline{\Lambda }n},\overline{\Lambda }=(\lambda ,+),(\lambda ,-),(i\sigma ,0),(i\sigma ,{\displaystyle \frac{1}{2}}),a=1,\dots ,dim\mathfrak{g}\right\}\hfill \end{array}$$

and in the Losert basis (LB), we get

$$\begin{array}{ccc}\hfill \mathfrak{g}\left(\mathrm{SL}\right(2,\mathbb{R}\left)\right)& =& \left\{T_{mnk}^a=T^a{\varphi }_{mnk}(\rho ,{\phi }_1,{\phi }_2),m,n\in \mathbb{Z}+ϵ,k\in \mathbb{N},ϵ=0,{\displaystyle \frac{1}{2}}\right\}.\hfill \end{array}$$

The Lie brackets take the form

$$\begin{array}{c}\hfill \begin{array}{cccc}\left[T_{m\overline{\Lambda }n}^a,T_{m^{\prime }{\overline{\Lambda }}^{\prime }{n}^{\prime }}^{a^{\prime }}\right]\hfill & =\hfill & i{f^{a{a}^{\prime }}}_{a^{″}}{\displaystyle \sum _{{\overline{\Lambda }}^{″}}}\int {C_{\overline{\Lambda },{\overline{\Lambda }}^{\prime }}^{{\overline{\Lambda }}^{″}}}_{m,m^{\prime },n,n^{\prime }}{T}_{m+m^{\prime }{\overline{\Lambda }}^{″}n+n^{\prime }}^{a^{″}}\hfill & \left(\mathrm{PB}\right)\hfill \\ \left[T_{mnk}^a,T_{m^{\prime }{n}^{\prime }{k}^{\prime }}^{a^{\prime }}\right]\hfill & =\hfill & i{f^{a{a}^{\prime }}}_{a^{″}}{\displaystyle \sum _{k^{″}}}{C_{k{k}^{\prime }}^{k^{″}}}_{m,m^{\prime },n,n^{\prime }}{T}_{m+m^{\prime }n+n^{\prime }{k}^{″}}^{a^{″}}\hfill & \left(\mathrm{LB}\right)\hfill \end{array}\end{array}$$

(102)

according to (100) and (101). Of course, since we can express the $T_{mnk}^a$ in terms of the $T_{m\overline{\Lambda }n}^a$ and vice versa, because of (99) and (98), the two presentations of the algebra are isomorphic. In (102), we explicitly write the symbol ∑ in the LB (i.e., we do not use the Einstein summation convention) in order to emphasize the different presentation of the algebra in the PB and in the LB bases. Indeed, in the former basis, the algebra involves an integral, while in the latter, it involves the sum.

The next step in the construction is to introduce Hermitian operators and compatible two-cocycles. The construction is similar to the SU$\left(2\right)$ case. We first introduce the two commuting operators $L_0,R_0$. In the Losert basis, the action of the commuting Hermitian operators is given in (89), and in the Plancherel basis, the action of Hermitian operators is given in (81) with the corresponding value of $\Lambda$. We then associate to $L_0,R_0$ compatible two-cocycles ${\omega }_L,{\omega }_R$:

$$\begin{array}{ccc}\hfill {\omega }_L(X,Y)& =& -{\displaystyle \frac{k_L}{4{\pi }^2}}\underset{{\mathbb{H}}_{2,2}}{\int }\mathrm{d}\rho sinh\rho cosh\rho \mathrm{d}{\phi }_1\mathrm{d}{\phi }_2{⟨X,L_{0}Y⟩}_0\hfill \\ \hfill {\omega }_R(X,Y)& =& -{\displaystyle \frac{k_R}{4{\pi }^2}}\underset{{\mathbb{H}}_{2,2}}{\int }\mathrm{d}\rho sinh\rho cosh\rho \mathrm{d}{\phi }_1\mathrm{d}{\phi }_2{⟨X,R_{0}Y⟩}_0.\hfill \end{array}$$

Thus, using the orthogonality conditions (82) and (84) (resp., (95)) and the conjugation property (85) (resp., (94)) in the Plancherel (resp., the Losert) basis, we obtain

$$\begin{array}{ccc}\hfill {\omega }_L(T_{an\lambda \eta m},T_{a^{\prime }{n}^{\prime }{\lambda }^{\prime }{\eta }^{\prime }{m}^{\prime }})& =& n{k}_L{k}_{a{a}^{\prime }}{\delta }_{\lambda ,{\lambda }^{\prime }}{\delta }_{\eta ,-{\eta }^{\prime }}{\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }}\hfill \\ \hfill {\omega }_L(T_{ani\sigma ϵm},T_{a^{\prime }{n}^{\prime }i{\sigma }^{\prime }{ϵ}^{\prime }{m}^{\prime }})& =& n{k}_L{k}_{a{a}^{\prime }}{\displaystyle \frac{\delta (\sigma -{\sigma }^{\prime })}{\sigma tanh\pi (\sigma +iϵ)}}{\delta }_{ϵ,{ϵ}^{\prime }}{\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }}\hfill \end{array}$$

(103)

in the PB basis, and

$$\begin{array}{ccc}\hfill {\omega }_L{(}_{anmk},T_{a^{\prime }{n}^{\prime }{m}^{\prime }{k}^{\prime }})& =& n{k}_L{k}_{a{a}^{\prime }}{\delta }_{k{k}^{\prime }}{\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }}\hfill \end{array}$$

(104)

in the LB basis. Defining

$$\begin{array}{c}\hfill \delta (\overline{\Lambda },{\overline{\Lambda }}^{\prime })=\left\{\begin{array}{cc}{\delta }_{\lambda ,{\lambda }^{\prime }}{\delta }_{\eta ,-{\eta }^{\prime }}& \overline{\Lambda }=(\lambda ,\eta ),{\overline{\Lambda }}^{\prime }=({\lambda }^{\prime },{\eta }^{\prime })\\ {\displaystyle \frac{\delta (\sigma -{\sigma }^{\prime })}{\sigma tanh\pi (\sigma +iϵ)}}{\delta }_{ϵ,{ϵ}^{\prime }}& \overline{\Lambda }=(i\sigma ,ϵ),{\overline{\Lambda }}^{\prime }=(i{\sigma }^{\prime },{ϵ}^{\prime })\\ 0& \mathrm{elsewhere}\end{array}\right.\end{array}$$

we obtain

$$\begin{array}{ccc}\hfill {\omega }_L(T_{an\overline{\Lambda }m},T_{a^{\prime }{n}^{\prime }{\overline{\Lambda }}^{\prime }{m}^{\prime }})& =& n{k}_L{k}_{a{a}^{\prime }}\delta (\overline{\Lambda },{\overline{\Lambda }}^{\prime }){\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }}\hfill \end{array}$$

in the PB basis. We have a similar expression for ${\omega }_R$.

With the same notations as before, we define the KM algebra associated to SL$(2,\mathbb{R})$:

$$\begin{array}{c}\hfill \tilde{\mathfrak{g}}\left(\mathrm{SL}(2,\mathbb{R})\right)=\left\{\begin{array}{cc}\left\{{\mathcal{T}}_{m\overline{\Lambda }n}^a,L_0,R_0,k_L,k_L\right\}\hfill & \left(\mathrm{PB}\right)\hfill \\ \left\{{\mathcal{T}}_{mnk}^a,L_0,R_0,k_L,k_L\right\}\hfill & \left(\mathrm{LB}\right)\hfill \end{array}\right.\end{array}$$

(105)

From (102), (103), and (82) and (84), the Lie brackets are

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{am\overline{\Lambda }n},{\mathcal{T}}_{a^{\prime }{m}^{\prime }{\overline{\Lambda }}^{\prime }{n}^{\prime }}\right]& =& i{f_{a{a}^{\prime }}}^{a^{″}}{\displaystyle \sum _{{\overline{\Lambda }}^{″}}}\int {C_{\overline{\Lambda },{\overline{\Lambda }}^{\prime }}^{{\overline{\Lambda }}^{″}}}_{m,m^{\prime },n,n^{\prime }}{\mathcal{T}}_{a^{″}m+m^{\prime }{\overline{\Lambda }}^{″}n+n^{\prime }}+(m{k}_L+n{k}_R)h_{a{a}^{\prime }}\delta (\overline{\Lambda },{\overline{\Lambda }}^{\prime }){\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }},\hfill \\ \hfill \left[L_0,{\mathcal{T}}_{am\overline{\Lambda }n}\right]& =& m{\mathcal{T}}_{am\overline{\Lambda }n},\hfill \\ \hfill \left[R_0,{\mathcal{T}}_{am\overline{\Lambda }n}\right]& =& n{\mathcal{T}}_{am\overline{\Lambda }n},\hfill \end{array}$$

(106)

in the Plancherel basis, and from (102), (103), and (84), the Lie brackets are

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{amnk},{\mathcal{T}}_{a^{\prime }{m}^{\prime }{n}^{\prime }{k}^{\prime }}\right]& =& i{f_{a{a}^{\prime }}}^{a^{″}}\sum _{k^{″}}{C_{k{k}^{\prime }}^{k^{″}}}_{m,m^{\prime },n,n^{\prime }}{\mathcal{T}}_{a^{″}m+m^{\prime }n+n^{\prime }{k}^{″}}+(m{k}_L+n{k}_R)h_{a{a}^{\prime }}{\delta }_{k{k}^{\prime }}{\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }},\hfill \\ \hfill \left[L_0,{\mathcal{T}}_{amnk}\right]& =& m{\mathcal{T}}_{amnk},\hfill \\ \hfill \left[R_0,{\mathcal{T}}_{amnk}\right]& =& n{\mathcal{T}}_{amnk},\hfill \end{array}$$

(107)

in the Losert basis. This algebra was obtained in [45].

Since we can expand the Losert basis in the Plancherel basis (98) and the Plancherel basis in the Losert basis (99), the two presentations of the algebra $\tilde{\mathfrak{g}}\left(\mathrm{SL}(2,\mathbb{R})\right)$ (106) and (107) are equivalent. Each presentation has its own advantages. In the former case, elements of $\tilde{\mathfrak{g}}\left(\mathrm{SL}(2,\mathbb{R})\right)$ are expressed by means of the unitary irreducible representations of $\mathfrak{sl}(2,\mathbb{R})$, but the Lie brackets involve integrals and Dirac $\delta -$distributions as a consequence of the continuous basis. In the latter case, the Lie brackets involve only sums and Kronecker symbols, but the elements of $\tilde{\mathfrak{g}}\left(\mathrm{SL}(2,\mathbb{R})\right)$ are expressed by means of a reducible representation of $\mathfrak{sl}(2,\mathbb{R})$ associated to the Hilbert basis of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$.

This algebra is very different from its analog of the KM algebra of compact Lie groups. Indeed, for a compact Lie group, we have $\mathfrak{g}\subset \tilde{\mathfrak{g}}\left(G_c\right)$ because the trivial representation is a unitary representation of $G_c$. However, in the non-compact case $G_{nc}$, $\mathfrak{g}$ is not included in $\tilde{\mathfrak{g}}\left(G_{nc}\right)$ because the trivial representation is non-normalizable on the $\mathrm{SL}(2,\mathbb{R})-$manifold.

We conclude this section by mentioning that one can associate a Virasoro algebra to $\mathrm{SL}(2,\mathbb{R})$. This construction follows the same lines as the corresponding construction in Section 3.5. We introduce

$$\begin{array}{c}\hfill \begin{array}{cccc}{\mathsf{\ell }}_{m\overline{\Lambda }n}\hfill & =\hfill & -{\psi }_{m\overline{\Lambda }n}(\theta ,{\phi }_1,{\phi }_2)L_0\hfill & \left(\mathrm{PB}\right)\hfill \\ {\mathsf{\ell }}_{mnk}\hfill & =\hfill & -{\Phi }_{mnk}(\theta ,{\phi }_1,{\phi }_2)L_0\hfill & \left(\mathrm{LB}\right)\hfill \end{array}\end{array}$$

which are compatible with ${\omega }_L$. The commutation relations read

$$\begin{array}{c}\hfill \begin{array}{cccc}\left[{\mathsf{\ell }}_{m\overline{\Lambda }n},{\mathsf{\ell }}_{m^{\prime }{\overline{\Lambda }}^{\prime }{n}^{\prime }}\right]\hfill & =\hfill & (m-m^{\prime }){\displaystyle \sum _{{\overline{\Lambda }}^{″}}}\int {C_{\overline{\Lambda },{\overline{\Lambda }}^{\prime }}^{{\overline{\Lambda }}^{″}}}_{m,m^{\prime },n,n^{\prime }}{\mathsf{\ell }}_{m+m^{\prime }{\overline{\Lambda }}^{″}n+n^{\prime }}\hfill & \left(\mathrm{PB}\right)\hfill \\ \left[{\mathsf{\ell }}_{mnk},{\mathsf{\ell }}_{m^{\prime }{n}^{\prime }{k}^{\prime }}\right]\hfill & =\hfill & (m-m^{\prime }){\displaystyle \sum _{k^{″}}}{C_{k{k}^{\prime }}^{k^{″}}}_{m,m^{\prime },n,n^{\prime }}{\mathsf{\ell }}_{m+m^{\prime }n+n^{\prime }{k}^{″}}\hfill & \left(\mathrm{LB}\right)\hfill \end{array}\end{array}$$

This algebra admits a central extension, exactly along the lines of Section 3.4, and with the same notations, we introduce the generators:

$$\begin{array}{c}\hfill \mathrm{Vir}\left(\mathrm{SL}(2,\mathbb{R})\right)=\left\{\begin{array}{cc}\left\{L_{m\overline{\Lambda }n},c\right\}\hfill & \left(\mathrm{PB}\right)\hfill \\ \\ \left\{L_{mnk},c\right\}\hfill & \left(\mathrm{LB}\right)\hfill \end{array}\right.\end{array}$$

with Lie brackets in the PB basis

$$\begin{array}{ccc}\hfill \left[L_{m\overline{\Lambda }n},L_{m^{\prime }{\overline{\Lambda }}^{\prime }{n}^{\prime }}\right]& =& (m-m^{\prime })\sum _{{\overline{\Lambda }}^{″}}\int {C_{\overline{\Lambda },{\overline{\Lambda }}^{\prime }}^{{\overline{\Lambda }}^{″}}}_{m,m^{\prime },n,n^{\prime }}{L}_{m+m^{\prime }{\overline{\Lambda }}^{″}n+n^{\prime }}+{\displaystyle \frac{c}{12}}(m^3-m){\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }}\delta (\overline{\Lambda },{\overline{\Lambda }}^{\prime })\hfill \end{array}$$

(108)

and in the LB basis

$$\begin{array}{ccc}\hfill \left[L_{mnk},{\mathsf{\ell }}_{m^{\prime }{n}^{\prime }{k}^{\prime }}\right]& =& (m-m^{\prime })\sum _{k^{″}}{C_{k{k}^{\prime }}^{k^{″}}}_{m,m^{\prime },n,n^{\prime }}{L}_{m+m^{\prime }n+n^{\prime }{k}^{″}}+{\displaystyle \frac{c}{12}}(m^3-m){\delta }_{m,-m^{\prime }}{\delta }_{n,-n^{\prime }}{\delta }_{k{k}^{\prime }}.\hfill \end{array}$$

(109)

Thus, ${\tilde{\mathfrak{g}}}^{\prime }\left(\mathrm{SL}(2,\mathbb{R})\right)⋊\mathrm{Vir}\left(\mathrm{SL}(2,\mathbb{R})\right)$ (similarly to SU$\left(2\right)$, we only introduce one central charge, say $k_L$) has a semidirect structure with the action of the Virasoro algebra on the KM algebra, with the remaining part being

$$\begin{array}{c}\hfill \begin{array}{cccc}\left[L_{m\overline{\Lambda }n},{\mathcal{T}}_{m^{\prime }{\overline{\Lambda }}^{\prime }{n}^{\prime }}^a\right]\hfill & =\hfill & -m^{\prime }{\displaystyle \sum _{{\overline{\Lambda }}^{″}}}\int {C_{\overline{\Lambda },{\overline{\Lambda }}^{\prime }}^{{\overline{\Lambda }}^{″}}}_{m,m^{\prime },n,n^{\prime }}{\mathcal{T}}_{m+m^{\prime }{\overline{\Lambda }}^{″}n+n^{\prime }}^a\hfill & \left(\mathrm{PB}\right)\hfill \\ \left[L_{mnk},{\mathcal{T}}_{m^{\prime }{n}^{\prime }{k}^{\prime }}^a\right]\hfill & =\hfill & -m^{\prime }{\displaystyle \sum _{k^{″}}}{C_{k{k}^{\prime }}^{k^{″}}}_{m,m^{\prime },n,n^{\prime }}{\mathcal{T}}_{m+m^{\prime }n+n^{\prime }{k}^{″}}^a\hfill & \left(\mathrm{LB}\right)\hfill \end{array}.\end{array}$$

(110)

The algebra ${\tilde{\mathfrak{g}}}^{\prime }\left(\mathrm{SL}(2,\mathbb{R})\right)⋊\mathrm{Vir}\left(\mathrm{SL}(2,\mathbb{R})\right)$ is, thus, defined by (106)/(107) (with the central charge being $k_R=0$), (108)/(109), and (110). Again, the Virasoro algebra of $\mathrm{SL}(2,\mathbb{R})$ is very different from the Virasoro algebra of SU$\left(2\right)$. Recall that for the Virasoro algebra of SU$\left(2\right)$, we have $L_0=-L_{000}$ (see (77)), but here, as the trivial representation is not normalizable, we cannot relate $L_0$ to a specific element of Vir$\left(\mathrm{SL}\right(2,\mathbb{R}\left)\right)$. However, one can extend the Virasoro algebra to ${\mathrm{Vir}}^{\prime }\left(\mathrm{SL}(2,\mathbb{R})\right)=$ Vir$\left(\mathrm{SL}(2,\mathbb{R})\right)\times \left\{L_0\right\}$, where the action of $L_0$ is given by

$$\begin{array}{c}\hfill \left[L_0,L_{nsm}\right]=n{L}_{nsm}.\end{array}$$

We finish this section by mentioning that a KM algebra on $\mathrm{SL}(2,\mathbb{R})/U\left(1\right)$ can be easily deduced from the KM algebra on $\mathrm{SL}(2,\mathbb{R})$. The only representations that appear in the harmonic analysis on $\mathrm{SL}(2,\mathbb{R})/U\left(1\right)$ are the representations that are chargeless with respect to $R_0$. The only representations that survive this condition in the PB basis are the matrix elements of the bosonic principal continuous series ${\psi }_{ni\sigma 0}^0$, and in the LB basis, the elements of $W_{m0}:{\Phi }_{m0k}$ (LB). More details and the explicit brackets can be found in [45].

5. Soft Manifolds

This section is devoted to the construction of KM algebras on soft group manifolds; that is, on group manifolds with a soft deformation.

KM Algebras on Soft Manifolds

In Section 3 and Section 4, we consider KM algebras associated to (compact and non-compact) group manifolds. These manifolds have a large isometry group (associated to the left and right actions of the group itself). Furthermore, for these manifolds, the Vielbein satisfies the Maurer–Cartan equation (37) or is a left invariant one-form. Moreover, for these manifolds, the metric tensor is naturally deduced from the Vielbein (38).

Let ${\mathrm{G}}_m$ be a group (compact or non-compact); we now consider a smooth, ‘softening’ deformation $G_m^{\mu }$ of the Lie group $G_m$, locally diffeomorphic to $G_m$ itself (see, e.g., [15] and the references therein). We further assume that the manifold $G_m^{\mu }$ has the same parameterization $m^M$ (see Section 3 for a ${\mathrm{G}}_m=G_c$ compact manifold, and see Section 4 for a ${\mathrm{G}}_m=G_{nc}$ non-compact manifold), the only difference between $G_m$ and $G_m^{\mu }$ being at the level of the metric tensor. Thus, we assume that the Vielbein $\mu$ is an intrinsic one-form (valued in the Lie algebra ${\mathfrak{g}}_m$ of $G_m$)

$${\mu }^A\left(m\right)={{\mu }_M}^A\left(m\right)\mathrm{d}{m}^M$$

i.e., it is not a Maurer–Cartan one-form (it does not satisfy Equation (37)):

$$\mathrm{d}\mu +\mu \wedge \mu =R,$$

(111)

where R is the curvature two-form of $\mu$. In other words, $\mu$ is not left-invariant (i.e., it is a ‘soft’, intrinsic one-form). It is in this sense that we consider that the ‘soft’ group manifold $G_m^{\mu }$ is a deformation of $G_m$. The metric tensor is now defined by

$$g_{MN}^{\mu }\left(m\right)={{\mu }_M}^A\left(m\right){{\mu }_N}^B\left(m\right){\eta }_{AB},$$

(112)

where ${\eta }_{AB}$ is the Killing form of the Lie algebra ${\mathfrak{g}}_m$ (positive definite for compact manifolds and indefinite for non-compact manifolds). Taking the exterior derivative of both sides of Equation (111), one obtains the Bianchi identity for the curvature of $\mu$,

$$dR+2R\wedge \mu =0\iff \nabla R=0,$$

where the covariant derivative operator ∇ on $G_m^{\mu }$ has been introduced. For instance, if $G_m=\mathrm{ISO}(1,d-1)/\mathrm{SO}(1,d-1)$, where SO$(1,d-1)$ (resp., ISO$(1,d-1)$) corresponds to the Lorentz (resp., Poincaré) transformations in $D-$dimensions, $G_m$ is the $D-$dimensional Minkowski space-time of Special Relativity, and its deformation $G_m^{\mu }$ is the $D-$dimensional Riemann space-time of General Relativity [12].

The fact that the metric tensors differ for the manifold $G_m$ (see (38) or its analog for a non-compact manifold) and its deformation $G_m^{\mu }$ (see (112)) is not the only difference between $G_m$ and $G_m^{\mu }$. Indeed, as seen previously, the manifold $G_m$ has a large isometry group, namely ${\left(G_m\right)}_L\times {\left(G_m\right)}_R$, but in general, the isometry group of $G_m^{\mu }$ is reduced with respect to its undeformed analog. Stated differently, solving Killing Equation (1) leads to less invariant vector fields. Consequently, if we follow the construction of KM algebras in Section 2.1, we will obtain a less rich structure. For this reason, we associate a KM algebra to $G_m$ following a different strategy. We endow $G_m^{\mu }$ with the scalar product $G_m^{\mu }$

$${(f,g)}_{\mu }={\displaystyle \frac{1}{C}}{\int }_{G_m}\sqrt{g^{\mu }}{\mathrm{d}}^{n}m\overline{f\left(m\right)}g\left(m\right),$$

(113)

where $g^{\mu }=|det\left(g_{MN}^{\mu }\right)|$, and n is the dimension of $G_m$. The coefficient C depends on the manifold. For a compact manifold $G_c$, we take $C=V$, the volume of $G_c$. For a non-compact manifold, see (80) for $G_m=\mathrm{SL}(2,\mathbb{R})$. Notice that, since the parameterization of $G_m^{\mu }$ and $G_m$ is the same, the limits of integration are, again, $G_m$ in this case. As a final hypothesis concerning the deformed manifold, we assume that $G_m^{\mu }$ is non-singular and well defined at any point of $G_m^{\mu }$ (see (115) below).

Let $L^2\left(G_m\right)$ be the set of square integrable functions on $G_m$. We would like to identify a Hilbert basis of $L^2\left(G_m^{\mu }\right)$ from a Hilbert basis of $L^2\left(G_m\right)$ (for $G_m=G_{nc}$ a non-compact manifold, the Plancherel theorem involves normalizable and non-normalizable functions (see Section 4). However, the technique presented here extends to non-normalizable functions. For $G_m=G_{nc}$, we will consider two explicit examples, with an obvious generalization to any $G_{nc}$). The results presented here are due to Mackey [73] (see also [45]). Let $\mathcal{M}$ be a manifold (not necessarily a group manifold), and let $L^2\left(\mathcal{M}\right)$ be the set of square integrable functions defined on $\mathcal{M}$. Assume that the manifold $\mathcal{M}$ is endowed with two different scalar products with measures $\mathrm{d}\alpha$ and $\mathrm{d}\beta$, respectively:

$$\begin{array}{c}\hfill \begin{array}{cc}(\mathcal{M},\mathrm{d}\alpha ):\hfill & {(f,g)}_{\alpha }={\int }_{\mathcal{M}}\mathrm{d}\alpha \overline{f\left(m\right)}g\left(m\right),\hfill \\ (\mathcal{M},\mathrm{d}\beta ):\hfill & {(f,g)}_{\beta }={\int }_{\mathcal{M}}\mathrm{d}\beta \overline{f\left(m\right)}g\left(m\right).\hfill \end{array}\end{array}$$

We assume further that there exists a mapping $T_{\beta \alpha }$

$$\begin{array}{c}\hfill T_{\beta \alpha }:L^2(\mathcal{M},\mathrm{d}\beta )\to L^2(\mathcal{M},\mathrm{d}\alpha ),\end{array}$$

such that

$$\begin{array}{c}\hfill {\int }_{\mathcal{M}}\mathrm{d}\alpha ={\int }_{\mathcal{M}}\mathrm{d}\beta T_{\beta \alpha }.\end{array}$$

For an n-dimensional Riemannian manifold $\mathcal{M}$ parameterized by $m_1,\dots ,m_n$ with metric $g_{\alpha }$ (resp., $g_{\beta }$), we have $\mathrm{d}\alpha =\sqrt{\left|\det g_{\alpha }\right|}{\mathrm{d}}^{n}m$ (resp., $\mathrm{d}\beta =\sqrt{\left|\det g_{\beta }\right|}{\mathrm{d}}^{n}m$), and thus, $T_{\beta \alpha }=\sqrt{\left|\det g_{\alpha }\right|/\left|\det g_{\beta }\right|}$. This means that if $\{f_i^{\beta },i\in \mathbb{N}\}$ is a Hilbert basis of $L^2(\mathcal{M},\mathrm{d}\beta )$, then $\{f_i^{\alpha }={\displaystyle \frac{f_i^{\beta }}{\sqrt{T_{\beta \alpha }}}},i\in \mathbb{N}\}$ is a Hilbert basis for $L^2(\mathcal{M},\mathrm{d}\alpha )$, and we obviously have

$$\begin{array}{c}\hfill {(f_i^{\beta },f_j^{\beta })}_{\beta }={\delta }_{ij}⟺{(f_i^{\alpha },f_j^{\alpha })}_{\alpha }={\delta }_{ij}\end{array}$$

(114)

and the map $T_{\beta \alpha }$ is unitary.

Returning to our softly deformed manifold, let $g_{MN}$ be the metric of $G_m$, let $g_{MN}^{\mu }$ be the metric on $G_m^{\mu }$, and let $g=|det(g_{MN}\left)\right|$, $g^{\mu }=|det\left(g_{MN}^{\mu }\right)|$. Let $\mathcal{B}=\{{\rho }_I\left(m\right),I\in \mathcal{I}\}$ be a Hilbert basis of $L^2\left(G_m\right)$; according to (114),

$$\begin{array}{c}\hfill {\mathcal{B}}_{\mu }=\left\{{\rho }_I^{\mu }\left(m\right)=\sqrt{T^{\mu }}{\rho }_I\left(m\right),I\in \mathcal{I}\right\},\end{array}$$

where $T=\sqrt{{\displaystyle \frac{g}{g^{\mu }}}}$ is a Hilbert basis of $L^2\left(G_m^{\mu }\right)$, and we have

$$\begin{array}{c}\hfill {({\rho }_I^{\mu },{\rho }_J^{\mu })}_{\mu }=({\rho }_I,{\rho }_J)={\delta }_{IJ},\end{array}$$

where, in the second equality, we have used the scalar product over the manifold $G_m$. We assume that the transition function is non-degenerate, namely that

$$\begin{array}{c}\hfill t_{\min }\le T^{\mu }\left(m\right)\le t_{\max },\forall m\in G_m^{\mu }\end{array}$$

(115)

with $t_{\min },t_{\max }\in {\mathbb{R}}_{+}\setminus \left\{0\right\}$.

This construction can be adapted easily for non-normalizable functions. As an illustration, consider the Minkowski space-time in D dimensions. By the Fourier theorem, the set of plane waves

$$\begin{array}{c}\hfill \mathcal{B}=\left\{{\Phi }_{\mathbf{p}}\left(\mathbf{x}\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{D/2}}}{e}^{i\mathbf{p}·\mathbf{x}},\mathbf{p}\in {\mathbb{R}}^{1,D-1}\right\}\end{array}$$

(116)

where $\mathbf{p}·\mathbf{x}$ is the usual Minkowski scalar product, enables the expansion of any Schwartz function. Indeed, in this case, by using Fourier transformation, we have

$$\begin{array}{ccc}\hfill \Psi \left(\mathbf{x}\right)& =& \underset{{\mathbb{R}}^{1,D-1}}{\int }\mathrm{d}\mathbf{p}\tilde{\Psi }\left(\mathbf{p}\right){\Phi }_{\mathbf{p}}\left(\mathbf{x}\right),\hfill \\ \hfill \tilde{\Psi }\left(\mathbf{p}\right)& =& ({\Phi }_{\mathbf{p}},\Psi )=\underset{{\mathbb{R}}^{1,D-1}}{\int }\mathrm{d}\mathbf{x}\overline{{\Phi }_{\mathbf{p}}\left(\mathbf{x}\right)}\Psi \left(\mathbf{x}\right),\hfill \end{array}$$

and

$$\begin{array}{c}\hfill ({\Phi }_{\mathbf{p}},{\Phi }_{\mathbf{q}})=\underset{{\mathbb{R}}^{1,D-1}}{\int }\mathrm{d}\mathbf{p}\overline{{\Phi }_{\mathbf{p}}\left(\mathbf{x}\right)}{\Phi }_{\mathbf{p}}\left(\mathbf{x}\right)={\delta }^D(\mathbf{p}-\mathbf{q}).\end{array}$$

The procedure of Mackey can be extended easily in this case and

$$\begin{array}{c}\hfill {\mathcal{B}}_{\mu }=\left\{{\Phi }_{\mathbf{p}}^{\mu }\left(\mathbf{x}\right)={\displaystyle \frac{1}{{\left(g^{\mu }\right)}^{1/4}}}{\Phi }_{\mathbf{p}}\left(\mathbf{x}\right),\mathbf{p}\in {\mathbb{R}}^{1,D-1}\right\}\end{array}$$

with

$$\begin{array}{c}\hfill {({\Phi }_{\mathbf{p}}^{\mu },{\Phi }_{\mathbf{q}}^{\mu })}_{\mu }=({\Phi }_{\mathbf{p}},{\Phi }_{\mathbf{q}})={\delta }^D(\mathbf{p}-\mathbf{q}).\end{array}$$

Finally, the closure relation of the wave functions ${\Phi }_{\overrightarrow{p}}$ implies

$$\begin{array}{c}\hfill \int \mathrm{d}\mathbf{p}\overline{{\Phi }_{\mathbf{p}}^{\mu }\left(\mathbf{x}\right)}{\Phi }_{\mathbf{p}}^{\mu }\left(\mathbf{y}\right)={\displaystyle \frac{{\delta }^D(\mathbf{x}-\mathbf{y})}{{\sqrt{g}}^{\mu }}},\end{array}$$

thus, the set ${\mathcal{B}}_{\mu }$ is complete, and these elements enable the expansion of any Schwartz function of the Riemannian space-time:

$$\begin{array}{c}\hfill \Psi \left(\mathbf{x}\right)=\underset{{\mathbb{R}}^{1,D-1}}{\int }\mathrm{d}\mathbf{p}\tilde{\Psi }\left(\mathbf{p}\right){\Phi }_{\mathbf{p}}^{\mu }\left(\mathbf{x}\right),\tilde{\Psi }\left(\mathbf{p}\right)={({\Phi }_{\mathbf{p}}^{\mu },\Psi )}_{\mu }.\end{array}$$

Note, however, that the functions ${\Phi }_{\mathbf{p}}^{\mu }$ in ${\mathcal{B}}_{\mu }$ are defined by means of the scalar product in the Minkowski space-time and not the scalar product in the Riemann space-time. As a second example, we now consider a deformation of $\mathrm{SL}(2,\mathbb{R})$. In this case, the Mackey procedure leads to

$$\begin{array}{ccc}\hfill {\psi }_{n\lambda m}^{\eta }(\rho ,{\phi }_1,{\phi }_2)& \to & {\psi }_{n\lambda m}^{\mu \eta }(\rho ,{\phi }_1,{\phi }_2)={\left({\displaystyle \frac{cosh\rho sinh\rho }{g^{\mu }}}\right)}^{1/4}{\psi }_{n\lambda m}^{\eta }(\rho ,{\phi }_1,{\phi }_2)\hfill \\ \hfill {\psi }_{ni\sigma m}^{ϵ}(\rho ,{\phi }_1,{\phi }_2)& \to & {\psi }_{ni\sigma m}^{\mu ϵ}(\rho ,{\phi }_1,{\phi }_2)={\left({\displaystyle \frac{cosh\rho sinh\rho }{g^{\mu }}}\right)}^{1/4}{\psi }_{ni\sigma m}^{ϵ}(\rho ,{\phi }_1,{\phi }_2),\hfill \end{array}$$

(117)

for, respectively, the matrix elements of the discrete and principal continuous series. These matrix elements satisfy the first equation in (82) and Equation (84), with the scalar product (113), instead of the scalar product (80). In a similar manner, as seen for the previous example, the set of functions in (117) is a complete set. Of course, only the former functions are normalized, while the latter functions are not normalizable. Because of hypothesis (115), even the functions ${\psi }_{n\lambda m}^{\mu \eta }$ are Schwartzian. This set of functions enables us to have a ‘Plancherel’ decomposition on $G_{nc}^{\mu }$. Actually, if f is a Schwartz function, we have

$$\begin{array}{cc}& f(\rho ,{\phi }_1,{\phi }_2)=\sum _{\Lambda ,m,n}\int f^{n\Lambda m}{\psi }_{n\Lambda ,m}^{\mu }(\rho ,{\phi }_1,{\phi }_2)\\ & f_{\pm }^{n\lambda m}={({\psi }_{n\lambda m}^{\mu \pm },f)}_{\mu },f_{nm}^{ϵ}\left(\sigma \right)={({\psi }_{ni\sigma m}^{\mu ϵ},f)}_{\mu }.\end{array}$$

with the notations of (88) and (87).

Furthermore, if $G_m=G_{nc}$ is non-compact, it is always possible to identify a Hilbert basis, i.e., a complete set of orthonormal functions. The Mackey procedure follows easily in this case. Again, for $G_{nc}=\mathrm{SL}(2,\mathbb{R})$, this leads to

$$\begin{array}{c}\hfill {\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2)\to {\Phi }_{nmk}^{\mu }(\rho ,{\phi }_1,{\phi }_2)={\left({\displaystyle \frac{cosh\rho sinh\rho }{g^{\mu }}}\right)}^{1/4}{\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2),\end{array}$$

(118)

which form a Hilbert basis of $L^2\left(\mathrm{SL}{(2,\mathbb{R})}^{\mu }\right)$ and are also Schwartzian.

Following the notations of Section 3.1 (similar results hold for the label states of non-compact manifolds, so we take the same notations in both cases) and, in particular, the results due to Racah [57], we denote ${\Psi }_{LQR}$ (see (41), for $G_m=G_c$) the set of matrix elements, which allows us to expand any Schwartz functions on $G_m$, and define

$$\begin{array}{c}\hfill {\mathcal{B}}_{\mu }=\left\{{\Psi }_{LQR}^{\mu }\left(m\right)=\sqrt{T^{\mu }}{\Psi }_{LQR}\left(m\right),\left(LQR\right)\in \mathcal{I}\right\},\end{array}$$

(119)

the corresponding set in $G_m^{\mu }$ through the Mackey procedure.

