Basic Classical Lie Superalgebras Research Articles

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

Bounded weight modules for basic classical Lie superalgebras at infinity

Bounded weight modules for basic classical Lie superalgebras at infinity

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.

The Modular Representations of Basic Classical Restricted Lie Superalgebras

The Modular Representations of Basic Classical Restricted Lie Superalgebras

Centralizers of Nilpotent Elements in Basic Classical Lie Superalgebras in Good Characteristic

AbstractLet $$\mathfrak {g}=\mathfrak {g}_{\bar{0}}\oplus \mathfrak {g}_{\bar{1}}$$ g = g 0 ¯ ⊕ g 1 ¯ be a basic classical Lie superalgebra over an algebraically closed field $$\mathbb {K}$$ K whose characteristic $$p>0$$ p > 0 is a good prime for $$\mathfrak {g}$$ g . Let $$G_{\bar{0}}$$ G 0 ¯ be the reductive algebraic group over $$\mathbb {K}$$ K such that $$\textrm{Lie}(G_{\bar{0}})=\mathfrak {g}_{\bar{0}}$$ Lie ( G 0 ¯ ) = g 0 ¯ . Suppose $$e\in \mathfrak {g}_{\bar{0}}$$ e ∈ g 0 ¯ is nilpotent. Write $$\mathfrak {g}^{e}$$ g e for the centralizer of e in $$\mathfrak {g}$$ g and $$\mathfrak {z}(\mathfrak {g}^{e})$$ z ( g e ) for the centre of $$\mathfrak {g}^{e}$$ g e . We calculate a basis for $$\mathfrak {g}^{e}$$ g e and $$\mathfrak {z}(\mathfrak {g}^{e})$$ z ( g e ) by using associated cocharacters $$\tau :\mathbb {K}^{\times }\rightarrow G_{\bar{0}}$$ τ : K × → G 0 ¯ of e. In addition, we give the classification of e which are reachable, strongly reachable or satisfy the Panyushev property for exceptional Lie superalgebras $$D(2,1;\alpha )$$ D ( 2 , 1 ; α ) , G(3) and F(4).

Ghost Distributions on Supersymmetric Spaces II: Basic Classical Superalgebras

Abstract We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both $({\mathfrak{g}},{\mathfrak{k}})$ and $({\mathfrak{g}},{\mathfrak{k}}^{\prime})$ admit Iwasawa decompositions. For such pairs, we define a ghost algebra, generalizing the subalgebra of ${\mathcal{U}}{\mathfrak{g}}$ defined by Gorelik. We realize this algebra as an algebra of $G$-equivariant operators on the supersymmetric space itself, and for certain pairs, the “special” ones, we realize our operators as twisted-equivariant differential operators on $G/K$. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when $({\mathfrak{g}},{\mathfrak{k}})$ is interlaced.

Reachable elements in basic classical Lie superalgebras

Let g=g0¯⊕g1¯ be a basic classical Lie superalgebra over C, e∈g0¯ a nilpotent element and ge the centralizer of e in g. We study various properties of nilpotent elements in g, which have previously only been considered in the case of Lie algebras. In particular, we prove that e is reachable if and only if e satisfies the Panyushev property for g=sl(m|n), m≠n or psl(n|n) and osp(m|2n). For exceptional Lie superalgebras g=D(2,1;α), G(3), F(4), we give the classification of e which are reachable, strongly reachable or satisfy the Panyushev property. In addition, we give bases for ge and its centre z(ge) for g=psl(n|n), which completes results of Han on the relationship between dim⁡ge, dim⁡z(ge) and the labelled Dynkin diagrams for all basic classical Lie superalgebras.

Representations of affine Lie superalgebras and their quantization in type A

We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra Uqslