Search Results (22)

This work presents a graded investigation of $\mathrm{\Theta }$-superderivations within the framework of Lie superalgebras, generalizing the $\mathrm{\Theta }$-derivation concept from ordinary Lie algebras to graded settings. For a given Lie superalgebra $\mathfrak{g}$, a linear mapping $\phi$ qualifies as a $\mathrm{\Theta }$-superderivation where $\mathrm{\Theta }$ is a superderivation such that $\phi \left([\eta ,\xi ]\right)=[\phi \left(\eta \right),\xi ]+{(-1)}^{\left|\phi \right|\left|\eta \right|}[\eta ,\mathrm{\Theta }\left(\xi \right)]$ for all homogeneous elements $\eta ,\xi \in \mathfrak{g}$. This formulation simultaneously encompasses ordinary superderivations and even components of graded centroids. We demonstrate that the collection ${sDer}^{*}\left(\mathfrak{g}\right)$ of all $\mathrm{\Theta }$-superderivations naturally carries the structure of a Lie superalgebra and admits the decomposition ${sDer}^{*}\left(\mathfrak{g}\right)=sDer\left(\mathfrak{g}\right)+C_{\overline{0}}\left(\mathfrak{g}\right)$, where $sDer\left(\mathfrak{g}\right)$ denotes the superderivation algebra and $C_{\overline{0}}\left(\mathfrak{g}\right)$ represents the even part of the graded centroid. For perfect or centerless Lie superalgebras, this sum becomes direct. In the particular case of finite-dimensional simple Lie superalgebras over algebraically closed fields of characteristic zero, we establish ${sDer}^{*}\left(\mathfrak{g}\right)=ad\left(\mathfrak{g}\right)\oplus \mathbb{F}·{id}_{\mathfrak{g}}$. Furthermore, a semidirect product decomposition ${sDer}^{*}\left(\mathfrak{g}\right)\cong sDer\left(\mathfrak{g}\right)⋉C_{\overline{0}}\left(\mathfrak{g}\right)$ holds whenever the center vanishes. Concrete illustrations involving the Heisenberg superalgebra, the super-Virasoro algebra, and low-dimensional examples are provided, complete with explicit matrix representations. Our findings extend classical derivation and centroid theories to the superalgebraic realm, laying groundwork for future implications in deformation theory and supersymmetric quantum mechanics. Full article

We construct a finite-dimensional Z₃-graded Lie superalgebra of dimensions (12,4,3), featuring a grade-2 sector that obeys a cubic bracket relation with the fermionic sector. This induces an emergent triality symmetry cycling the three components. The full set of graded Jacobi identities is verified analytically in low dimensions and numerically in a faithful 19-dimensional matrix representation, with residuals $\le 8\times {10}^{-13}$ over ${10}^7$ random tests. Explicit quadratic and cubic Casimir operators are computed, with proofs of centrality, and the adjoint representation is shown to be anomaly-free. The algebra provides a minimal, closed extension beyond conventional ${\mathrm{Z}}_2$ supersymmetry and may offer an algebraic laboratory for models with ternary symmetries. Full article

We survey the recent developments on quantum invariants of 3-manifolds and links: $\widehat{Z}$ and $F_L$. They are q-series invariants originated from mathematical physics, inspired by the categorification of a numerical quantum invariant—the Witten–Reshetikhin–Turaev (WRT) invariant—of 3-manifolds. They exhibit rich features, for example, quantum modularity, infinite-dimensional Verma module structures, and knot–quiver correspondence. Furthermore, they have connections to the 3d-3d correspondence and other topological invariants. We also provide a review of an extension of the above series invariants to Lie superalgebras. Full article

Lie-algebraic Poisson structures, related to the superalgebra of super-pseudodifferential operators on the circle over the even component of the ${\mathbb{Z}}_2$-graded Grassmann algebra, have been studied in detail; the corresponding coadjoint orbits, generated by the Casimir invariants, regarding the different superalgebra splittings into the subalgebras, are analyzed. The related Lax-type completely integrable Hamiltonian flows are constructed on suitably defined functional manifolds with respect to the canonical super-Lie–Poisson structures on them. An approach was proposed allowing the extension of the related coadjoint flows by means of the respectively constructed super-evolution flows on the adjoint super-subalgebras, specially deformed by means of super-pseudodifferential operator elements, depending on the generalized eigenfunctions of the corresponding super-linear Lax-type spectral problem. As a consequence, it is stated that all constructed new coadjoint superflows generate on the suitably extended supermanifolds Lax-type integrable Hamiltonian systems. The centrally extended super-Lie-algebraic structures have been analyzed and the related coadjoint orbits described, generated by the corresponding Casimir invariants and coinciding with integrable Hamiltonian systems on suitably defined supermanifolds. Full article

Let R be a 2-torsion-free and $n!$-torsion-free commutative ring with unity, and let X be a locally finite preordered set. We endow the incidence algebra $I(X,R)$ with a superalgebra structure via a nontrivial idempotent, which decomposes $I(X,R)$ into even and odd parts $A_0\oplus A_1$. Our main result shows that if any two directed edges in each connected component of the complete Hasse diagram $(X,D)$ lie in one cycle, then every supercommuting map on $I(X,R)$ is proper. A supercommuting map $\theta :I(X,R)\to I(X,R)$ is defined by the condition ${[\theta \left(x\right),x]}_s=0$ for all $x\in I(X,R)$, where ${[a,b]}_s=ab-{(-1)}^{\left|a\right|\left|b\right|}ba$ is the supercommutator. We prove that such maps must take the form $\theta \left(x\right)=\lambda x+\mu \left(x\right),$ where $\lambda \in Z_s\left(I(X,R)\right)$ (the supercenter) and $\mu :I(X,R)\to Z_s\left(I(X,R)\right)$ is an R-linear map. This generalizes the known results on commuting maps of incidence algebras and other associative algebras. Full article

Let $n\ge 2$ be a fixed integer. The aim of this paper is to investigate the properties of n-derivations within the framework of perfect Lie superalgebras over a commutative ring R. The main result shows that if the base ring contains ${\displaystyle \frac{1}{n-1}}$, and L is a perfect Lie superalgebra with a center equal to zero, then any n-derivation of L is necessarily a derivation. Additionally, every n-derivation of the derivation algebra $Der\left(L\right)$ is an inner derivation. Moreover, we extend the concept of n-homomorphisms to mappings between Lie superalgebras L and $L^{\prime }$ and prove that under specific assumptions, homomorphisms, anti-homomorphisms, and their combinations are all n-homomorphisms. Finally, we conclude our paper with some open problems. Full article

We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they are different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of $2\le m<n$ and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m, we obtain a tower of higher-order (as differential operators) even Hamiltonians, while for m odd we obtain a tower of higher-order odd supercharges, and the corresponding algebra consists of the odd sector only. Full article

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called “naive” approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis. Full article

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional $N=1,2,3$- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable $N=2,3$-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra $\mathcal{K}(1|3)$ and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the $N=1,2,3$-supercircle. Full article

This paper performed an investigation into the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1\mid 1})$, a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( $S\psi D\odot$) residing on the super-circle $S^{1|1}$. We also introduce and rigorously define the central charge within the framework of $\overrightarrow{(}{S}^{1\mid 1})$, leveraging the canonical central extension of $S\psi D\odot$. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra $\overrightarrow{(}{S}^{1|1})$ is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge. Full article

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and ${\alpha }^k$-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology. Full article

Over a field of characteristic $p>3$, let $KO(n,n+1;\underline{t})$ denote the odd contact Lie superalgebra. In this paper, the super-biderivations of odd Contact Lie superalgebra $KO(n,n+1;\underline{t})$ are studied. Let $T_{KO}$ be a torus of $KO(n,n+1;\underline{t})$, which is an abelian subalgebra of $KO(n,n+1;\underline{t})$. By applying the weight space decomposition approach of $KO(n,n+1;\underline{t})$ with respect to $T_{KO}$, we show that all skew-symmetric super-biderivations of $KO(n,n+1;\underline{t})$ are inner super-biderivations. Full article

We construct certain modified highest weight modules which are called quasi highest weight modules in this paper. Using the quasi highest weight modules, we introduce a new category of modules over an affine Lie superalgebra which contains projective covers. We also prove that both these projective covers and the quasi highest weight modules satisfy the vanishing property of BRST cohomology. Full article

We studied the Gaudin models with $\mathfrak{gl}\left(1\right|1)$ symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation $\mathfrak{gl}\left(1\right|1\left)\right[t]$-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}\left(1\right|1\left)\right[t]$-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also gave dimensions of the generalized eigenspaces. Full article

In our investigation on quantum gravity, we introduce an infinite dimensional complex Lie algebra ${\mathfrak{g}}_{\mathsf{u}}$ that extends ${\mathbf{e}}_{\mathbf{9}}$. It is defined through a symmetric Cartan matrix of a rank 12 Borcherds algebra. We turn ${\mathfrak{g}}_{\mathsf{u}}$ into a Lie superalgebra ${\mathfrak{sg}}_{\mathsf{u}}$ with no superpartners, in order to comply with the Pauli exclusion principle. There is a natural action of the Poincaré group on ${\mathfrak{sg}}_{\mathsf{u}}$, which is an automorphism in the massive sector. We introduce a mechanism for scattering that includes decays as particular resonant scattering. Finally, we complete the model by merging the local ${\mathfrak{sg}}_{\mathsf{u}}$ into a vertex-type algebra. Full article