The next step in our construction is to decompose the product ${\Psi }_{LQR}^{\mu }\left(m\right){\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }\left(m\right)$ in the basis (119). We first analyze the case where $G_m=G_c$ is a compact Lie group. As the metric tensor of the soft manifold satisfies (115), this, in particular, means that any function ${\Psi }_{LQR}^{\mu }$ can be expanded in the Hilbert basis of $L^2\left(G_c\right)$ (41). Conversely, any function ${\Psi }_{LQR}$ can be expanded in the Hilbert basis of $L^2\left(G_c^{\mu }\right)$:

$$\begin{array}{c}\hfill {\Psi }_{LQR}^{\mu }\left(m\right)={P_{LQR}}^{L^{\prime }{Q}^{\prime }{R}^{\prime }}{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}\left(m\right),{\Psi }_{LQR}\left(m\right)={{\left(P^{-1}\right)}_{LQR}}^{L^{\prime }{Q}^{\prime }{R}^{\prime }}{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }\left(m\right).\end{array}$$

(120)

Therefore, the product ${\Psi }_{L_1{Q}_1{R}_1}^{\mu }\left(m\right){\Psi }_{L_2{Q}_2{R}_2}^{\mu }\left(m\right)$ decomposes as

$$\begin{array}{c}\hfill {\Psi }_{L_1{Q}_1{R}_1}^{\mu }\left(m\right){\Psi }_{L_2{Q}_2{R}_2}^{\mu }\left(m\right)={C^{\mu }}_{L_1{Q}_1{R}_1;L_2{Q}_2{R}_2}^{L_3{Q}_3{R}_3}{\Psi }_{L_3{Q}_3{R}_3}^{\mu }\left(m\right),\end{array}$$

(121)

where

$$\begin{array}{c}\hfill {C^{\mu }}_{L_1{Q}_1{R}_1;L_2{Q}_2{R}_2}^{L_3{Q}_3{R}_3}={P_{L_1{Q}_1{R}_1}}^{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime }}{P_{L_2{Q}_2{R}_2}}^{L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}{{\left(P^{-1}\right)}_{L_3^{\prime }{Q}_3^{\prime }{R}_3^{\prime }}}^{L_3{Q}_3{R}_3}{C}_{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime };L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}^{L_3^{\prime }{Q}_3^{\prime }{R}_3^{\prime }}.\end{array}$$

(122)

If $G_{nc}$ is a non-compact manifold, this procedure extends naturally to any Hilbert basis, such as (117) for $\mathrm{SL}(2,\mathbb{R})$. In [47], we proceeded in a different but equivalent way to obtain the decomposition of the product ${\Psi }_{L_1{Q}_1{R}_1}^{\mu }\left(m\right){\Psi }_{L_2{Q}_2{R}_2}^{\mu }\left(m\right)$. However, the proof given in [47] does not extend to the case where $G_m=G_{nc}$ is a non-compact manifold. We now consider non-compact manifolds, and we consider $\mathrm{SL}{(2,\mathbb{R})}_{\mu }$ as an illustration. Since the matrix elements ${\psi }_{n\lambda m}^{\eta }$ are Schwartz functions, and the metric of $\mathrm{SL}{(2,\mathbb{R})}_{\mu }$ satisfies (115), the functions ${\psi }_{n\lambda m}^{\mu \eta }$ (see (117)) are also Schwartzian and can, thus, be decomposed in the Hilbert basis of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{\perp }$ (in this analysis, in order to simplify the presentation, we assume that ${\psi }_{n\lambda m}^{\mu \eta }$ decompose in the set of functions ${\psi }_{n\lambda m}^{\eta }$; more general situations can be encountered.) (see Section 4.2.3):

$$\begin{array}{ccc}\hfill {\psi }_{n\lambda m}^{\mu \eta }(\rho ,{\phi }_1,{\phi }_2)& =& {P_{n\lambda m}}^{n^{\prime }{\lambda }^{\prime }{m}^{\prime }}{\psi }_{n^{\prime }{\lambda }^{\prime }{m}^{\prime }}^{\eta }(\rho ,{\phi }_1,{\phi }_2),\hfill \\ \hfill {\psi }_{n\lambda m}^{\eta }(\rho ,{\phi }_1,{\phi }_2)& =& {{\left(P^{-1}\right)}_{n\lambda m}}^{n^{\prime }{\lambda }^{\prime }{m}^{\prime }}{\psi }_{n^{\prime }{\lambda }^{\prime }{m}^{\prime }}^{\mu \eta }(\rho ,{\phi }_1,{\phi }_2).\hfill \end{array}$$

(123)

We now turn to the decomposition of ${\psi }_{ni\sigma m}^{\mu ϵ}$. We recall that, in the case of $\mathrm{SL}(2,\mathbb{R})$, the Plancherel theorem enables us to express the matrix elements of the continuous series in terms of the Losert basis of $L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$ and vice versa [45] (see (98) and (99)). Thus, in the case of $\mathrm{SL}{(2,\mathbb{R})}_{\mu }$, because of (117) and (118), we have exactly the same expansion, and (117) and (118) reduce to

$$\begin{array}{ccc}\hfill {\Phi }_{nmk}^{\mu }(\rho ,{\phi }_1,{\phi }_2)& =& \underset{0}{\overset{+\infty }{\int }}\mathrm{d}\sigma \sigma tanh\pi (\sigma +iϵ)f_{nmk}\left(\sigma \right){\psi }_{ni\sigma m}^{\mu ϵ}(\rho ,{\phi }_1,{\phi }_2),k\ge k_{\min }\hfill \\ \hfill {\psi }_{ni\sigma m}^{\mu ϵ}(\rho ,{\phi }_1,{\phi }_2)& =& \sum _{k\ge k_{\min }}\overline{f^{nmk}}\left(\sigma \right){\Phi }_{nmk}^{\mu }(\rho ,{\phi }_1,{\phi }_2).\hfill \end{array}$$

Observe that we have only a sum over k in the second equality and an integral over $\sigma$ in the first equality. We also have (as before, we assume for simplicity that if ${\Phi }_{nmk}\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$, then ${\Phi }_{nmk}^{\mu }\in L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$)

$$\begin{array}{ccc}\hfill {\Phi }_{nmk}^{\mu }(\rho ,{\phi }_1,{\phi }_2)& =& {P_{nmk}}^{n^{\prime }{m}^{\prime }{k}^{\prime }}{\Phi }_{n^{\prime }{m}^{\prime }{k}^{\prime }}(\rho ,{\phi }_1,{\phi }_2),\hfill \\ \hfill {\Phi }_{nmk}(\rho ,{\phi }_1,{\phi }_2)& =& {{\left(P^{-1}\right)}_{nmk}}^{n^{\prime }{m}^{\prime }{k}^{\prime }}{\Phi }_{n^{\prime }{m}^{\prime }{k}^{\prime }}^{\mu }(\rho ,{\phi }_1,{\phi }_2)\hfill \end{array}$$

where, for the sum over $k^{\prime }$, we have $k^{\prime }\ge k_{\min }$, and the sum over $n,m$ is unconstrained (note that if ${\Phi }_{nmk}^{\mu }\notin L^2{\left(\mathrm{SL}(2,\mathbb{R})\right)}^{{\mathrm{d}}^{\perp }}$, we have no restriction on the summation over $k^{\prime }$). Thus,

$$\begin{array}{ccc}\hfill {\psi }_{ni\sigma m}^{\mu ϵ}(\rho ,{\phi }_1,{\phi }_2)& =& \overline{f^{nmk}}\left(\sigma \right){P_{nmk}}^{n^{\prime }{m}^{\prime }{k}^{\prime }}{\Phi }_{n^{\prime }{m}^{\prime }{k}^{\prime }}(\rho ,{\phi }_1,{\phi }_2)\hfill \\ & =& \underset{0}{\overset{+\infty }{\int }}\mathrm{d}{\sigma }^{\prime }{\sigma }^{\prime }tanh\pi ({\sigma }^{\prime }+iϵ)\overline{f^{nmk}}\left(\sigma \right){P_{nmk}}^{n^{\prime }{m}^{\prime }{k}^{\prime }}{f}_{n^{\prime }{m}^{\prime }{k}^{\prime }}\left({\sigma }^{\prime }\right){\psi }_{n^{\prime }i{\sigma }^{\prime }{m}^{\prime }}^{ϵ}(\rho ,{\phi }_1,{\phi }_2)\hfill \\ & \equiv & \int \mathrm{d}{\sigma }^{\prime }{P}_{nm}^{ϵ}(\sigma ,{\sigma }^{\prime }){\psi }_{ni{\sigma }^{\prime }m}^{ϵ}(\rho ,{\phi }_1,{\phi }_2),\hfill \end{array}$$

(124)

where

$$\begin{array}{c}\hfill P_{nm}^{ϵ}(\sigma ,{\sigma }^{\prime })={\sigma }^{\prime }tanh\pi ({\sigma }^{\prime }+iϵ)\overline{f^{nmk}}\left(\sigma \right){P_{nmk}}^{n^{\prime }{m}^{\prime }{k}^{\prime }}{f}_{n^{\prime }{m}^{\prime }{k}^{\prime }}\left({\sigma }^{\prime }\right).\end{array}$$

Note that there is no summation over n and m. This relation can be inverted

$$\begin{array}{ccc}\hfill {\psi }_{ni\sigma m}^{ϵ}(\rho ,{\phi }_1,{\phi }_2)& =& \int \mathrm{d}{\sigma }^{\prime }{\left(P^{-1}\right)}_{nm}^{ϵ}(\sigma ,{\sigma }^{\prime }){\psi }_{ni{\sigma }^{\prime }m}^{ϵ}(\rho ,{\phi }_1,{\phi }_2),\hfill \end{array}$$

(125)

with

$$\begin{array}{c}\hfill {\left(P^{-1}\right)}_{nm}^{ϵ}(\sigma ,{\sigma }^{\prime })={\sigma }^{\prime }tanh\pi ({\sigma }^{\prime }+iϵ)\overline{f^{nmk}}\left(\sigma \right){{\left(P^{-1}\right)}_{nmk}}^{n^{\prime }{m}^{\prime }{k}^{\prime }}{f}_{n^{\prime }{m}^{\prime }{k}^{\prime }}\left({\sigma }^{\prime }\right).\end{array}$$

Then, the product ${\psi }_{n_1{\overline{\Lambda }}_1{m}_1}^{\mu }{\psi }_{n_2{\overline{\Lambda }}_2{m}_2}^{\mu }$ decomposes as

$$\begin{array}{ccc}\hfill {\psi }_{n_1{\overline{\Lambda }}_1{m}_1}^{\mu }(\rho ,{\phi }_1,{\phi }_2){\psi }_{n_2{\overline{\Lambda }}_2{m}_2}^{\mu }(\rho ,{\phi }_1,{\phi }_2)& =& \sum _{\overline{\Lambda }}\int {{C^{\mu }}_{{\overline{\Lambda }}_1,{\overline{\Lambda }}_2}^{\overline{\Lambda }}}_{m_1,m_2,m_1^{\prime },m_2^{\prime }}{\psi }_{m_1+m_2\overline{\Lambda }{m}_1^{\prime }+m_2^{\prime }}^{\mu }(\rho ,{\phi }_1,{\phi }_2)\hfill \end{array}$$

(126)

where the coefficients ${{C^{\mu }}_{{\overline{\Lambda }}_1,{\overline{\Lambda }}_2}^{\overline{\Lambda }}}_{m_1,m_2,m_1^{\prime },m_2^{\prime }}$ can be deduced either from the transformations properties (123), (126), and (125) or from the relations (100) involving the Clebsch–Gordan coefficients of $\mathrm{SL}(2,\mathbb{R})$.

To summarize, for any soft manifold $G_m^{\mu }$, we have

$$\begin{array}{c}\hfill {\Psi }_{LQR}^{\mu }\left(m\right)=\sum _{LQR}\int {P_{LQR}}^{L^{\prime }{Q}^{\prime }{R}^{\prime }}{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }\left(m\right),{\Psi }_{LQR}\left(m\right)=\sum _{LQR}\int {{\left(P^{-1}\right)}_{LQR}}^{L^{\prime }{Q}^{\prime }{R}^{\prime }}{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }\left(m\right)\end{array}$$

(127)

The decomposition of the product ${\Psi }_{LQR}^{\mu }{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }$ follows at once by (127) and (44). The Lie algebra

$$\begin{array}{c}\hfill \mathfrak{g}\left(G_m^{\mu }\right)=\left\{T_{aLQR}^{\mu }\left(m\right)=T_a{\psi }_{LQR}^{\mu }\left(m\right),a=1,\dots ,d,(L,Q,R)\in \mathcal{I}\right\}\end{array}$$

has Lie brackets

$$\begin{array}{c}\hfill \left[T_{a_1{L}_1{Q}_1{R}_1}^{\mu }\left(m\right),T_{a_2{L}_2{Q}_2{R}_2}^{\mu }\left(m\right)\right]=i{f_{a_1{a}_2}}^{a_3}\sum \int {C^{\mu }}_{L_1{Q}_1{R}_1;L_2{Q}_2{R}_2}^{L_3{Q}_3{R}_3}{T}_{a_3{L}_3{Q}_3{R}_3}^{\mu }\left(m\right),\end{array}$$

according to (52) and (127). Note that, due to relation (127), the Lie algebras $\mathfrak{g}\left(G_m^{\mu }\right)$ and $\mathfrak{g}\left(G_m\right)$ are isomorphic.

We now turn our attention to the identification of Hermitian operators. As mentioned previously, the isometry group of the soft manifold $G_m^{\mu }$ is smaller than its undeformed analog. Thus, if we solve Killing Equation (1) for $G_m^{\mu }$, we have fewer Killing vectors than those for $G_m$. In order to have a number of generators larger than the dimension of the isometry group of $G_m^{\mu }$, define [47] (see also [74] for an analogous definition)

$$\begin{array}{c}\hfill L_A^{\mu }=\sqrt{T^{\mu }}{L}_A{\displaystyle \frac{1}{\sqrt{T^{\mu }}}},R_A^{\mu }=\sqrt{T^{\mu }}{R}_A{\displaystyle \frac{1}{\sqrt{T^{\mu }}}}.\end{array}$$

(128)

In general, these generators do not generate the isometry group of $G_m^{\mu }$, but it follows at once that

$$\begin{array}{c}\hfill \left[L_A^{\mu },L_B^{\mu }\right]=i{c_{AB}}^C{L}_C^{\mu },\left[R_A^{\mu },R_B^{\mu }\right]=i{c_{AB}}^C{R}_C^{\mu },\left[L_A^{\mu },R_B^{\mu }\right]=0,\end{array}$$

and

$$\begin{array}{ccc}\hfill L_A^{\mu }{\Psi }_{LQR}^{\mu }\left(m\right)& =& {{\left(M_A^Q\right)}_L}^{L^{\prime }}{\Psi }_{L^{\prime }QR}^{\mu }\left(m\right)\hfill \\ \hfill R_A^{\mu }{\Psi }_{LQR}\left(m\right)& =& {{\left(M_A^Q\right)}_R}^{R^{\prime }}{\Psi }_{LQ{R}^{\prime }}^{\mu }\left(m\right)\hfill \end{array}$$

(129)

with ${{\left(M_A^Q\right)}_L}^{L^{\prime }}$ the matrix elements of the representation of $G_m$ associated to the eigenvalues of the Casimir operators Q. Thus, $\{L_A^{\mu },A=1,\dots ,n\}$ and $\{R_A^{\mu },A=1,\dots ,n\}$ generate the Lie algebra ${\mathfrak{g}}_m$, and ${\Psi }_{LQR}^{\mu }$ are the corresponding matrix elements of $G_m$ (but not of $G_m^{\mu }$, which is not a group). This means that, locally, $G_m^{\mu }$ and $G_m$ are diffeomorphic. Moreover, since the operators $L_A,R_A$ are Hermitian with respect to the scalar product on $G_m$, the operators $L_A^{\mu },R_A^{\mu }$ will be Hermitian with respect to the scalar product on $G_m^{\mu }$. Note, however, that the generators $L_A^{\mu },R_A^{\mu }$ are certainly not linear combinations of the generators $L_A,R_A$, simply because, in general, the soft manifold $G_m^{\mu }$ has fewer Killing vectors than the manifold $G_m$.

One observation is required. As we have seen, ${\psi }_{LQR}^{\mu }$ are in the left and right representations of $G_m$ (see (129)). However, since the metric tensor is deformed by the parameter $T^{\mu }$, we have to take into account this deformation parameter when considering tensor products of representations. In particular, if we define

$$\begin{array}{ccc}\hfill {\Psi }_{LQR}^{\mu }\left(m\right){\otimes }_{\mu }{\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }\left(m\right)& \equiv & {\displaystyle \frac{1}{\sqrt{T^{\mu }}}}{\Psi }_{LQR}^{\mu }\left(m\right){\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}^{\mu }\left(m\right)=\sqrt{T^{\mu }}{\Psi }_{LQR}\left(m\right){\Psi }_{L^{\prime }{Q}^{\prime }{R}^{\prime }}\left(m\right)\hfill \\ & =& \sum _{L^{″}{Q}^{″}{R}^{″}}\int C_{LQRL;L^{\prime }{Q}^{\prime }{R}^{\prime }}^{L^{″}{Q}^{″}{R}^{″}}{\Psi }_{L^{″}{Q}^{″}{R}^{″}}^{\mu }\left(m\right),\hfill \end{array}$$

thus, we recover the usual results, instead of (121) (for a compact manifold).

The last step in our construction is to identify relevant central extensions. First, observe that two-cocycles can be defined equivalently to the case $G_m$ according to (21) or (22). However, because of (128), this construction is more involved. We identify relevant central extensions following a different strategy. Indeed, we have already obtained central extensions for the algebra $\mathfrak{g}\left(G_m\right)$ denoted $k_L^i,k_R^i$-associated and to the two-cocycles ${\omega }_i^L,{\omega }_i^R$ (see (54) for $G_m=G_c$ a compact Lie group, and for a similar relation for $G_m=G_{nc}$, a non-compact Lie group).

Before going further, recall some well-known properties about two-cocycles. Let $(\{X_a,a=1,\dots ,\},[,])$ be a Lie algebra with Lie brackets $[X_a,X_b]=i{h_{ab}}^c{X}_c$. Then, if we define a new algebra $(\{X_a,a=1,\dots ,\},{[,]}^{\prime })$ with a new bracket, ${[X_a,X_b]}^{\prime }=[X_a,X_b]+\omega (X_a,X_b)$ ($\omega (X_a,X_b)$ belongs to $\mathbb{R}$ or $\mathbb{C}$, depending on whether we consider real or complex Lie algebras), then this algebra is endowed with the structure of a Lie algebra if $\omega$ is a two-cocycle. This is equivalent to saying that the Jacobi identity is satisfied, i.e., that we have

$$\begin{array}{c}\hfill {h_{bc}}^d\omega (X_a,X_d)+{h_{ca}}^d\omega (X_b,X_d)+{h_{ab}}^d\omega (X_c,X_d)=0.\end{array}$$

Obviously, if we perform a change of basis: $X_a^{\prime }={P_a}^b{X}_b$, the two-cocycle is given by

$$\begin{array}{c}\hfill \omega (X_a^{\prime },X_b^{\prime })={P_a}^c{P_b}^c\omega (X_a,X_b),\end{array}$$

(130)

and, the Jacoby identity above is satisfied with the new two-cocycle.

