Lie Superalgebra Research Articles

Central extensions, derivations, and automorphisms of the super Heisenberg-Virasoro algebra

In this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group.

Smooth modules of the super [formula omitted]-algebra [formula omitted] of Neveu-Schwarz type

In this paper, we explore the super W-algebra SW(32,32) of Neveu-Schwarz type denoted by L, which serves as a supersymmetric conformal algebra. Our investigation focuses on the analysis of simple smooth modules including the Whittaker modules and the highest weight modules over L. Specifically, we utilize simple modules from finite-dimensional solvable Lie superalgebras to construct a multitude of simple smooth modules over L. Additionally, we provide several equivalent descriptions for simple smooth modules over L.

Representations of the Restricted Lie Superalgebra p(n)

Representations of the Restricted Lie Superalgebra p(n)

Some results on anti-pre-Lie superalgebras and admissible Novikov superalgebras

In this paper, we extend the notion of anti-pre-Lie algebras to the ℤ2-graded version, and introduce the notion of anti-pre-Lie superalgebras. They can be characterized as a class of Lie-admissible superalgebras that satisfy its negative left multiplication operators are the representations of the corresponding sub-adjacent Lie superalgebras. And we give the classification of 2-dimensional anti-pre-Lie superalgebras. We introduce the notion of anti-super -operators on Lie superalgebras to explore the relationships between anti-super -operators and anti-pre-Lie superalgebras. We show that nondegenerate super-commutative 2-cocycles on Lie superalgebras can obtain a class of compatible anti-pre-Lie superalgebra structures. In addition, we introduce a subclass of anti-pre-Lie superalgebras, namely admissible Novikov superalgebras, which correspond to Novikov superalgebras through q-superalgebras. Finally, we introduce the notions of anti-pre-Lie Poisson superalgebras and admissible Novikov-Poisson superalgebras, extending the correspondence to the level of Poisson type structures, the correspondence of the Novikov-Poisson superalgebras and the admissible Novikov-Poisson superalgebras are realized through admissible pairs.

The first cohomology of lie superalgebra P̃(2) with coefficients in kac modules

Over a field of characteristic p > 2, firstly, the structure of Kac modules of Lie superalgebra P ˜ ( 2 ) and the weight space decompositions are given. Secondly, the weight-derivations of P ˜ ( 2 ) to its Kac modules are computed. Finally, the first cohomology of P ˜ ( 2 ) with coefficients in Kac modules is determined.

A Fock space approach to socle series of Grothendieck group over Lie superalgebra o s p ( 2 | 2 )

We explore two modules F and K over a subalgebra U of the quantum group U q ( g l ( ∞ ) ) . The first U -module is the Fock space related to the natural module over the quantum group. If we let O + to be the category of finite-dimensional o s p ( 2 | 2 ) -modules of integral weights, the second U -module K is the vector space obtained by tensoring the Grothendieck group K ( O + ) by the field Q . Using thick tensor ideals and a U -isomorphism, we compute the socles of F and K .

Representations of the super-Yangian of type B(n,m)

We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras osp2n+1|2m. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case n=1 and arbitrary m⩾1.

Integrable Systems on Rectangular W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Superalgebras via Super Adler-Type Operators

In this paper, we introduce a class of super Adler-type operators associated with the Lie superalgebra gl(m|n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {gl}(m|n)$$\\end{document}. We show that these operators generate Poisson vertex superalgebras which are isomorphic to the classical W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-superalgebras associated with gl(m|n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {gl}(m|n)$$\\end{document} and some rectangular nilpotent elements. We use this isomorphism to construct integrable hierarchies on these rectangular W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-superalgebras.

Harmonic analysis of compact Lie supergroups

We realize the irreducible representations of a compact Lie supergroup G, with a contragredient simple Lie superalgebra, in the space of square integrable (in the sense of Berezin) holomorphic sections on X=GA, A is the real torus in the complexification of G. We give an explicit realization of unitary representations when G=SU(1|1).

Odd annular Bar-Natan category and 𝔤𝔩(1|1)

In this paper, we introduce two monoidal supercategories: the odd dotted Temperley–Lieb category [Formula: see text], which is a generalization of the odd Temperley–Lieb category studied by Brundan and Ellis in [Monoidal Supercategories, Commun. Math. Phys. 351 (2017) 1045–1089], and the odd annular Bar-Natan category [Formula: see text], which generalizes the odd Bar-Natan category studied by Putyra in [A [Formula: see text]-category of chronological cobordisms and odd Khovanov homology, Banach Center Publ. 103 (2014) 291–355]. We then show there is an equivalence of categories between them if [Formula: see text]. We use this equivalence to better understand the action of the Lie superalgebra [Formula: see text] on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and the second author in [An action of [Formula: see text] on odd annular Khovanov homology, Math. Res. Lett. 27(3) (2020) 711–742].

Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Simple Harish-Chandra modules over the superconformal current algebra

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra gˆ, which is the semi-direct sum of the N=1 superconformal algebra with the affine Lie superalgebra g˙⊗A⊕CC1, where g˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N=1 Heisenberg-Virasoro algebra.

The centre of the Dunkl total angular momentum algebra

For a finite dimensional representation V of a finite reflection group W, we consider the rational Cherednik algebra Ht,c(V,W) associated with (V,W) at the parameters t≠0 and c. The Dunkl total angular momentum algebra Ot,c(V,W) arises as the centraliser algebra of the Lie superalgebra osp(1|2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of Ht,c(V,W) and the Clifford algebra generated by V.We show that when dim⁡V≥3 and for every value of the parameter c, the centre of Ot,c(V,W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (−1)V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into Ot,c(V,W), we establish results analogous to “Vogan's conjecture” for a family of operators depending on suitable elements of the double cover W˜.

On the Infinite-Dimensional Modular Lie Superalgebras of O -Type

The natural filtrations of infinite-dimensional Lie superalgebras of [Formula: see text]-type over a field of positive characteristic are proved to be invariant under automorphisms by studying ad-nilpotent elements. We observe that two Lie superalgebras of [Formula: see text]-type are isomorphic if and only if the integers in the definition of these Lie superalgebras are intrinsic. Thereby, all the Lie superalgebras of [Formula: see text]-type are classified in the sense of isomorphism.

An Elliptic Solution of the Classical Yang–Baxter Equation Associated with the Queer Lie Superalgebra

A solution of the classical Yang–Baxter equation associated with the queer Lie superalgebra is constructed in terms of Hermite theta functions.

The centralizer of~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra over fields of prime characteristic

This paper considers centralizers of the Lie superalgebra~$\frak{sl}(0,n)$ over prime characteristic fields. Using homological methods, the centralizers of the even and odd parts of ~$\frak{sl}(0,n)$ in the generalized Witt Lie superalgebra are calculated and a summary of their structural properties is provided.

It takes two spectral sequences

We study the representation theory of the Lie superalgebra gl(1|1), constructing two spectral sequences which eventually annihilate precisely the superdimension 0 indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on Repgl(1|1). These two spectral sequences are related by contragredient duality, and from their limits we construct explicit semisimplification functors, which we explicitly prove are isomorphic up to a twist. We use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to gl(1|1)-subalgebras.

The description of solvable Lie superalgebras of maximal rank

In the paper we give some basic properties of the superderivations of Lie superalgebras. Under certain condition, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to the dimensions of complementary subspaces to the nilradicals. Moreover, under the condition that an analogue of Lie's theorem is true, the description of solvable Lie superalgebras of maximal rank is obtained. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank under the mentioned condition is isomorphic to the maximal solvable extension of nilradical of maximal rank. Finally, along with effective method of construction of solvable Lie superalgebras of maximal rank we present the description of special type of maximal solvable extension of nilpotent Lie superalgebras.

Howe Duality of Type P

AbstractWe establish classical and categorical Howe dualities between the Lie superalgebras $$\mathfrak {p}(m)$$ p ( m ) and $$\mathfrak {p}(n)$$ p ( n ) , for $$m,n \ge 1$$ m , n ≥ 1 . We also describe a presentation via generators and relations as well as a Kostant $$\mathbb {Z}$$ Z -form for the universal enveloping superalgebra $$U(\mathfrak {p}(m))$$ U ( p ( m ) ) .

The refined solution to the Capelli eigenvalue problem for [formula omitted] and [formula omitted]

Let g be either the Lie superalgebra gl(V)⊕gl(V) where V≔ℂm|n or the Lie superalgebra gl(V) where V≔ℂm|2n. Furthermore, let W be the g-module defined by W≔V⊗V∗ in the former case and W≔S2(V) in the latter case. Associated to (g,W) there exists a distinguished basis of Capelli operators{Dλ}λ∈Ω, naturally indexed by a set of hook partitions Ω, for the subalgebra of g-invariants in the superalgebra PD(W) of superdifferential operators on W.Let b be a Borel subalgebra of g. We compute eigenvalues of the Dλ on the irreducible g-submodules of P(W) and obtain them explicitly as the evaluation of the interpolation super Jack polynomials of Sergeev–Veselov at suitable affine functions of the b-highest weight. While the former case is straightforward, the latter is significantly more complex. This generalizes a result by Sahi, Salmasian and Serganova for these cases, where such formulas were given for a fixed choice of Borel subalgebra.