Search Results (16)

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

In supersymmetric extensions of the Standard Model, the observed particles come in fermion–boson pairs necessary for the realization of supersymmetry (SUSY). In spite of the expected abundance of super-partners for all the known particles, not a single supersymmetric pair has been reported to date. Although a hypothetical SUSY breaking mechanism, operating at high energy inaccessible to current experiments cannot be ruled out, this reduces SUSY’s predictive power and it is unclear whether SUSY, in its standard form, can help reducing the remaining puzzles of the standard model (SM). Here we argue that SUSY can be realized in a different way, connecting spacetime and internal bosonic symmetries, combining bosonic gauge fields and fermionic matter particles in a single gauge field, a Lie superalgebra-valued connection. In this unconventional representation, states do not come in SUSY pairs, avoiding the doubling of particles and fields and SUSY is not a fully off-shell invariance of the action. The resulting systems are remarkably simple, closely resembling a standard quantum field theory and SUSY still emerges as a contingent symmetry that depends on the features of the vacuum/ground state. We illustrate the general construction with two examples: (i) A 2 + 1 dimensional system based on the $osp(2,2|2)$ superalgebra, including Lorentz and $u\left(1\right)$ generators that describe graphene; (ii) a supersymmetric extension of 3 + 1 conformal gravity with an $SU(2,2|2)$ connection that describes a gauge theory with an emergent chiral symmetry breaking, coupled to gravity. The extensions to higher odd and even dimensions, as well as the extensions to accommodate more general internal symmetries are also outlined. Full article

We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of $(m,n)$ -block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras. Full article

In this paper, a supersymmetric extension of the minimal surface equation is formulated. Based on this formulation, a Lie superalgebra of infinitesimal symmetries of this equation is determined. A classification of the one-dimensional subalgebras is performed, which results in a list of 143 conjugacy classes with respect to action by the supergroup generated by the Lie superalgebra. The symmetry reduction method is used to obtain invariant solutions of the supersymmetric minimal surface equation. The classical minimal surface equation is also examined and its group-theoretical properties are compared with those of the supersymmetric version. Full article

In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of the subalgebras, the invariants have a non-standard structure. For nine selected subalgebras, we use the symmetry reduction method to find invariants, orbits and reduced systems. Through the solutions of these reduced systems, we obtain solutions of the supersymmetric Euler system. The obtained solutions include bumps, kinks, multiple wave solutions and solutions expressed in terms of arbitrary functions. Full article

An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z plays an important role in constructing the gravity duals for non-relativistic field theories. Supersymmetric extensions of symmetry algebras including the anisotropic scaling are very helpful to consider holographic relations accurately. We give a short summary on the classification of superalgebras with the anisotropic scaling as subalgebras of the following Lie superalgebras, psu(2,2|4), osp(8|4) and osp (8*|4), which appear in the study of AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra. Full article