Returning to $G_m^{\mu }$, relation (127) allows for defining the two-cocycles of $G_m^{\mu }$ from the two-cocycles of $G_m$ because of (130). In the case of a compact Lie group $G_m=G_c$ (the case of a non-compact Lie group is easily obtained with the substitution $\sum \to \sum \int$), the two-cocycles take the form

$$\begin{array}{c}\hfill {\omega }_i^{\mu L}(T_{a_1{L}_1{Q}_1{R}_1}^{\mu },T_{a_2{L}_2{Q}_2{R}_2}^{\mu })=k_{a_1{a}_2}{P_{L_1{Q}_1{R}_1}}^{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime }}{P_{L_2{Q}_2{R}_2}}^{L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}{L}_2^{\prime }\left(i\right){\eta }_{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime },L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }},\end{array}$$

(131)

(with a similar relation for ${\omega }_i^R$) because of (120). It is important to emphasize that the two-cocycles above are defined with integration on the manifold $G_m$, not on $G_m^{\mu }$. This observation is very important when we identify the series of operators that are compatible with these cocycles. It turns out that the operators compatible with ${\omega }_i^{\mu L},{\omega }_i^{\mu R}$ are the operators $D_i^L,D_i^R$, satisfying (30), not the operators $D_i^{\mu L},D_i^{\mu R}$.

The KM algebra associated to $G_c^{\mu }$ (with $G_c$ a compact Lie group) is then defined by

$$\begin{array}{c}\hfill \tilde{\mathfrak{g}}\left(G_c^{\mu }\right)=\{{\mathcal{T}}_{aLQR}^{\mu },a=1\dots ,d,\left(LQR\right)\in \mathcal{I},k_L^i,k_R^i,i=1,\dots {\mathsf{\ell }}^{\prime }\}⋊\{D_i^L,D_i^R,i=1,\dots ,{\mathsf{\ell }}^{\prime }\}\end{array}$$

with Lie brackets

$$\begin{array}{ccc}\hfill \left[{{\mathcal{T}}^{\mu }}_{a{L}_1{Q}_1{R}_1},{{\mathcal{T}}^{\mu }}_{a_2{L}_2{Q}_2{R}_2}\right]& =& i{f_{a_1{a}_2}}^{a_3}{C^{\mu }}_{L_1{Q}_1{R}_1;L_2{Q}_2{R}_2}^{L_3{Q}_3{R}_3}{{\mathcal{T}}^{\mu }}_{a_3{L}_3{Q}_3{R}_3}\hfill \\ & & +k_{a_1{a}_2}{P_{L_1{Q}_1{R}_1}}^{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime }}{P_{L_2{Q}_2{R}_2}}^{L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}{\eta }_{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime },L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}\left(k_L^i{L}_2^{\prime }\left(i\right)+k_R^i{R}_2^{\prime }\left(i\right)\right),\hfill \\ \hfill \left[D_i^L,{\mathcal{T}}_{aLRQ}\right]& =& {P_{LQR}}^{L^{\prime }{Q}^{\prime }{R}^{\prime }}{L}^{\prime }\left(i\right){{\mathcal{T}}^{\mu }}_{a{L}^{\prime }{Q}^{\prime }{R}^{\prime }},\hfill \\ \hfill \left[D_i^R,{\mathcal{T}}_{aLRQ}\right]& =& {P_{LQR}}^{L^{\prime }{Q}^{\prime }{R}^{\prime }}{R}^{\prime }\left(i\right){{\mathcal{T}}^{\mu }}_{aLQ{R}^{\prime }},\hfill \end{array}$$

(132)

where ${C^{\mu }}_{L_1{Q}_1{R}_1;L_2{Q}_2{R}_2}^{L_3{Q}_3{R}_3}$ is defined in (122). Analogous expressions hold for $G_m=G_{nc}$ a non-compact Lie group.

Because the algebras $\mathfrak{g}\left(G_c\right)$ and $\mathfrak{g}\left(G_c^{\mu }\right)$ are isomorphic, and because of (131), it seems that, at a first glance, the algebras $\tilde{\mathfrak{g}}\left(G_c\right)$ and $\tilde{\mathfrak{g}}\left(G_c^{\mu }\right)$ are also isomorphic. However, for the algebra associated to $G_c$, the differential operator $D_i^{L,R}$ is associated to a Killing vector of $G_c$, while for the algebra $G_c^{\mu }$, we consider the same differential operators, but they are not associated to any Killing vector. This observation simply means that the algebras $\tilde{\mathfrak{g}}\left(G_c\right)$ and $\tilde{\mathfrak{g}}\left(G_c^{\mu }\right)$ are only diffeomorphic locally, as $G_m$ and $G_m^{\mu }$ are locally diffeomorphic. There is a case where $G_m=$ U$\left(1\right)$ is very specific. Indeed, if we consider a soft manifold U${\left(1\right)}^{\mu }$ with scalar product

$$\begin{array}{c}\hfill {(f,g)}_F=\underset{0}{\overset{2\pi }{\int }}\mathrm{d}\theta F\left(\theta \right)\overline{f}\left(\theta \right)g\left(\theta \right),\end{array}$$

a change in variables $F\left(\theta \right)\mathrm{d}\theta =\mathrm{d}\phi$ leads to the isomorphism $\tilde{g}($U${\left(1\right)}^{\mu })\cong \tilde{g}($U$\left(1\right))$, and there are no non-trivial soft deformations of affine Lie algebras [47]. This is a consequence of the property that any one-dimensional manifold without a boundary is homeomorphic to the one-dimensional sphere ${\mathbb{S}}^1$.

It is also possible, in some cases, to associate a Virasoro algebra of $G_m^{\mu }$. We briefly comment on this point. Let $D_i^L,D_i^R$ be the set of generators of the Cartan subalgebra of ${\left({\mathfrak{g}}_m\right)}_L\oplus {\left({\mathfrak{g}}_m\right)}_R$ satisfying condition (30). The de Witt algebra (with obvious relations for $L\to R$) reads

$$\begin{array}{c}\hfill {\mathrm{Witt}}_i\left(G_m^{\mu }\right)=\left\{{\mathsf{\ell }}_{iLQR}^{\mu }=-{\psi }_{LQR}^{\mu }{D_i}^L,\left(LQR\right)\in \mathcal{I}\right\}\end{array}$$

(133)

and has Lie brackets

$$\begin{array}{c}\hfill \left[{\mathsf{\ell }}_{i{L}_1{Q}_1{R}_1}^{\mu },{\mathsf{\ell }}_{i{L}_2{Q}_2{R}_2}^{\mu }\right]={P_{L_1{Q}_1{R}_1}}^{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime }}{P_{L_2{Q}_2{R}_2}}^{L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}\left(L_1^{\prime }\left(i\right)-L_2^{\prime }\left(i\right)\right)C_{L_1^{\prime }{Q}_1^{\prime }{R}_1^{\prime };L_2^{\prime }{Q}_2^{\prime }{R}_2^{\prime }}^{L^{″}{Q}^{″}{R}^{″}}{\mathsf{\ell }}_{i{L}^{″}{Q}^{″}{R}^{″}}^{\mu }\end{array}$$

because of (20) and (127). On the one hand, if the corresponding operators ${\mathsf{\ell }}_{iLQR}$ in ${\mathrm{Witt}}_i\left(G_m\right)$ are compatible with the cocycle ${\omega }_i^L$, the operators ${\mathsf{\ell }}_{iLQR}^{\mu }$ are compatible with the cocycle ${\omega }_i^{\mu L}$. On the other hand, if ${\mathrm{Witt}}_i\left(G_m\right)$ admits a central extension, then ${\mathrm{Witt}}_i\left(G_m^{\mu }\right)$ also admits a central extension because (130) holds.

In this review, we will not consider any example of KM algebras associated to soft deformations of (compact) Lie groups, since their construction essentially moves along the same lines as the compact group manifolds. The interested reader may consult [47] for more details.

6. Roots Systems and Some Elements of Representation Theory

In this section, we would like to introduce a system of roots for the Kac–Moody algebras considered in previous sections. In the second part, we shall introduce some elements of representation theory, and, in particular, we shall see that the situation is very different from the corresponding situation for affine Lie algebras. Notwithstanding that the results of the previous section (i.e., the construction of the KM algebra $\tilde{\mathfrak{g}}(\mathcal{M})$) can easily be extended to any complex or real Lie algebra $\mathfrak{g}$, in this section, we assume that $\mathfrak{g}$ is a compact Lie algebra.

6.1. Roots System

For simplicity, we now consider Kac–Moody algebras associated to compact Lie groups; that is, $\mathcal{M}=G_c$ or $G_c/H$ with H a subgroup of $G_c$ [40]. The other cases can also be considered, but with more technical difficulties that are irrelevant for the analyses below—see [45] for the case where $\mathcal{M}$ is a non-compact Lie group.

Let $\mathfrak{g}$ be a compact Lie algebra (not to be confused with the compact Lie algebra ${\mathfrak{g}}_c$ of Section 3.1), and assume that $\mathfrak{g}$ is of rank r. Let $\{H^i,i=1,\dots ,r\}\subset \mathfrak{g}$ be a Cartan subalgebra of $\mathfrak{g}$, let $\Sigma$ be the root system of $\mathfrak{g}$, and let $E_{\alpha },\alpha \in \Sigma$ be the corresponding root vector. To ease the notations, we denote $I=\left(LQR\right)$ with the notations of Section 3.1; then, using the Cartan–Weyl basis, the Kac–Moody algebra is generated by

$$\begin{array}{c}\hfill \tilde{\mathfrak{g}}\left(\mathcal{M}\right)=\left\{H_I^i,E_{\alpha I},i=1,\dots ,r,\alpha \in \Sigma ,I\in \mathcal{I},D_i,k_i,i=1\dots ,{\mathsf{\ell }}^{\prime }\right\}\end{array}$$

and the non-vanishing Lie brackets are

$$\begin{array}{ccc}\hfill \left[H_I^i,H_{I^{\prime }}^{i^{\prime }}\right]& =& {\eta }_{I{I}^{\prime }}{h}^{i{i}^{\prime }}\sum _{i=1}^{{\mathsf{\ell }}^{\prime }}{I}^{\prime }\left(i\right)k_i,\hfill \\ \hfill \left[H_I^i,E_{\alpha J}\right]& =& {c_{IJ}}^K{\alpha }^i{E}_{\alpha K},\hfill \\ \hfill \left[E_{{\alpha }_I},E_{{\beta }_J}\right]& =& \left\{\begin{array}{cc}ϵ(\alpha ,\beta ){c_{IJ}}^K{E}_{\alpha +\beta K},\hfill & \alpha +\beta \in \Sigma ,\hfill \\ {c_{IJ}}^K\alpha ·H_K+{\eta }_{IJ}{\displaystyle \sum _{i=1}^{{\mathsf{\ell }}^{\prime }}}J\left(i\right)k_i\hfill & \alpha +\beta =0,\hfill \\ 0,\hfill & \left\{\begin{array}{c}\alpha +\beta \ne 0,\hfill \\ \alpha +\beta \notin \Sigma ,\hfill \end{array}\right.\hfill \end{array}\right.\hfill \\ \hfill \left[D_i,E_{\alpha J}\right]& =& J\left(i\right)E_{{\alpha }_J},\hfill \\ \hfill [D_i,H_J^j]& =& J\left(i\right)H_J^i,\hfill \end{array}$$

(134)

where

$$\begin{array}{c}\hfill h^{ij}={⟨H^i,H^j⟩}_0,h_{ij}={\left(h^{-1}\right)}_{ij},\alpha ·H_I=h_{ij}{\alpha }^i{H}_I^j\end{array}$$

with the Killing form ${⟨·,·⟩}_0$ defined in Section 3.1, and the operators associated to roots of $\mathfrak{g}$ are normalized as

$$\begin{array}{c}\hfill {⟨E_{\alpha },E_{\beta }⟩}_0={\delta }_{\alpha ,-\beta }.\end{array}$$

The coefficients $ϵ(\alpha ,\beta )$ depend on $\mathfrak{g}$ and are equal to $\pm 1$ if $\mathfrak{g}$ is simply laced. Note the hermiticity relations:

$$\begin{array}{c}\hfill {\left(H_I^i\right)}^{†}=H_{I^c}^i,{\left(E_{\alpha I}\right)}^{†}=E_{-\alpha I^c},{\left(k_i\right)}^{†}=k_i,\end{array}$$

where $I^c$ means $I^c\left(i\right)=-I\left(i\right),i=1,\dots ,{\mathsf{\ell }}^{\prime }$.

In a way entirely analogous to the usual affine Lie algebra (see, e.g., [5], pp. 343–344) [40], we can extend the Killing form to $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ by

$$\begin{array}{ccc}\hfill {⟨{\mathcal{T}}_{aI},{\mathcal{T}}_{bJ}⟩}_1& =& {\eta }_{IJ}{k}_{ab},\hfill \\ \hfill {⟨D_j,{\mathcal{T}}_{aI}⟩}_1& =& {⟨k_j,{\mathcal{T}}_{aI}⟩}_1=0,\hfill \\ \hfill {⟨k_i,k_j⟩}_1& =& {⟨D_i,D_j⟩}_1=0,\hfill \\ \hfill {⟨D_i,k_j⟩}_1& =& {\delta }_{ij}.\hfill \end{array}$$

(135)

Let $\mathcal{Q}$ be the possible eigenvalues of the Casimir operators, let $q\in \mathcal{Q}$, and let ${\mathcal{D}}^q$ be the corresponding representation. Define ${\mathcal{D}}_0^q=\{{\Psi }_{q,1},\dots ,{\Psi }_{q,n_q}\}\subset {\mathcal{D}}^q$ to be the set of vectors with zero weight. Let $\{{\mathcal{H}}_{q,1}^i,\dots ,{\mathcal{H}}_{q,n_q}^i,i=1,\dots ,r\}\subset \{H_I^i,I,i=1,\dots ,r\in \mathcal{I}\}$ be the corresponding elements in $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$. These elements are obviously commuting. (for instance, if $\mathfrak{g}=\mathfrak{su}\left(2\right)$, all bosonic, or integer spin, representations have exactly one zero-weight vector, and in this case, $\tilde{\mathfrak{g}}\left(\mathrm{SU}\left(2\right)\right)$ is an infinite-rank Lie algebra.) Thus, from (134), the maximal set of commuting operators is given by

$$\begin{array}{c}\hfill \tilde{\mathfrak{h}}=\{{\mathcal{H}}_{n,q}^i,i=1,\dots ,r,q\in \mathcal{Q},n=1,\dots ,n_q,D_i,k_i,i=1,\dots ,{\mathsf{\ell }}^{\prime }\}.\end{array}$$

This means that the Kac–Moody algebra $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ is an infinite-rank Lie algebra. Due to the difficulty of dealing with infinitely many commuting generators being diagonalized simultaneously, we define the root-space of $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ by considering the finite-dimensional subalgebra ${\tilde{\mathfrak{h}}}_0\subset \tilde{\mathfrak{h}}$ defined by

$$\begin{array}{c}\hfill {\tilde{\mathfrak{h}}}_0=\{H_{\mathbf{0}}^i,i=1,\dots ,r,D_i,k_i,i=1,\dots ,{\mathsf{\ell }}^{\prime }\},\end{array}$$

(with the notations of Section 3.1 for the definition of ${\mathsf{\ell }}^{\prime }$) where $H_{\mathbf{0}}^i$ correspond to the generators of the ‘loop’ algebra, and $\mathfrak{g}\left(\mathcal{M}\right)$ is associated to the trivial representation ${\mathcal{D}}^0$ of $\mathfrak{g}$. The corresponding root-spaces are given by

$$\begin{array}{ccc}\hfill {\mathfrak{g}}_{(\alpha ,n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }})}& =& \left\{E_{\alpha I}\mathrm{with}I\left(1\right)=n_1,\dots ,I\left({\mathsf{\ell }}^{\prime }\right)=n_{{\mathsf{\ell }}^{\prime }}\right\},\alpha \in \Sigma ,n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }}\in \mathbb{Z},\hfill \\ \hfill {\mathfrak{g}}_{(0,n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }})}& =& \left\{H_I^i\mathrm{with}I\left(1\right)=n_1,\dots ,I\left({\mathsf{\ell }}^{\prime }\right)=n_{{\mathsf{\ell }}^{\prime }}\right\},n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }}\in \mathbb{Z}.\hfill \end{array}$$

(136)

Since, in general, the number ${\mathsf{\ell }}^{\prime }$ of compatible commuting operators $D_I$ is such that ${\mathsf{\ell }}^{\prime }<n=dim\mathcal{M}$, unlike the usual Kac–Moody algebras, the root spaces associated to the roots are infinite-dimensional, and we have

$$\begin{array}{ccc}\hfill \left[{\mathfrak{g}}_{(0,\mathbf{n})},{\mathfrak{g}}_{(\alpha ,\mathbf{m})}\right]& \subset & {\mathfrak{g}}_{(\alpha ,\mathbf{m}+\mathbf{n})},\hfill \\ \hfill \left[{\mathfrak{g}}_{(\alpha ,\mathbf{m})},{\mathfrak{g}}_{(\beta ,\mathbf{n})}\right]& \subset & {\mathfrak{g}}_{(\alpha +\beta ,\mathbf{m}+\mathbf{n})},\alpha +\beta \in \Sigma \hfill \end{array}$$

with $\mathbf{n}=(n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }})$. There is one notable exception. The Kac–Moody algebra associated to U${\left(1\right)}^n$ admits exactly ${\mathsf{\ell }}^{\prime }=n$ compatible commuting operators [40] (these algebras are called toroidal algebras in [35] and quasi-simple Lie algebras in [32]). Thus, in these cases, the root-space ${\mathfrak{g}}_{(\alpha ,n_1,\dots ,n_n)}$ is one-dimensional. Finally, it is important to observe that the Lie brackets between two elements involve not only the root structure, but also the representation theory of $G_c$ in the form of the Clebsch–Gordan coefficients ${c_{IJ}}^K$ (see (134)).

Introduce $\mathbf{0}=(0,\dots ,0)$; then, a root of $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ is defined by $\tilde{\alpha }=(\alpha ,\mathbf{0},\mathbf{n})$, where $\alpha \in \Sigma$ corresponds to the root of the simple compact Lie algebra $\mathfrak{g}$; the following ${\mathsf{\ell }}^{\prime }$ entries correspond to the vanishing eigenvalues of the central charges $k_1,\dots ,k_{{\mathsf{\ell }}^{\prime }}$, and $\mathbf{n}\in {\mathbb{Z}}^{{\mathsf{\ell }}^{\prime }}$ are the eigenvalues of $D_1,\dots ,D_{{\mathsf{\ell }}^{\prime }}$.

