Lie Superalgebra Research Articles

Codimensions of identities of solvable Lie superalgebras

We study identities of Lie superalgebras over a field of characteristic zero. We construct a series of examples of finite-dimensional solvable Lie superalgebras with non-nilpotent commutator subalgebra for which the PI-exponent of codimension growth exists and is an integer number.

Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs

Abstract We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over ${\mathbb {C}}$ with a triangular decomposition and a character $\zeta $ of the nilpotent radical, the associated Backelin functor $\Gamma _\zeta $ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category ${\mathcal {O}}$ . In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $\Gamma _\zeta $ and its target category, respectively, categorify a q-symmetrizing map and the corresponding q-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type $AIII$ .

Lie superalgebras based on ᵓ0ᵒ9(2,ᵓd)

Lie superalgebras based on ᵓ0ᵒ9(2,ᵓd)

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let M be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra G, where G is gl(m|n), osp(2m|2n) or spo(2m|2n). We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for G on M can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for G are diagonalizable on the space spanned by singular vectors of M and hence on M. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra so(2n) and the symplectic Lie algebra sp(2n), on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

Topological semiinfinite tensor (super)modules

We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl(V⊕ΠV) and osp(V⊕ΠV) associated with a Tate space V. Here V⊕ΠV is a Z/2Z-graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories Tˆgl(V), Tˆo(V) and Tˆsp(V), and show that these categories are anti-equivalent to respective previously studied categories Tgl(V), To(V), Tsp(V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to Tˆgl(V), Tˆo(V) and Tˆsp(V). Moreover, the categories To(V) and Tsp(V) are known to be equivalent, and this implies the equivalence of the categories Tˆo(V) and Tˆsp(V). After introducing a supersymmetric setting, we establish the equivalence of the category Tˆgl(V) with the category Tˆgl(V⊕ΠV), and the equivalence of both categories Tˆo(V) and Tˆsp(V) with Tˆosp(V⊕ΠV).

S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

This paper performed an investigation into the s-embedding of the Lie superalgebra (→S1∣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 ( SψD⊙) residing on the super-circle S1|1. We also introduce and rigorously define the central charge within the framework of (→S1∣1), leveraging the canonical central extension of SψD⊙. 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 (→S1|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.

Hom-Lie Superalgebras in Characteristic 2

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 αk-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.

The Weyl groupoids of [formula omitted] and [formula omitted

We provide a convenient formulation of the definition of Cartan graphs and Weyl groupoids introduced by Heckenberger and Schneider, and construct Cartan graphs for regular symmetrizable contragredient Lie superalgebras. For sl(m|n), osp(2m+1|2n) and osp(2m|2n) we explicitly describe the Cartan graph in terms of partitions and determine the relations between the generators of their Weyl groupoids.

Irreducible modules of modular Lie superalgebras and super version of the first Kac–Weisfeiler conjecture

Abstract Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p>2$ . In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$ , as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

A Note on Finite Dimensional Odd Contact Lie Superalgebra in Prime Characteristic

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

Formula omitted]-Stirling numbers in type [formula omitted

Stirling numbers of the first and second kind, which respectively count permutations in the symmetric group and partitions of a set, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of the second kind have recently appeared in conjectures of Zabrocki and of Swanson–Wallach concerning the Hilbert series of certain super coinvariant algebras. We provide conjectural bases for these algebras and show that they have the correct Hilbert series.

ℤ2 × ℤ2-graded Lie algebras and supseralgebras and related physical models

In this paper we construct physical systems with ℤ2×ℤ2-graded symmetries. There are two different structures:ℤ2×ℤ2-graded Lie algebras andℤ2×ℤ2-graded Lie superalgebras. Physical models with the latter symmetries can be seen as generalizations of supersymmetric models. The systems described by ℤ2 × ℤ2-graded Lie algebras have not been investigated yet (up to our knowledge). We present examples of physical models invariant under each one of both kinds of graded structures.

Super-skewsymmetric super-biderivations on the N = 1 super Heisenberg-Virasoro algebra

In this paper, the super-skewsymmetric super-biderivations on the N = 1 super Heisenberg-Virasoro algebra are completely determined. Furthermore, we find that every super-skewsymmetric super-biderivation of the N = 1 super Heisenberg-Virasoro algebra is inner.

The First Cohomology of o s p ( 1,2 ) with Coefficients in Baby Verma Modules and Simple Modules

Over a field of characteristic [Formula: see text], the first cohomology of the orthogonal symplectic Lie superalgebra [Formula: see text] with coefficients in baby Verma modules and simple modules is determined by use of the weight space decompositions of these modules relative to a Cartan subalgebra of [Formula: see text]. As a byproduct, the first cohomology of [Formula: see text] with coefficients in the restricted enveloping algebra (under the adjoint action) is not trivial.

Left-symmetric superalgebra structures on the super Galilean conformal algebra

The compatible left-symmetric superalgebra structures on the super Galilean conformal algebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric superalgebra structures on the infinite-dimensional Lie superalgebra g studied by Chen and Hassan play an essential role in determining these compatible structures.

A Lie conformal superalgebra and duality of representations for E(4,4)

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4,4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn't known. This is achieved by constructing a Lie conformal superalgebra of type (4,4) for which E(4,4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4,4) by vector fields on a (4|4)-dimensional supermanifold.

Construction of matryoshka nested indecomposable N-replications of Kac-modules of quasi-reductive Lie superalgebras, including the sl(m/n) and osp(2/2n) series

We construct a new class of finite dimensional indecomposable representations of simple superalgebras which may explain, in a natural way, the existence of the heavier elementary particles. In type I Lie superalgebras sl(m/n) and osp(2/2n) , one of the Dynkin weights labeling the finite dimensional irreducible representations is continuous. Taking the derivative, we show how to construct indecomposable representations recursively embedding N copies of the original irreducible representation, coupled by generalized Cabibbo angles, as observed among the three generations of leptons and quarks of the standard model. The construction is then generalized in the appendix to quasi-reductive Lie superalgebras.

Super-biderivations and linear super-commuting maps on a super Virasoro type algebras

Let [Formula: see text] signify the super-Virasoro type algebra over the field of the complex numbers [Formula: see text]. This paper mainly shows that any super-skewsymmetric super-biderivations of the centerless super-Virasoro type algebras [Formula: see text] are inner super-biderivations. Then, we will show that any linear super commuting maps on the centerless super-Virasoro type algebras [Formula: see text] are standard.

Characterization of contact Lie superalgebras

We present the concept of finite-dimensional complex homogeneous contact Lie superalgebra. The Z2-graded homogeneous cases are studied in detail producing some relevant examples. We characterize homogeneous contact Lie superalgebras in terms of their Berezinian and their structure matrix. These Lie superalgebras are also characterized by means of deformation theory, and as an application we obtain the complete classification of low dimensional Lie superalgebras of this type.

A conjecture of Dekking on the dimensions of the lower central series factors of a certain just infinite Lie algebra

Revisiting just infinite nil graded fractal Lie superalgebras studied by de Morais Costa and Petrogradsky, and more precisely a Lie algebra R in characteristic 2, we give a simple combinatorial construction of the sequence of dimensions of the lower central series factors of R in terms of automatic and morphic sequences. In particular we prove a conjecture of Dekking about this sequence of dimensions.