A root $(\alpha ,\mathbf{0},n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }})$ is said to be positive if it satisfies

$$\begin{array}{c}\hfill (\alpha ,\mathbf{0},n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }})>0\mathrm{if}\left\{\begin{array}{cc}\mathrm{either}& \left\{\begin{array}{c}\exists k\in \{1,\dots ,{\mathsf{\ell }}^{\prime }\}\mathrm{s}.\mathrm{t}.\hfill \\ n_{{\mathsf{\ell }}^{\prime }}=\dots =n_{k+1}=0\mathrm{and}{n}_k>0\hfill \end{array}\right.\\ \mathrm{or}& n_{{\mathsf{\ell }}^{\prime }}=\dots =n_1=0\mathrm{and}\alpha >0.\end{array}\right.\end{array}$$

(137)

According to (135), we can endow the weight-space with the scalar product:

$$\begin{array}{c}\hfill (\alpha ,c_1,\dots ,c_{{\mathsf{\ell }}^{\prime }},n_1,\dots ,n_{{\mathsf{\ell }}^{\prime }})·({\alpha }^{\prime },c_1^{\prime },\dots ,c_{{\mathsf{\ell }}^{\prime }}^{\prime },n_1^{\prime },\dots ,n_{{\mathsf{\ell }}^{\prime }}^{\prime })=\alpha ·{\alpha }^{\prime }+\sum _{j=1}^{{\mathsf{\ell }}^{\prime }}\left(n_j{c}_j^{\prime }+n_J^{\prime }{c}_j\right),\end{array}$$

where $\alpha ·{\alpha }^{\prime }$ is the usual scalar product in $\Sigma$.

As it happens, for usual Kac–Moody algebras [3], we have two types of roots. The set of roots $(\alpha ,\mathbf{0},\mathbf{n})$ of ${\mathfrak{g}}_{(\alpha ,\mathbf{n})}$ with $\alpha \in \Sigma ,\mathbf{n}\in {\mathbb{Z}}^{{\mathsf{\ell }}^{\prime }}$ satisfy

$$\begin{array}{c}\hfill (\alpha ,\mathbf{0},\mathbf{n})·(\alpha ,\mathbf{0},\mathbf{n})=\alpha ·\alpha >0,\end{array}$$

and are called real roots, while the set $(0,\mathbf{0},\mathbf{n})$ of ${\mathfrak{g}}_{(0,\mathbf{n})}$ with $\mathbf{n}\in {\mathbb{Z}}^{{\mathsf{\ell }}^{\prime }}$, satisfying

$$\begin{array}{c}\hfill (0,\mathbf{0},\mathbf{n})·(0,\mathbf{0},{\mathbf{n}}^{\prime })=0,\end{array}$$

is called the set of imaginary roots. The imaginary root-space is ${\mathsf{\ell }}^{\prime }-$dimensional.

Recall that ${\mathsf{\ell }}^{\prime }$ denotes the number of central charges, which we call the order of centrality. We now show that unless ${\mathsf{\ell }}^{\prime }=1$, we cannot find a system of simple roots for $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$. To this extent, introduce ${\alpha }_i,i=1,\dots ,r$ the simple roots of $\mathfrak{g}$. If ${\mathsf{\ell }}^{\prime }=1$, and we denote by $\psi$ the highest root of $\mathfrak{g}$, it is easy to see that

$$\begin{array}{c}\hfill {\widehat{\alpha }}_i=({\alpha }_i,0,0),i=1,\dots ,r,{\widehat{\alpha }}_{r+1}=(-\psi ,0,1)\end{array}$$

(138)

is a system of simple roots of $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$. Now, if we suppose that ${\mathsf{\ell }}^{\prime }=2$, as the positive roots are given by (i) $(\alpha ,0,0,0,0)$ with $\alpha >0$, (ii) $(\alpha ,0,0,n_1,0)$ with $\alpha \in \Sigma ,n_1>0$, or (iii) $(\alpha ,0,0,n_1,n_2),\alpha \in \Sigma ,n_1\in \mathbb{Z},n_2>0$, and since the roots $(\alpha ,0,0,n_1,0)$ are neither bounded from below nor from above (because $n_1\in \mathbb{Z}$), we cannot define a simple root of the form $(-\psi ,0,0,-n_{\max },1)$, where $n_{max}$ corresponds to the highest possible value of $n_1$ (or $-n_{max}$ the lowest possible value of $n_1$). This means that for ${\mathsf{\ell }}^{\prime }\ge 2$, we cannot construct a system of simple roots. In other words, the only generalized Kac–Moody algebras that admit simple roots are (obviously) the usual affine algebras, as well as the Kac–Moody algebras associated to $SU\left(2\right)/U\left(1\right)$ studied in [40].

Note that there is an alternative generalization of affine Lie algebras also called Kac–Moody algebras, introduced independently by Kac and Moody. These algebras admit a system of simple roots and are defined by a generalized Cartan matrix [1,3,75]. However, both generalizations of affine Lie algebras, i.e., Kac–Moody algebras associated to a manifold $\mathcal{M}$, and Kac–Moody algebras associated to a generalized Cartan matrix, exhibit fundamentally different features. Indeed, the former is of infinite rank—except for the algebra $\tilde{\mathfrak{g}}\left(\mathrm{U}{\left(1\right)}^n\right)$, see Section 6.3, but all its roots and the corresponding generators are explicitly known, whereas the latter has finite rank, but its generators are only iteratively known in terms of the Chevalley–Serre relations.

6.2. Some Elements of Representation Theory

Let $G_c$ be a compact Lie group, and let $H\subset G_c$ be a subgroup. Let $\mathcal{M}=G_c$ or $\mathcal{M}=G_c/H$, and let $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ be the corresponding Kac–Moody algebra. The representation theory of $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ is very different from that of usual affine Lie algebras when $dim\mathcal{M}>1$. In fact, we will show that, firstly, unitary representations of $\tilde{\mathfrak{g}}\left(\mathcal{M}\right)$ exist iff there is only one non-vanishing central charge. Further unitary representations forbid the highest weight representations.

To begin, consider the Kac–Moody algebra $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$. In the Cartan–Weyl basis, we have

$$\begin{array}{c}\hfill \tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)=\left\{H_{\mathbf{m}}^i,E_{\alpha ,\mathbf{m}},\alpha \in \Sigma ,\mathbf{m}\in {\mathbb{Z}}^n,d_i,k_i,i=1,\dots ,n\right\},\end{array}$$

and the Lie brackets are given by ($\mathbf{m}=(m_1,\dots ,m_n)$) (to be in accordance with the literature on affine Lie algebras, please note the change in sign in the central charges $k_i$ compared to (134)).

$$\begin{array}{ccc}\hfill \left[H_{\mathbf{m}}^i,H_{{\mathbf{m}}^{\prime }}^{i^{\prime }}\right]& =& h^{i{i}^{\prime }}\sum _{i=1}^n{m}_i{k}_i{\delta }_{\mathbf{m},-{\mathbf{m}}^{\prime }},\hfill \\ \hfill \left[H_{\mathbf{m}}^i,E_{\alpha \mathbf{n}}\right]& =& {\alpha }^i{E}_{\alpha \mathbf{m}+\mathbf{n}},\hfill \\ \hfill \left[E_{\alpha \mathbf{m}},E_{\beta \mathbf{n}}\right]& =& \left\{\begin{array}{cc}ϵ(\alpha ,\beta )E_{\alpha +\beta \mathbf{m}+\mathbf{n}},\hfill & \alpha +\beta \in \Sigma ,\hfill \\ \alpha ·H_{\mathbf{m}+\mathbf{n}}+{\displaystyle \sum _{i=1}^n}{m}_i{k}_i{\delta }_{\mathbf{m},-\mathbf{n}},\hfill & \alpha +\beta =0,\hfill \\ 0,\hfill & \left\{\begin{array}{c}\alpha +\beta \ne 0,\hfill \\ \alpha +\beta \notin \Sigma ,\hfill \end{array}\right.\hfill \end{array}\right.\hfill \\ \hfill \left[d_i,E_{\alpha \mathbf{m}}\right]& =& m_i{E}_{\alpha \mathbf{m}},\hfill \\ \hfill [d_i,H_{\mathbf{m}}^j]& =& m_i{H}_{\mathbf{m}}^i.\hfill \end{array}$$

(139)

We stress again that, in this case, the root-space ${\mathfrak{g}}_{(\alpha ,n_n,\dots ,m_n)},\alpha \in \Sigma ,m_1,\dots ,m_n\in \mathbb{Z}$ is one-dimensional. Moreover, because of this last property, the Cartan subalgebra, in contrast to that for generic cases ($G_c\ne U{\left(1\right)}^n$), is not infinite-dimensional, but its dimension is equal to $\mathrm{rk}\mathfrak{g}+2n$.

Recall that a root $(\alpha ,0,\mathbf{m})$ is said to be positive if it satisfies (137). Assume that rk $\mathfrak{g}=r$, and let ${\alpha }_i,i=1,\dots ,r$ be the simple roots of $\mathfrak{g}$. Introduce further ${\mu }^i,i=1,\dots ,r$ the fundamental weights of $\mathfrak{g}$, satisfying

$$\begin{array}{c}\hfill 2{\displaystyle \frac{{\alpha }_i·{\mu }^j}{{\alpha }_i·{\alpha }_i}}={{\delta }_i}^j.\end{array}$$

Since $\mathfrak{g}\subset \tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$, if $\mathcal{D}$ is a unitary representation of $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$, it decomposes into $\mathcal{D}=\oplus {\mathcal{D}}_i$, where ${\mathcal{D}}_i$ are unitary representations of $\mathfrak{g}$. However, unitary representations of $\mathfrak{g}$ are in one-to-one correspondence with highest weight vectors ${\mu }_0={\sum }_{i=1}^r{p}_i{\mu }^i,p_i\in \mathbb{N}$.

Proposition 2 

([40]). Let $n>1$, and let $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$ be a generalized Kac–Moody algebra. Let ${\mu }_0=p_i{\mu }^i,p_i\in \mathbb{N}$, and suppose that $|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩$ is such that

$$\begin{array}{ccc}\hfill E_{\alpha ,\mathbf{m}}|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩& =& 0\mathit{if}(\alpha ,\mathbf{0},\mathbf{m})>0,\hfill \\ \hfill H_{\mathbf{m}}^i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩& =& 0\mathit{if}(0,\mathbf{0},\mathbf{m})>0,\hfill \\ \hfill H_{\mathbf{0}}^i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩& =& {\mu }_0^i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩,\hfill \\ \hfill k_i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩& =& c_i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩,\hfill \\ \hfill d_i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩& =& {\left(m_0\right)}_i|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩.\hfill \end{array}$$

(140)

Define ${\mathcal{D}}_{{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}}$ to be the highest weight representation obtained by the action of $E_{\alpha ,\mathbf{m}}$ with $(\alpha ,\mathbf{0},\mathbf{m})<0$ on the vacuum state $|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩$.

(i)

If ${\mathcal{D}}_{{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}}$ is a unitary representation, all central charges vanishes, except one.

(ii)

If $|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩$ is the highest weight satisfying (140), then ${\mathcal{D}}_{{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}}$ is not a unitary representation.

Proof. 

(i) Let $(-\alpha ,\mathbf{0},\mathbf{m})$ be a root of $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$. The generators

$$\begin{array}{c}\hfill X_{\alpha ,\mathbf{m}}^{\pm }=\sqrt{{\displaystyle \frac{2}{\alpha ·\alpha }}}{E}_{\mp \alpha ,\pm \mathbf{m}},h_{\alpha }={\displaystyle \frac{2}{\alpha ·\alpha }}\left(-\alpha ·H_{\mathbf{0}}+\sum _{i=1}^n{m}_i{k}_i\right),\end{array}$$

span an $\mathfrak{su}\left(2\right)-$subalgebra. Assume $(\alpha ,\mathbf{0},\mathbf{m})>0$; thus, $X_{\alpha ,\mathbf{m}}^{+}|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩=0$. Unitarity condition $({\left(X_{\alpha ,\mathbf{m}}^{+}\right)}^{†}=X_{\alpha ,\mathbf{m}}^{-}$)

$$\begin{array}{ccc}\hfill \left|\right|X_{\alpha ,\mathbf{m}}^{-}|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}{⟩\left|\right|}^2& =& ⟨{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}\left|\left[X_{\alpha ,\mathbf{m}}^{+},X_{\alpha ,\mathbf{m}}^{-}\right]\right|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩=\hfill \\ & =& ⟨{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}|h_{\alpha }|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}⟩={\displaystyle \frac{2}{\alpha ·\alpha }}\left(-\alpha ·{\mu }_0+\sum _{i=1}^n{c}_i{m}_i\right)\ge 0\hfill \end{array}$$

implies

$$\begin{array}{c}\hfill \sum _{i=1}^n{c}_i{m}_i\ge \alpha ·{\mu }_0.\end{array}$$

(141)

Suppose $\alpha >0$; then, $\alpha ·{\mu }_0>0$. In this case, (141) is very strong. Indeed, since $(-\alpha ,\mathbf{0},\mathbf{m})>0$, this means that $\mathbf{m}=(m_1,\dots ,m_{k-1},m_k,0,\dots ,0)$ with $0<k\le n,m_k>0$ and $m_1,\dots ,m_{k-1}\in \mathbb{Z}$. Condition (141), which must be satisfied for any $m_i\in \mathbb{Z},i=1,\dots ,k-1$, is equivalent to imposing that only one central charge is non-vanishing. This proves (i).

(ii): Since only one central charge is non-vanishing, without loss of generality, we can suppose $c_n=c\ne 0$ and $c_i=0,i=1,\dots ,n-1$. Assume, again, $\alpha >0$, and let $\left(-\alpha ,\mathbf{0},m_1,\dots ,m_k,0,\dots 0\right)>0$ with $0<k<n$, $m_k>0$ and $(m_1,\dots ,m_{k-1})\in {\mathbb{Z}}^{k-1}$. The operators (with $\mathbf{m}=(m_1,\dots ,m_k,0,\dots ,0)$)

$$\begin{array}{c}\hfill Y_{\alpha ,\mathbf{m}}^{\pm }=\sqrt{{\displaystyle \frac{2}{\alpha ·\alpha }}}{E}_{\mp \alpha ,\pm \mathbf{m}},h_{\alpha }^{\prime }=-{\displaystyle \frac{2}{\alpha ·\alpha }}\alpha ·H_{\mathbf{0}},\end{array}$$

(142)

also generate an $\mathfrak{su}\left(2\right)-$subalgebra. As before, the condition $\left|\right|Y_{\alpha ,\mathbf{n}}^{-}|{\mu }_0,\mathbf{c},{\mathbf{m}}_{\mathbf{0}}{⟩\left|\right|}^2\ge 0$ holds, from which we deduce that

$$\begin{array}{c}\hfill \sum _{p=1}^k{m}_i{c}_i\ge \alpha ·{\mu }_0.\end{array}$$

(143)

However, as $c_1=\dots =c_{n-1}=0$, this cannot be satisfied if $\alpha >0$. Thus, the representation ${\mathcal{D}}_{{\mu }_0,\mathbf{c}}$ is not unitary, which proves (ii). □

Corollary 1. 

Let $\mathfrak{g}$ be a simple compact Lie algebra; then, the Kac–Moody Lie algebra $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right),n>1$ does not have any unitary highest weight representation.

These results can be extended to the case of a more general Kac–Moody algebra, such as $\tilde{\mathfrak{g}}\left(G_c\right)$ or $\tilde{\mathfrak{g}}(G_c/H)$. Some elements of the proof are given in [40]. It is important to observe that, in the absence of symmetries between the generators $D_i$, we can have ${\mathsf{\ell }}^{\prime }$ different possibilities, given by (eventually reordering the eigenvalues of the operators $D_i$ to define positive roots, see Equation (137))

$$\begin{array}{c}\hfill \mathbf{c}=(0,\dots ,0,c_p,0,\dots ,0),p\in \{1,\dots ,{\mathsf{\ell }}^{\prime }\}.\end{array}$$

These results were obtained in a different manner in [76]. Let $G_c$ be a compact Lie group, and let $\mathfrak{g}\left(G_c\right)$ and $\tilde{\mathfrak{g}}\left(G_c\right)$ be, respectively, the ‘loop algebra’ (see (52)) and the Kac–Moody algebra (see Section 3.3). On purpose, the authors introduced spinors of $G_c$ which is assumed to be a spin manifold). Then, they observed that it is easy to obtain a unitary representation of the ‘loop-’algebra $\mathfrak{g}\left(G_c\right)$; however, this representation is not a highest weight representation. In other words, we get a ‘first quantized’ representation with no vacuum states annihilated by all fermion annihilation operators. They then showed that if we try to define a ‘second quantized’ or a highest weight representation, some divergences do appear (see p. 380, [76]), and the operators are ill-defined. However, by considering bosonic condensate fermions, they were able to construct a highest weight representation, but no invariant inner product was identified.

To illustrate the situation, let us consider an explicit example. Let $\mathfrak{g}$ be a compact Lie algebra, and let $\mathcal{D}$ be a real unitary $d-$dimensional representation. Denote the generators of $\mathfrak{g}$ in the representation $\mathcal{D}$ by the Hermitian matrices $M_a,a=1\dots ,dim\mathfrak{g}$. Introduce $H^i,i=1,\dots ,d$ real fermions in the representation $\mathcal{D}$.

First, consider the case where $\mathcal{M}={\mathbb{S}}^1$. We have the following decomposition (to ease the presentation, we only assume here Neveu–Schwarz (NS)—not Ramond—fermions), i.e., with anti-periodic boundary conditions:

$$\begin{array}{c}\hfill H^i\left(\theta \right)=\sum _{n\in \mathbb{Z}+{\displaystyle \frac{1}{2}}}{b}_n^i{e}^{-in\theta }.\end{array}$$

We further assume the quantization relations (with $\{a,b\}=ab+ba$)

$$\begin{array}{c}\hfill \{b_n^i,b_m^j\}={\delta }^{ij}{\delta }_{m,-n},\end{array}$$

and the reality condition

$$\begin{array}{c}\hfill {\left(b_m^i\right)}^{†}=b_{-m}^i.\end{array}$$

The vacuum of the representation is defined by

$$\begin{array}{c}\hfill b_n^i|0⟩=0,n>0,\end{array}$$

and in other words, $b_n^i,n<0/n>0$ are creation/annihilation operators. Consider now the current

$$\begin{array}{c}\hfill T_a\left(\theta \right)={\displaystyle \frac{1}{2}}{H}^i\left(\theta \right){\left(M_a\right)}_{ij}{H}^j\left(\theta \right).\end{array}$$

Since $T_a$ is bilinear in the fermion fields, in order to have well-defined quantities, a normal ordering prescription must be defined. As usual, we put the creation operators to the right:

$$\begin{array}{c}\hfill {}_{\circ }{}^{\circ }{b}_n^i{b}_m^j_{\circ }^{\circ }=\left\{\begin{array}{ccc}-b_m^i{b}_n^j\hfill & \mathrm{if}\hfill & n>0\hfill \\ \phantom{-}{b}_m^i{b}_n^j\hfill & \mathrm{if}\hfill & n<0\hfill \end{array}\right.\end{array}$$

If we decompose $T_a\left(\theta \right)={\sum }_{M\in \mathbb{Z}}{T}_{aM}{e}^{-iM\theta }$, we obtain

$$\begin{array}{c}\hfill T_{aM}={\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}\sum _{m\in \mathbb{Z}}{}_{\circ }{}^{\circ }{b}_m^i{b}_{M-m}^j_{\circ }^{\circ }=-{\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}\sum _{m>M}{b}_{M-m}^j{b}_m^i+{\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}\sum _{m<M}{b}_m^i{b}_{M-m}^j.\end{array}$$

(144)

This normal ordering prescription has two advantages: (i) the vacuum is well-defined $⟨0\left|T_{aM}\right|0⟩=0$, and (ii) $\parallel T_{aM}|0⟩\parallel$ is finite. For the second condition (ii), observe that, in (144), in the second sum, the term $b_{M-m}^j$ is an annihilation operator and, thus, annihilates the vacuum. In the first sum, the second term $b_m^i$ is a creation operator if $M<m<0$. Thus, if $M<0$, there are $-M-2$ possible values of m, where $b_{M-m}^j{b}_m^i$ acts non-trivially on $|0⟩$, and consequently, $\parallel T_{aM}|0⟩\parallel$ is finite. So the operators $T_{aM}$ are always well-defined. Furthermore, the normal ordering prescription is also of crucial importance. Indeed, using the Wick theorem, one can show that, in fact, the operators $T_{aM}$ generate the affine Lie algebra $\tilde{\mathfrak{g}}\left({\mathbb{S}}^1\right)$ with a well-defined central extension expressed in terms of the quadratic Casimir operator in the representation $\mathcal{D}$ [5], and thus, we obtain a unitary highest weight representation of $\tilde{\mathfrak{g}}\left({\mathbb{S}}^1\right)$.

We now briefly analyze the case where $\mathcal{M}={\mathbb{S}}^1\times {\mathbb{S}}^1$. Again, considering (NS,NS) modes, we get

$$\begin{array}{c}\hfill H^i({\theta }_1,{\theta }_2)=\sum _{m_1,m_2\in \mathbb{Z}+{\displaystyle \frac{1}{2}}}{b}_{m_1{m}_2}^i{e}^{-i(m_1{\theta }_1+m_2{\theta }_2)}.\end{array}$$

(145)

As for the circle, we assume the quantization relations

$$\begin{array}{c}\hfill \{b_{m_1{m}_2}^i,b_{n_1{n}_2}^j\}={\delta }^{ij}{\delta }_{m_1,-n_1}{\delta }_{m_2,-n_2},\end{array}$$

and the reality condition

$$\begin{array}{c}\hfill {\left(b_{m_1{m}_2}^i\right)}^{†}=b_{-m_1-m_2}^i.\end{array}$$

(146)

To define annihilation and creation operators, we introduce the following order relation (since we consider only (NS,NS) fermions, m and p cannot be equal to zero. For R-fermions, we have to account for the possibility that m or p is equal to zero to define our order relation):

$$\begin{array}{c}\hfill (m,p)>0⟺m>0.\end{array}$$

The vacuum is defined by

$$\begin{array}{c}\hfill b_{m_1{m}_2}^i|0⟩=0,m_1>0,\forall m_2\in \mathbb{Z}+{\displaystyle \frac{1}{2}},\end{array}$$

and the normal ordering prescription by

$$\begin{array}{c}\hfill {}_{\circ }{}^{\circ }b_{mp}^i{b}_{nq\circ }^{j\circ }=\left\{\begin{array}{ccc}-b_{nq}^j{b}_{mp}^i& \mathrm{if}m>0\hfill & \forall p\in \mathbb{Z}+{\displaystyle \frac{1}{2}}\hfill \\ \phantom{-}{b}_{mp}^i{b}_{nq}^j& \mathrm{if}m<0\hfill & \forall p\in \mathbb{Z}+{\displaystyle \frac{1}{2}}\hfill \end{array}\right.\end{array}$$

Decomposing for the circle,

$$\begin{array}{c}\hfill T_a({\theta }_1,{\theta }_2)={\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}{}_{\circ }{}^{\circ }{H}^i({\theta }_1,{\theta }_2)H^j({\theta }_1,{\theta }_2)_{\circ }^{\circ }\end{array}$$

we get

$$\begin{array}{ccc}\hfill T_{aMP}& =& {\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}\sum _{m,p\in \mathbb{Z}+{\displaystyle \frac{1}{2}}}{}_{\circ }{}^{\circ }{b}_{mp}^i{b}_{M-mP-p}^j_{\circ }^{\circ }\hfill \\ & =& -{\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}\sum _{m>Mp\in \mathbb{Z}+\left({\displaystyle \frac{1}{2}}\right)}{b}_{M-mP-p}^j{b}_{mp}^i+{\displaystyle \frac{1}{2}}{\left(M_a\right)}_{ij}\sum _{m<Mp\in \mathbb{Z}+\left({\displaystyle \frac{1}{2}}\right)}{b}_{mp}^i{b}_{M-mP-p}^j\hfill \end{array}$$

As for the circle, the vacuum is well-defined, but now $\parallel T_{aMP}|0⟩\parallel$ diverges because when $M<0$, the first sum above involves infinitely many terms (because of the sum over the second index, which belongs to $\mathbb{Z}$). This situation has been analyzed explicitly in [42,43], where, in order to have well-defined generators beyond the usual normal ordering prescription, a regulator was introduced, and infinite sums were regularized by means of a Riemann $\zeta -$function. Next, the normal ordering prescription also leads to a representation of $\tilde{\mathfrak{g}}({\mathbb{S}}^1\times {\mathbb{S}}^1)$ but with only one central extension, and again, the central charge can be expressed in terms of the quadratic Casimir operator of $\mathfrak{g}$ in the representation $\mathcal{D}$ [43]. Even if it were possible to construct fermion [43] or boson [42] representations of $\tilde{\mathfrak{g}}({\mathbb{S}}^1\times {\mathbb{S}}^1)$, this procedure is not fully satisfactory because some cut-off is needed in order to have well-defined quantities. This construction is not in contradiction with Corollary 1 because we did not consider any cut-off or regularization prescription in Proposition 2.

6.3. Some Results on Toroidal Algebras

In this section, we would like to study the details of the Kac–Moody algebra $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$ more. We first consider the case where $n=2$, keeping in mind that the general case $n>2$ follows easily. Starting from the ‘loop-’algebra $\mathfrak{g}\left(U{\left(1\right)}^2\right)=\{T_{a{m}_1{m}_2}=T_a{e}^{i{m}_1{\theta }_1+i{m}_2{\theta }_2},a=1,\dots ,dim\mathfrak{g},m_1,m_2\in \mathbb{Z}\}$, its central extensions are characterized by closed one-forms. Since $H^1({\mathbb{S}}^1\times {\mathbb{S}}^1)=\mathbb{Z}\oplus \mathbb{Z}$, a general closed one-form reads

$$\begin{array}{c}\hfill \gamma =i{k}_1\mathrm{d}{\theta }_2-i{k}_2\mathrm{d}{\theta }_1-i\mathrm{d}h:=k_1{\gamma }_1+k_2{\gamma }_2+{\gamma }_h,\end{array}$$

where $k_1,k_2\in \mathbb{R}$ and h is an arbitrary periodic function on the two-torus. Let $h_{m_1,m_2}({\theta }_1,{\theta }_2)=e^{i{m}_1{\theta }_1+i{m}_2{\theta }_2},(m_1,m_2)\ne (0,0)$, and let ${\gamma }_{m_1,m_2}=-k_{m_1,m_1}d{h}_{m_1,m_2}$ be the corresponding exact one-form. The differential operators associated to the one-forms ${\gamma }_1,{\gamma }_2,{\gamma }_{m_1,m_2}$ are, respectively,

$$\begin{array}{ccc}\hfill d_1& =& -i{\partial }_1,\hfill \\ \hfill d_2& =& -i{\partial }_2,\hfill \\ \hfill d_{n_1,n_2}& =& e^{i{m}_1{\theta }_1+i{m}_2{\theta }_2}\left(-i{m}_2{\partial }_1+i{m}_1{\partial }_2\right).\hfill \end{array}$$

A generic two-cocycle is, thus, given by

$$\begin{array}{ccc}\hfill \omega (T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})& =& -k_1{\omega }_1(T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})-k_2{\omega }_2(T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})\hfill \\ & & -\sum _{(m_1,m_2)\in \mathbb{Z}\setminus \{0,0\}}{k}_{m_1,m_2}{\omega }_{m_1,m_2}(T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})\hfill \end{array}$$

where (see (26))

$$\begin{array}{ccc}\hfill {\omega }_1(T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})& =& n_1^{\prime }{k}_{a{a}^{\prime }}{\delta }_{n_1,-n_1^{\prime }}{\delta }_{n_2,-n_2^{\prime }},\hfill \\ \hfill {\omega }_2(T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})& =& n_2^{\prime }{k}_{a{a}^{\prime }}{\delta }_{n_1,-n_1^{\prime }}{\delta }_{n_2,-n_2^{\prime }},\hfill \\ \hfill {\omega }_{m_1,m_2}(T_{a{n}_1{n}_2},T_{a^{\prime }{m}_1^{\prime }{n}_2^{\prime }})& =& k_{a{a}^{\prime }}(m_2{n}_1^{\prime }-m_1{n}_2^{\prime }){\delta }_{n_1+n_1^{\prime },-m_1}{\delta }_{n_2+n_2^{\prime },-m_2}\hfill \\ & =& k_{a{a}^{\prime }}(n_1{n}_2^{\prime }-n_2{n}_1^{\prime }){\delta }_{n_1+n_1^{\prime },-m_1}{\delta }_{n_2+n_2^{\prime },-m_2}.\hfill \end{array}$$

(147)

These cocycles were also obtained in [33]. It is direct to observe (from (28)) that the differential operators $d_1$ and $d_2$ are compatible with the cocycles ${\omega }_1,{\omega }_2,{\omega }_{m_1,m_2}$.

In [35], using Kähler differentials [77,78], Moody and his collaborators obtained the universal central extension of the ‘loop’ algebra, which they called toroidal algebra. It turns out that, in fact, the central extensions in [35] coincide with the two-cocycle above (147). Indeed, the first cases (resp., second cases) in Equation (3) of [78] correspond to the cocycle ${\omega }_{m_1,m_2}$—after an appropriate rescalling—(resp., ${\omega }_1,{\omega }_2$). Moreover, in [78], the authors also proved that their formulæ coincide with the formulæ obtained in [77] used by Moody and their collaborators. This means that to centrally extend the ‘loop-’algebra $\mathfrak{g}\left(U{\left(1\right)}^2\right)$, we can equivalently use the cocycle (147) or Kähler differentials, as in [35].

For completeness and for further use, we briefly recall the main feature of the Moody et al. construction [35]. Let $\mathbb{C}[z_1,z_1^{-1},z_2,z_2^{-1}]$ be the Laurent polynomial ring in two variables, and let $\mathfrak{g}\otimes \mathbb{C}[z_1,z_1^{-1},z_2,z_2^{-1}]$ be the ‘loop’ algebra (see (52) in our notations). To define the universal central extension of the ‘loop’ algebra, we introduce ${\Omega }_1$ the set of one-forms of $\mathbb{C}[z_1,z_1^{-1},z_2,z_2^{-1}]$. Finally, let $¯:{\Omega }_1\to {\Omega }_1/\mathrm{Imd}$. Thus, for any $F\in \mathbb{C}[z_1,z_1^{-1},z_2,z_2^{-1}]$, we have $\overline{\mathrm{d}F}=0$. The universal central extension of the ‘loop’ algebra is given by [35]

$$\begin{array}{c}\hfill {[F\otimes x,G\otimes y]}^{\prime }=FG\otimes [x,y]+\overline{\left(\mathrm{d}F\right)G}{⟨x,y⟩}_0,\end{array}$$

for any $x,y\in \mathfrak{g},F,G\in \mathbb{C}[z_1,z_1^{-1},z_2,z_2^{-1}]$ with $[,]$ (resp., ${⟨,⟩}_0$) the Lie brackets (the Killing form) of $\mathfrak{g}$. Since

$$\begin{array}{c}\hfill \overline{\mathrm{d}\left(z_1^{k_1}{z}_2^{k_2}\right)}=k_1\overline{z_1^{k_1}{z}_2^{k_2}{z}_1^{-1}\mathrm{d}{z}_1}+k_2\overline{z_1^{k_1}{z}_2^{k_2}{z}_2^{-1}\mathrm{d}{z}_2}=0,\end{array}$$

a basis of ${\Omega }_1/\mathrm{Imd}$ is given by

$$\begin{array}{c}\hfill \begin{array}{ccc}{C}_{k_1{k}_2}=\overline{z_1^{k_1}{z}_2^{k_2}{z}_2^{-1}\mathrm{d}{z}_2}\hfill & \mathrm{if}\hfill & k_1\ne 0\hfill \\ C_{0k_2}=\overline{z_2^{k_2}{z}_1^{-1}\mathrm{d}{z}_1}\hfill & \mathrm{if}\hfill & k_1=0,k_2\ne 0\hfill \\ c_1=\overline{z_1^{-1}\mathrm{d}{z}_1},c_2=\overline{z_2^{-1}\mathrm{d}{z}_2}\hfill & \mathrm{if}\hfill & k_1=k_2=0\hfill \end{array}\end{array}$$

and

$$\begin{array}{c}\hfill \overline{\mathrm{d}\left(z_1^{n_1}{z}_2^{n_2}\right)z_1^{n_1^{\prime }}{z}_2^{n_2^{\prime }}}=\left\{\begin{array}{ccc}(n_2{n}_1^{\prime }-n_1{n}_2^{\prime }){\displaystyle \frac{C_{n_1+n_1^{\prime }{n}_2+n_2^{\prime }}}{n_1+n_1^{\prime }}}\hfill & \mathrm{if}\hfill & n_1+n_1^{\prime }\ne 0\hfill \\ (n_2{n}_1^{\prime }-n_1{n}_2^{\prime }){\displaystyle \frac{C_{0n_2+n_2^{\prime }}}{n_2+n_2^{\prime }}}\hfill & \mathrm{if}\hfill & n_1+n_1^{\prime }=0,n_2+n_2^{\prime }\ne 0\hfill \\ n_1{c}_1+n_2{c}_2\hfill & \mathrm{if}\hfill & n_1+n_1^{\prime }=n_2+n_2^{\prime }=0\hfill \end{array}\right.\end{array}$$

which corresponds exactly to the cocycles given in (147). Thus, the two constructions coincide.

Coming back to our notations, the toroidal algebra is $T_2\left(\mathfrak{g}\right)=\{{\mathcal{T}}_{a{n}_1{n}_2},d_1,d_2,k_1,k_2,k_{m_1{m}_2},$ $a=1,\dots ,dim\mathfrak{g},n_1,n_2\in \mathbb{Z},(m_1,m_2)\in {\mathbb{Z}}^2\setminus \{0,0\}\}$ with Lie brackets (in our notations):

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{a{n}_1{n}_2},{\mathcal{T}}_{a^{\prime }{n}_1^{\prime }{n}_2^{\prime }}\right]& =& {{i f}_{{aa}^{\prime }}}^b{\mathcal{T}}_{b{n}_1+n_1^{\prime }{n}_2+n_2^{\prime }}+(k_1{n}_1+k_2{n}_2)k_{a{a}^{\prime }}{\delta }_{n_1,-n_1^{\prime }}{\delta }_{n_2,-n_2^{\prime }}\hfill \\ & & +\sum _{(m_1,m_2)\in {\mathbb{Z}}^2\setminus \{0,0\}}{k}_{m_1{m}_2}(n_2{n}_1^{\prime }-n_1{n}_2^{\prime })k_{a{a}^{\prime }}{\delta }_{n_1+n_1^{\prime },-m_1}{\delta }_{n_2+n_2^{\prime },-m_2},\hfill \\ & =& {{i f}_{{aa}^{\prime }}}^b{\mathcal{T}}_{b{n}_1+n_1^{\prime }{n}_2+n_2^{\prime }}+(k_1{n}_1+k_2{n}_2)k_{a{a}^{\prime }}{\delta }_{n_1,-n_1^{\prime }}{\delta }_{n_2,-n_2^{\prime }}\hfill \\ & & +k_{-n_1-n_1^{\prime },-n_2-n_2^{\prime }}(n_2{n}_1^{\prime }-n_1{n}_2^{\prime })k_{a{a}^{\prime }},\hfill \\ \hfill \left[d_1,{\mathcal{T}}_{a{n}_1{n}_2}\right]& =& n_1{\mathcal{T}}_{a{n}_1{n}_2},\hfill \\ \hfill \left[d_2,{\mathcal{T}}_{a{n}_1{n}_2}\right]& =& n_2{\mathcal{T}}_{a{n}_1{n}_2}.\hfill \end{array}$$

(148)

This algebra admits an interesting subalgebra, where only two-central charges are non-vanishing, say $k_1,k_2$, and this corresponds to the Kac–Moody algebra considered throughout this review: $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)=\{{\mathcal{T}}_{a{m}_1,m_2},d_1,d_2,k_1,k_2,a=1,\dots ,dim\mathfrak{g},m_1,m_2\in \mathbb{Z}\}$. This algebra is also named the double affine algebra in [79]. All the results of the two-toroidal and the double affine algebras extend naturally for $n>2$ [80], i.e., to the $n-$toroidal and $n-$affine algebras.

Now, we turn to the representation theory of the double affine Lie algebra (or Kac–Moody algebra associated to $\mathcal{M}=(U{\left(1\right)}^2$). In Section 6.2, the results were given, considering the root system $\Sigma$ of $\mathfrak{g}$. In [35], non-unitary representations of $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)$ were explicitly obtained considering the roots of $\widehat{\mathfrak{g}}$, the affine extensions of $\mathfrak{g}$. The key observation is the following: even if $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)$ does not admit a system of simple roots, it is possible to have a Chevalley–Serre presentation of $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)$ (and, of course, also of the toroidal algebra $T_2\left(\mathfrak{g}\right)$) in two different but equivalent manners—(1) with the simple roots of $\mathfrak{g}$ or (2) with the simple roots of $\widehat{\mathfrak{g}}$. Now consider the double-affine algebra or the Kac–Moody algebra $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)$ with two central extensions $k_1,k_2$, whose Lie brackets are given by (139) with $n=2$ (note that all results presented below extend easily for the toroidal algebra with non-vanishing central charge $k_{m_1{m}_2}$).

Let ${\alpha }_i,i=1,\dots ,r$ be the simple roots of $\mathfrak{g}$, and let $A_{ij}$ be the corresponding Cartan matrix:

$$\begin{array}{c}\hfill A_{ij}=2{\displaystyle \frac{{\alpha }_i·{\alpha }_j}{{\alpha }_i·{\alpha }_i}},1\le i,j\le r.\end{array}$$

The matrix A is non-singular, i.e., $detA\ne 0$. When associated to any simple roots of the semisimple Lie algebra $\mathfrak{g}$, we define the three operators for $(m_1,m_2)\in {\mathbb{Z}}^2$ as

$$\begin{array}{c}\hfill h_{i{m}_1{m}_2}={\displaystyle \frac{2}{{\alpha }_i·{\alpha }_i}}{\alpha }_i·H_{m_1{m}_2},e_{i{m}_1{m}_2}^{\pm }=\sqrt{{\displaystyle \frac{2}{{\alpha }_i·{\alpha }_i}}}{E}_{\pm {\alpha }_i{m}_1,m_2},1\le i\le r.\end{array}$$

The Chevalley–Serre relations are (In [35] the authors were given a presentation of the toroidal (and not the double-affine Lie) algebra.)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left[k_1,h_{i{m}_1{m}_2}\right]& =\left[k_1,e_{i{m}_1{m}_2}^{\pm }\right]=\left[k_2,h_{i{m}_1{m}_2}\right]=\left[k_2,e_{i{m}_1{m}_2}^{\pm }\right]=0\hfill \\ \hfill \left[h_{i{m}_1{m}_2},h_{j{n}_1{n}_2}\right]& =(k_1{m}_1+k_2{m}_2){\alpha }_i^{\vee }·{\alpha }_j^{\vee }{\delta }_{m_1,-n_1}{\delta }_{m_2,-n_2}\hfill \\ \hfill \left[h_{i{m}_1{m}_2},e_{j{n}_1{n}_2}^{\pm }\right]& =\pm A_{ij}{e}_{j{m}_1+n_1{m}_2+n_2}^{\pm }\hfill \\ \hfill \left[e_{i{m}_1{m}_2}^{\pm },e_{i,n_1{n}_2}^{\pm }\right]& =0\hfill \\ \hfill \left[e_{i{m}_1{m}_2}^{+},e_{j{n}_1{n}_2}^{-}\right]& ={\delta }_{ij}\left(h_{i{m}_1+n_1{m}_2+n_2}+{\displaystyle \frac{2}{{\alpha }_i·{\alpha }_i}}(k_1{m}_1+k_2{m}_2){\delta }_{m_1,-n_1}{\delta }_{m_2,-n_2}\right)\hfill \\ \hfill {\mathrm{ad}}^{1-A_{ij}}\left(e_{imn}^{\pm }\right)·e_{jpq}^{\pm }& =0,i\ne j,\hfill \end{array}\end{array}$$

($\mathrm{ad}\left(x\right)·y=[y,x],{\mathrm{ad}}^2\left(x\right)·y=[[y,x],x]$, etc., and ${\alpha }^{\vee }=2\alpha /\alpha ·\alpha$ is the co-root ) enables the reproduction of the whole algebra (see (139) with $n=2$) using the Serre relation, i.e., the last equation.

The algebra $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)$ can be equivalently presented using the simple roots of the affine Lie algebra $\widehat{\mathfrak{g}}$ associated to $\mathfrak{g}$. Let ${\widehat{\alpha }}_{\left(i\right)},i=0,\dots ,r$ be the simple roots of $\widehat{\mathfrak{g}}$:

$$\begin{array}{c}\hfill \begin{array}{cccc}{\widehat{\alpha }}_{\left(i\right)}\hfill & =\hfill & ({\alpha }_i,0,0),\hfill & i=1,\dots ,r,\hfill \\ {\widehat{\alpha }}_0\hfill & =\hfill & (-\psi ,0,1),\hfill \end{array}\end{array}$$

(149)

where ${\alpha }_i,\psi$ are, respectively, the simple roots and the highest root of $\mathfrak{g}$. Let ${\widehat{A}}_{ij}$ be the Cartan matrix of $\widehat{\mathfrak{g}}$:

$$\begin{array}{c}\hfill {\widehat{A}}_{ij}=2{\displaystyle \frac{\widehat{{\alpha }_i}·\widehat{{\alpha }_j}}{\widehat{{\alpha }_i}·\widehat{{\alpha }_i}}},0\le i,j\le r\end{array}$$

where the scalar product (see for example [5]) is

$$\begin{array}{c}\hfill \widehat{\alpha }·{\widehat{\alpha }}^{\prime }=(\alpha ,k,m)·({\alpha }^{\prime },k^{\prime },m^{\prime })=\alpha ·{\alpha }^{\prime }+k{m}^{\prime }+k^{\prime }m.\end{array}$$

Note that the Cartan matrix is now singular and $dim\mathrm{Ker}A=1$, i.e., A is of corank one. Associated to any simple roots of $\widehat{\mathfrak{g}}$, for $m\in \mathbb{Z}$, we define

$$\begin{array}{c}\hfill \begin{array}{ccc}{\widehat{e}}_{im}^{\pm }=e_{i0m}^{\pm },\hfill & {\widehat{h}}_{im}=h_{i0m},\hfill & i=1,\dots ,r\hfill \\ {\widehat{e}}_{0m}^{\pm }=\sqrt{{\displaystyle \frac{2}{\psi ·\psi }}}{E}_{\mp \psi \pm 1m},\hfill & {\widehat{h}}_{0m}={\displaystyle \frac{2}{\psi ·\psi }}(-\psi ·H_{1m}+k_1).\hfill \end{array}\end{array}$$

(150)

This determines a presentation of the algebra as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left[k_2,{\widehat{h}}_{im}\right]& =\left[k_2,{\widehat{e}}_{im}^{\pm }\right]=0,\hfill \\ \hfill \left[{\widehat{h}}_{im},{\widehat{h}}_{jn}\right]& =k_{2}m{\widehat{\alpha }}_i^{\vee }·{\widehat{\alpha }}_j^{\vee }{\delta }_{m,-n},\hfill \\ \hfill \left[{\widehat{h}}_{im},{\widehat{e}}_{jn}^{\pm }\right]& =\pm {\widehat{A}}_{ij}{\widehat{e}}_{jm+n}^{\pm },\hfill \\ \hfill \left[{\widehat{e}}_{im}^{+},{\widehat{e}}_{jn}^{-}\right]& ={\delta }_{ij}\left({\widehat{h}}_{im+n}+{\displaystyle \frac{2}{{\widehat{\alpha }}_i·{\widehat{\alpha }}_i}}{k}_{2}m{\delta }_{m,-n}\right),\hfill \\ \hfill \left[{\widehat{e}}_{im}^{\pm },{\widehat{e}}_{in}^{\pm }\right]& =0,\hfill \\ \hfill {\mathrm{ad}}^{1-{\widehat{A}}_{ij}}\left({\widehat{e}}_{im}^{\pm }\right)·{\widehat{e}}_{jn}^{\pm }& =0,i\ne j.\hfill \end{array}\end{array}$$

Again, the last relation, i.e., the Serre relation, enables the reproduction of the whole algebra (139). Indeed, it has been shown in [44] that the two presentations (149) and (151) lead to isomorphic algebras (139). Similar presentations (in terms of the roots of $\mathfrak{g}$ or $\widehat{\mathfrak{g}}$) hold for the toroidal Lie algebra $T_n\left(\mathfrak{g}\right)$ and for the Kac–Moody algebra $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$ (or $n-$affine Lie algebra) when $n>2$ [80].

Even if the two presentations are isomorphic, they present some structural differences. Indeed, within the presentation of $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^n\right)$ with the roots of the semisimple Lie algebra $\mathfrak{g}$, the Cartan matrix is non-singular, whereas for the presentation with the roots of the affine Lie algebra $\widehat{\mathfrak{g}}$, the Cartan matrix is singular. The presentation of toroidal Lie algebras in terms of roots of affine Lie algebras has been used by several authors in order to construct explicit representations of $T_2\left(\mathfrak{g}\right)$ and $T_n\left(\mathfrak{g}\right),n>2$. In [35,80], a Vertex representation of the Toroidal Lie algebra $T_n\left(\mathfrak{g}\right)$ (when $\mathfrak{g}$ is simply laced) was obtained. In [79], a fermion realization of $\tilde{\mathfrak{g}}\left(U{\left(1\right)}^2\right)$, i.e., of the double affine algebra in the terminology of [79], when $\mathfrak{g}$ is a classical Lie algebra, was explicitly constructed. Note that the representation theory of toroidal Lie algebras has been studied extensively (see, e.g., [79] and the references therein). However, since the Cartan matrix of $\widehat{\mathfrak{g}}$ is singular, and since all these articles are based on the Chevalley–Serre presentation of the algebra in terms of the roots of $\widehat{\mathfrak{g}}$, all these representations are non-unitary.

6.4. Unitary Representations

Now, we turn to constructing explicit unitary representations. On purpose, we assume that the only non-vanishing central charges are $k_1$ and $c_m=-m{k}_{0,m},m\ne 0$. Let $\overline{\mathfrak{g}}=\{{\mathcal{T}}_{anm},k_1,c_k,a=1,\dots ,dim\mathfrak{g},m,n\in \mathbb{Z},k\in \mathbb{Z}\setminus \left\{0\right\}\}$. With these notations, the non-vanishing Lie brackets (148) reduce to

$$\begin{array}{ccc}\hfill \left[{\mathcal{T}}_{a{n}_1{n}_2},{\mathcal{T}}_{a^{\prime }{n}_1^{\prime }{n}_2^{\prime }}\right]& =& {{i f}_{{aa}^{\prime }}}^b{\mathcal{T}}_{b{n}_1+n_1^{\prime }{n}_2+n_2^{\prime }}+n_1\left(k_1{\delta }_{n_2,-n_2^{\prime }}+c_{n_2+n_2^{\prime }}\right)k_{a{a}^{\prime }}{\delta }_{n_1,-n_1^{\prime }}.\hfill \end{array}$$

(151)

Consider now the mapping

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{a{n}_1{n}_2}& \mapsto & e^{i{n}_2{\theta }_2}{\mathcal{T}}_{a{n}_1}\hfill \\ \hfill k_1& \mapsto & k_1\hfill \\ \hfill c_m& \mapsto & e^{im{\theta }_2}{k}_1\hfill \end{array}$$

where ${\mathcal{T}}_{am}$ are the generators of the affine Lie algebra $\widehat{\mathfrak{g}}$. Through this mapping, (151) reduces to

$$\begin{array}{c}\hfill \left[e^{i{n}_2{\theta }_2}{\mathcal{T}}_{a{n}_1},e^{i{n}_2^{\prime }{\theta }_2}{\mathcal{T}}_{a^{\prime }{n}_1^{\prime }}\right]={{if}_{{aa}^{\prime }}}^b{e}^{i(n_2+n_2^{\prime }){\theta }_2}{\mathcal{T}}_{b{n}_1+n_1^{\prime }}+n_1{k}_1{e}^{i(n_2+n_2^{\prime }){\theta }_2}{h}_{a{a}^{\prime }}{\delta }_{n_1,-n_1^{\prime }},\end{array}$$

(152)

and the two algebras (151) and (152) are isomorphic (see [35] Proposition 2.8 or [81] Theorem 3.3). It is immediate to observe that the latter algebra (152) is the loop algebra of the affine algebra, and thus, we have the following isomorphism $\overline{\mathfrak{g}}\cong \widehat{\mathfrak{g}}\left({\mathbb{S}}^1\right)\cong \widehat{\mathfrak{g}}\otimes {\mathbb{S}}^1$. Please note that $\left[d_2,c_m\right]=0$, but $\left[d_2,e^{im{\theta }_2}{k}_1\right]=m{e}^{im{\theta }_2}{k}_1$.

A certain class of unitary representations follows at once and is directly obtained from unitary representations of the affine Lie algebra $\widehat{\mathfrak{g}}$. Unitary highest weight representations of $\widehat{\mathfrak{g}}$ are well-known. The simple roots of $\widehat{\mathfrak{g}}$ are given in (149) and the highest weight read (recall that rk $\mathfrak{g}=r$)

$$\begin{array}{c}\hfill {\displaystyle \frac{\psi }{\psi ·\psi }}=\sum _{i=1}^r{q}^i{\displaystyle \frac{{\alpha }_i}{{\alpha }_i·{\alpha }_i}},q^i\in \mathbb{N}\end{array}$$

The corresponding fundamental weights [5] are

$$\begin{array}{c}\hfill {\widehat{\mu }}^0=(0,{\displaystyle \frac{1}{2}}{q}^0\psi ·\psi ,0)\mathrm{with}{q}^0=1,{\widehat{\mu }}^i=({\mu }^i,{\displaystyle \frac{1}{2}}{q}^i\psi ·\psi ,0),i=1,\dots ,r.\end{array}$$

Let ${\widehat{\mu }}_0={\sum }_{i=0}^r{p}_i{\widehat{\mu }}^i,p_i\in \mathbb{N}$, and let $|{\widehat{\mu }}_0⟩$ be a highest weight defined by (with the notations of Section 6.3, Equation (150))

$$\begin{array}{ccc}\hfill {\widehat{e}}_i^{+}|{\widehat{\mu }}_0⟩& =& 0,\hfill \\ \hfill {\widehat{h}}_i|{\widehat{\mu }}_0⟩& =& p_i|{\widehat{\mu }}_0⟩.\hfill \end{array}$$

Because of the expression of $h_0$ in terms of the central charge k (we denote c as its eigenvalue), a highest weight is equivalently specified by ${\mu }_0={\sum }_{i=1}^r{p}_i{\mu }^i$ (the highest weight of the semisimple Lie algebra $\mathfrak{g}$, with its associated fundamental weight ${\mu }^i$) and the eigenvalue of the central charge c (the Cartan subalgebra of $\widehat{\mathfrak{g}}$ is $\{h_i,i=0,\dots ,r,k,d\}$, but the eigenvalue of d for the highest state $|{\widehat{\mu }}_0⟩$ is irrelevant. Indeed, if $d|{\widehat{\mu }}_0⟩=n_0|{\widehat{\mu }}_0⟩$, redefining $d\to d-{\displaystyle \frac{n_0}{c}}k$, we have $d|{\widehat{\mu }}_0⟩=0$, and we can take $n_0=0$ [82]). Let $x=2{\displaystyle \frac{c}{\psi ·\psi }}$ be the level of the representation. The representation ${\mathcal{D}}_{{\widehat{\mu }}_0}$ is unitary if (see, e.g., [5])

$$\begin{array}{c}\hfill \left(x\in \mathbb{Z}\mathrm{and}c\ge \psi ·{\mu }_0\ge 0\right)\iff \left(x=\sum _{i=0}^r{p}_i{q}^i\in \mathbb{Z}\mathrm{and}x\ge \sum _{i=1}^r{p}_i{q}^i\right)\end{array}$$

Consider now

$$\begin{array}{c}\hfill \mathcal{R}=\left\{|m⟩,m\in \mathbb{Z}\right\},\end{array}$$

(153)

the set of unitary representations of $U\left(1\right)$:

$$\begin{array}{c}\hfill d_2|m⟩=m|m⟩.\end{array}$$

Using the harmonic expansion on $U\left(1\right)$

$$\begin{array}{c}\hfill ⟨{\theta }_2|m⟩=e^{im{\theta }_2},\end{array}$$

unitary representations of $\overline{\mathfrak{g}}$ are given by the tensor product

$$\begin{array}{c}\hfill {\overline{\mathcal{D}}}_{{\widehat{\mu }}_0}={\mathcal{D}}_{{\widehat{\mu }}_0}\otimes \mathcal{R}\end{array}$$

(154)

and correspond to a harmonic expansion of the unitary representation ${\mathcal{D}}_{{\widehat{\mu }}_0}$ of $\widehat{\mathfrak{g}}$ on the manifold $U\left(1\right)$. These results were anticipated in [40]. Note that while ${\mathcal{D}}_{{\widehat{\mu }}_0}$ is a highest weight representation of $\widehat{\mathfrak{g}}$, ${\overline{\mathcal{D}}}_{{\widehat{\mu }}_0}$ is not a highest weight representation of $\overline{\mathfrak{g}}$. A similar analysis holds for the algebra $\widehat{\mathfrak{g}}\otimes {\mathbb{T}}_{n-1},n>1$, where $T_{n-1}$ is the $(n-1)-$dimensional torus [40].

7. Applications in Physics

The generalized KM current algebra $\mathfrak{g}\left(\mathcal{M}\right)$, its semidirect symmetry actions, and its cohomological central extensions constitute a versatile toolkit that can be used to investigate various subfields of high-energy theoretical physics. This section surveys three arenas in which the mathematical framework developed in this review finds concrete applications in physics. Our emphasis will be on (i) two-dimensional current algebra and CFT, including WZW models, Sugawara stress tensors, and the Virasoro algebra; (ii) higher-dimensional compactifications and spectra in KK theory, and (iii) structures emerging in cosmological billiards and in the hidden symmetries of supergravity. We will attempt to keep the presentation self-contained and pedagogical, referring to the relevant literature for technical details and a broader background. Across all three contexts, the crucial inputs have a threefold nature: geometric (e.g., choice of $\mathcal{M}$ and its symmetry), analytic (e.g., harmonic analysis on $\mathcal{M}$), and algebraic (e.g., compatibility of cocycles and representations).

7.1. Two-Dimensional Current Algebra and CFT: WZW, Sugawara, and Virasoro

7.1.1. Affine Symmetry from Loops and Its Generalization

On a two-dimensional worldsheet, currents $J^a\left(z\right)$ valued in a finite-dimensional Lie algebra $\mathfrak{g}$ satisfy the operator product expansions (OPEs) that encode an affine KM algebra at a certain level k. This structure emerges canonically in WZW models, where the basic field $g(z,\overline{z})\in G$ is a group-valued map, and the action includes a Wess–Zumino topological term; see, e.g., [4,5,6]. The holomorphic currents realize a centrally extended loop algebra of maps ${\mathbb{S}}^1\to \mathfrak{g}$, and the stress tensor $T\left(z\right)$ is obtained by the Sugawara construction, which expresses the Virasoro generators as quadratic combinations of KM modes, producing a central charge $c={\displaystyle \frac{kdim\mathfrak{g}}{k+{\mathfrak{g}}^{\vee }}}$, where ${\mathfrak{g}}^{\vee }$ is the dual Coxeter number of $\mathfrak{g}$ [5,6].

Within the general framework presented in this review, the circle ${\mathbb{S}}^1$ is replaced by a higher-dimensional manifold $\mathcal{M}$, thus yielding to the Lie algebra $\mathfrak{g}\left(\mathcal{M}\right)$ of $\mathfrak{g}$-valued functions on $\mathcal{M}$, organized through a Hilbert basis adapted to the symmetries of $\mathcal{M}$ (by exploiting the Peter–Weyl theorem on compact groups/cosets or the Plancherel theorem on non-compact groups/cosets). Semidirect actions by Killing vector fields or diffeomorphism algebras may enrich this symmetry, and compatible two-cocycles yield central extensions that generalize the affine case. From the two-dimensional worldsheet perspective, this construction can be viewed as a mode expansion of currents along internal manifolds: the worldsheet current algebra fibers over $\mathcal{M}$ so that KM modes carry, in addition, the harmonic labels on $\mathcal{M}$ itself. This particularly manifests when $\mathcal{M}$ is a compact group manifold or a homogeneous space thereof because, in this case, the harmonic analysis is representation-theoretically well-known and studied [4,16,17]. Possible issues related to normal ordering prescriptions in manifolds of dimension $>1$ are discussed in [42,43]; we also briefly recalled them in Section 6.

7.1.2. Group Manifolds and Coset CFTs

Let $\mathcal{M}=G_c$ be a compact Lie group manifold. The Peter–Weyl theorem provides an orthonormal basis $\left\{{\Psi }_{LQR}\right\}$ in $L^2\left(G_c\right)$, built from matrix elements of irreducible unitary representations, and the product of two basis elements decomposes in terms of Clebsch–Gordan coefficients. The machinery developed in this review lifts the $\mathfrak{g}$-valued modes $T_{a;LQR}$ to generators of a current algebra $\mathfrak{g}\left(G_c\right)$ with structure constants directly expressed in terms of the representation theory of $G_c$. In the CFT context, this gives a natural language for coset models $G/H$, in which the surviving harmonics are the H-invariant components on the right (or left), and the operator content obeys the selection rules of the $G\to H$ branching. This perspective clarifies how the coset primaries and their fusion rules relate to the representation-theoretic decomposition on $G/H$, as well as to the geometry of the coset [6,13,17].

7.1.3. Diffeomorphism Algebras and Virasoro Analogs

For $\mathcal{M}={\mathbb{S}}^1$, the semidirect ‘partner’ of the affine algebra is the Witt algebra (and its central extension, the Virasoro algebra). On higher spheres ${\mathbb{S}}^n$ or other higher-dimensional manifolds $\mathcal{M}$, families of vector fields play an analogous role. For instance, for ${\mathbb{S}}^2$, the algebra of area-preserving diffeomorphisms is relevant, while, more generally, one can construct subalgebras of vector fields (generated by Killing vectors or selected modes) acting on $\mathfrak{g}\left(\mathcal{M}\right)$. This leads to semidirect products that generalize to $\mathcal{M}$ the affine–Virasoro ‘interplay’ on ${\mathbb{S}}^1$. Actually, this idea can be traced back to the early literature on generalized KM algebras related to diffeomorphism groups of closed surfaces [32,33]. It should also be recalled here that the appearance of generalized (Borcherds–Kac–Moody) algebras in CFT and moonshine phenomena may be regarded as a hint to the depth of algebraic structures that can emerge beyond the affine case [19,20,21].

7.2. Compactifications and KK Spectra

7.2.1. Mode Expansions and Mass Towers

Consider a d-dimensional field theory on a space-time of the form ${\mathbb{R}}^{1,d-1}\times \mathcal{M}$, where $\mathcal{M}$ is a compact internal manifold (group manifold, coset, or deformation thereof); the expansion of fields in an orthonormal basis of $L^2\left(\mathcal{M}\right)$ produces KK towers whose masses are determined by eigenvalues of Laplace-type operators on $\mathcal{M}$. This finds extensive application and crucial relevance in general KK theories, as well as in dimensional compactifications in (super)string theory or M-theory [8,9,10,11,12].

Within this framework, the generalization from ${\mathbb{S}}^1$ to $\mathcal{M}$ yields the following consequences: (i) the Noether charges associated to a compact gauge algebra $\mathfrak{g}$ become families of charges indexed by the harmonic labels on $\mathcal{M}$, thus realizing the Lie algebra $\mathfrak{g}\left(\mathcal{M}\right)$; (ii) the semidirect action of the isometries of $\mathcal{M}$ provides an interesting geometric perspective on the selection rules and spectral degeneracies; (iii) the central extensions encode anomalies that can appear in the commutators of KK currents when integrating over $\mathcal{M}$, with the relevant two-cocycles in one-to-one correspondence with closed $(\dim \mathcal{M}-1)$-currents on $\mathcal{M}$ (as developed in this review).

7.2.2. Compact Group Manifolds and Cosets

For $\mathcal{M}=G_c$, the Peter–Weyl theory organizes the KK modes of fields into representations of $G_c$. The product of modes is controlled by Clebsch–Gordan coefficients, and the associated KM algebra $\mathfrak{g}\left(G_c\right)$ underlies the interactions and selection rules among KK modes. For $\mathcal{M}=G_c/H$, the H-invariant modes in the right (or left) action are retained, thus reproducing the well-known truncations of coset compactifications and their spectra [12,13,17]. Here, we confine ourselves to adding that this intriguingly relates to generalized Scherk–Schwarz reductions, where group-theoretic data constrain and determine consistent truncations of physical theories.

7.2.3. Toroidal and Deformed (Soft) Manifolds

When the internal, higher-dimensional manifold is a torus, one recovers familiar Abelian current algebras and their higher-rank generalizations, while deformations of group manifolds (which go under the name of soft manifolds), as used in group-geometric approaches to supergravity, naturally fit into the same formalism [12,15]. The central extension analysis identifies which deformations admit non-trivial two-cocycles compatible with symmetry actions and, thus, when anomalous terms can affect the reduced dynamics. Related algebraic structures, such as toroidal Lie algebras, also arise in the analysis of currents on higher-dimensional tori, and they have been studied in the mathematical literature for quite a long time [32,35,80].

7.2.4. Kac–Moody Symmetries in KK Theories and Beyond

KM-like enhancements in the symmetry algebras of KK reductions and related stringy settings have been quite extensively investigated [12,29,62]. In this respect, we should stress that the present framework clarifies when and how such enhancements occur: the algebra $\mathfrak{g}\left(\mathcal{M}\right)$ organizes the towers of KK currents, thus allowing for compatible central extensions to emerge precisely when the geometry of $\mathcal{M}$ admits closed $(\dim \mathcal{M}-1)$-currents, in turn giving rise to the cohomological two-cocycles. Therefore, this framework naturally highlights the dependence on the topology and metric of $\mathcal{M}$, thereby providing a controlled setting to explore consistent truncations, dualities, and anomaly structures of higher-dimensional (super) gravity theories.

7.3. Cosmological Billiards and Hidden Symmetries of Supergravity

7.3.1. BKL Dynamics and Billiards

For near, space-like singularities, the Belinsky–Khalatnikov–Lifshitz (BKL) analysis reveals chaotic, piecewise Kasner regimes interrupted by curvature-induced reflections—the so-called ‘mixmaster’ behavior [22,83]. Remarkably, this dynamic can be geometrized as a billiard motion in a region of hyperbolic space bounded by walls associated to dominant gravitational and p-form contributions. Somewhat surprisingly, in many supergravity theories, the billiard domain coincides with the fundamental Weyl chamber of a hyperbolic KM algebra, thus suggesting that an underlying infinite-dimensional symmetry controls the near-singularity regime [23,29].

7.3.2. Hidden Symmetries and Very Extended KM Algebras

Dimensional reduction of supergravities gives rise to large, non-compact global symmetries, controlling the electric–magnetic duality of the resulting theory; in string/M-theory, such a symmetry is defined over discrete fields, and it is named U-duality. In general, the target space of scalar fields can be regarded as a manifold coordinatized by the scalars themselves. In a wide class of cases, the target space can be modeled as a sigma-model on cosets $G/H$, where G is a non-compact Lie group, and H a maximal compact subgroup. The emergence of indefinite or hyperbolic extensions in further dimensional reductions as well as in conjectural uplifts (‘$E_{10}$/$E_{11}$’-type structures) has been widely discussed, as it may be shown to encode ‘hidden’ symmetries of M-theory and cosmological dynamics (see, e.g., [24,25,26,27,28], and the references therein). In this respect, the framework presented in this review exhibits two remarkable features: (i) the sigma model structure on non-compact target spaces (e.g., $SL(2,\mathbb{R})$ and $SL(2,\mathbb{R})/U\left(1\right)$) is inherently consistent with the non-compact harmonic analysis used to build $\mathfrak{g}\left(\mathcal{M}\right)$; (ii) the billiard/wall identifications hint at the possibility of describing the asymptotic dynamics through KM root systems, again, consistent with the appearance of infinite-dimensional algebras in reductions and duality webs [23,29].

7.3.3. Cocycles, Anomalies, and Constraints

The analysis of the possible central extensions of the current algebras on $\mathcal{M}$ provides crucial insights into the existence of anomaly-like terms in the effective dynamics, as well as the relevant constraints associated to conserved charges. In the aforementioned cosmological settings, integrating currents over compact slices (or along suitable cycles) in $\mathcal{M}$ yields to Schwinger terms controlled by the cohomology of $L^2\left(\mathcal{M}\right)$; this essentially extends the Pressley–Segal cocycles in the affine case to the richer $(\dim \mathcal{M}-1)$-current data. Thus, while many open questions remain (e.g., the precise matching between very extended algebras and full dynamics), the formal machinery and tools presented in this review may allow for establishing when bona fide infinite-dimensional symmetries are actually compatible with the underlying geometry of the internal manifold and with the employed boundary conditions; at the same time, potential topological and/or flux obstructions may be detected.

8. Outlook: Conjectures and Novel Applications

The framework developed in this review—generalized KM current algebras $\mathfrak{g}\left(\mathcal{M}\right)$ on a manifold $\mathcal{M}$, semidirect actions by isometries/diffeomorphisms, and the cohomological classification of central extensions—suggests several avenues for new applications in high-energy theoretical and mathematical physics. In this final section, we will briefly mention some potential, partly conjectural, developments in supergravity theories, superstring/M-theory, AdS/CFT, and holography. The proposals below should be regarded as possible lines of investigation rather than established research venues; in fact, we will stress their conjectural nature, but we also indicate how they could, in principle, be tested and checked. We leave all this for further future work.

8.1. Supergravity

8.1.1. KM-Structured Charge Algebras in Consistent Truncations

Consistent truncations on group manifolds and cosets are organized in terms of geometric data (left-invariant frames and structure constants) and by harmonic analysis on $\mathcal{M}$ [12,15]. In light of the topics discussed in the present review, it may be conjectured that in any truncation admitting a global orthonormal frame (e.g., generalized parallelizations), the full tower of Noether charges of the truncated theory closes into a centrally extended current algebra of the form $\mathfrak{g}\left(\mathcal{M}\right)$, with two-cocycles determined by closed $(\dim \mathcal{M}-1)$-currents on $\mathcal{M}$ itself. This conjecture is nothing but an extension of the standard, affine construction on a circle ${\mathbb{S}}^1$, and it may potentially detect possible anomalies through the study of the cohomology on $\mathcal{M}$. Recent progress on consistent truncations in deformed/generalized settings provides us with a promising testing ground [13,84]; in this respect, a concrete test may consist of computing Schwinger terms for KK-reduced currents in gauged supergravities and matching them against the cohomological classification of the two-cocycles outlined in this review.

8.1.2. Soft Manifolds, Fluxes, and Anomaly Constraints

Group-geometric approaches embed supergravity in a soft deformation of group manifolds, encoding fluxes and torsion as geometric data [12,15]. In this framework, it may be conjectured that the existence of non-trivial central extensions of $\mathfrak{g}\left(\mathcal{M}\right)$ compatible with isometries imposes integrability conditions on the flux backgrounds; such integrability conditions should turn out to be equivalent to a subset of the Bianchi identities and tadpole/anomaly-cancellation conditions in the dimensionally reduced resulting physical theory. In practice, these conditions would translate into the co-closedness of certain $(\dim \mathcal{M}-1)$-currents, as well as into quantization constraints for the Wess–Zumino-type terms. Remarkably, this would provide a powerful, algebraic criterion for the consistency of flux compactifications, as well as of their gauged supergravity limits [12,13].

8.1.3. Hidden Symmetries Beyond Billiards

The appearance of hyperbolic KM structures in cosmological billiards suggests a more prominent role of infinite-dimensional symmetries in supergravity dynamics [23]. One may be led to naturally conjecture that, away from the near-singularity regime, a subset of these symmetries survives as an algebra of generalized currents on suitable slices (e.g., homogeneous spatial sections) of $\mathcal{M}$, with central terms fixed by boundary conditions and topological data. In principle, this conjecture should be testable/falsifiable within numerical relativity setups, or in analytic families of Bianchi cosmologies, by constructing conserved charges associated to divergence-free vector fields on $\mathcal{M}$ and checking their commutators against the two-cocycles discussed in this review. As another conjectural remark, we should not forget to mention potential connections to duality orbits in extended supergravities, which may further constrain the admissible cocycles [29].

8.2. Superstrings and M-Theory

8.2.1. Worldvolume Current Algebras with Internal Labels

In the WZW models and related worldsheet theories, the affine symmetry controls the spectrum and dynamics [5,6]. Therefore, a lift of such a framework may be conjectured, in which the currents acquire harmonic labels on an internal manifold $\mathcal{M}$, thus effectively realizing $\mathfrak{g}\left(\mathcal{M}\right)$ on the worldsheet. When $\mathcal{M}$ is a group manifold or coset, the Peter–Weyl theorem allows for the explicit construction of operator bases; for non-compact $\mathcal{M}$, the Plancherel theorem should provide the relevant distributions. All in all, this approach seemingly combines target-space geometry and worldsheet current algebras in a way that is compatible with string field theory and group-geometric formulations [12,15,85]. A possible test of this conjecture would concern heterotic compactifications on group manifolds/cosets, in which one should compute OPE coefficients across harmonic sectors.

8.2.2. Brane Boundaries and Defect Algebras

M2/M5 branes admit boundary/defect descriptions with current algebras on lower-dimensional intersections. Thus, one may put forward the conjecture that, in backgrounds with isometries along an internal manifold $\mathcal{M}$, the boundary algebra could be identified with a central extension of the generalized KM current $\mathfrak{g}\left(\mathcal{M}\right)$, with the two-cocycle determined by the pullback of fluxes to $(\dim \mathcal{M}-1)$-cycles. Interestingly, this would generalize level quantization in WZW models to higher-dimensional defects. Hints of such structures have already appeared in early works on membranes [65], as well within the systematic investigation of WZW-like terms in string/M-theory [85].

8.2.3. Non-Geometric Backgrounds and Doubled/Exceptional Geometry

In doubled and exceptional field theories, gauge symmetries generally mix diffeomorphisms and p-form transformations. In this context, we may conjecture that the algebra of generalized diffeomorphisms and gauge transformations on a compact manifold $\mathcal{M}$ can be organized as a centrally extended $\mathfrak{g}\left(\mathcal{M}\right)$ built from the relevant finite-dimensional algebra $\mathfrak{g}$ (e.g., $E_{d\left(d\right)}$) and an $L^2\left(\mathcal{M}\right)$ basis. However, we should stress that the compatibility of the central extensions with section/closure constraints would become a sharp algebraic condition, possibly hard to meet/check. However, recent progress on deformed generalized parallelizations and consistent truncations [84] seemingly supports the idea that constructing such algebras on explicit $\mathcal{M}$ may be feasible; in this context, flux quantization would, thus, arise out of the quantization of the central charge in the extended algebra. Among other aspects, this perspective could clarify the representation content of KK towers in exceptional field theory, as well as their coupling to the so-called ‘dual’ graviton sectors [11,29].

8.3. AdS/CFT and Holography

8.3.1. Asymptotic Symmetries and Boundary Current Algebras

In $Ad{S}_3$, the Virasoro symmetry arises as an asymptotic symmetry; in higher dimensions, boundary symmetries are more subtle. In this framework, it could be conjectured that, whenever the bulk includes an internal compact manifold $\mathcal{M}$ with a certain isometry algebra, the boundary CFT exhibits a tower of conserved currents organized by $\mathfrak{g}\left(\mathcal{M}\right)$, whose central terms should also be sensitive to holographic counterterms and global anomalies [86,87,88]. In practice, such a holographic renormalization supplies the bilinear form used to compute two-cocycles (in terms of the boundary two-point functions of currents), while the harmonic analysis on $\mathcal{M}$ would determine the multiplet structure. This is, in principle, testable in truncations whose consistent embeddings are known, as well as in top-down approaches to holography, in which the manifold $\mathcal{M}$ is explicitly specified [84].

8.3.2. Coset Holography and Harmonic Selection Rules

For compact $\mathcal{M}=G_c/H$, we expect a tight match between boundary operator algebras and the representation-theoretic decomposition on $\mathcal{M}$, governed by Clebsch–Gordan coefficients and H-invariance. We also expect the correlators to generally obey some selection rules, which should, in turn, mirror the product structure of $L^2\left(\mathcal{M}\right)$; in this way, central terms should therefore be predicted by the cohomology classes of $(\dim \mathcal{M}-1)$-currents. This seems to be potentially explorable in the holographic duals built from consistent truncations on $G_c/H$ and comparable with CFT data obtained via bootstrap or integrability methods [86,88].

8.3.3. Holographic Anomalies from Cocycles

One may also put forward the conjecture that certain boundary anomalies (e.g., of a mixed flavor–gravitational type) could be captured by the cohomological two-cocycles of $\mathfrak{g}\left(\mathcal{M}\right)$ computed from the bulk; in this sense, harmonic analysis and de Rham cohomology on $L^2\left(\mathcal{M}\right)$ [86,87] would become crucial. This would be especially remarkable, since it would provide an algebraic holographic dictionary between fluxes on $\mathcal{M}$ and the central extensions in the boundary current algebra.

All in all, one conceptual consequence of the program reviewed in this work is the determination of a uniform algebraic language for the description and investigation of charges, anomalies, and selection rules in string/M-theory, supergravity, and holography. Even partial validations of the several conjectures stated above would shed light on long-standing structural questions about hidden symmetries and consistent truncations; on the other hand, failures would be interesting too, since they would help establish the true scope of current algebraic methods beyond the circle, at least as concerns the aforementioned physical applications.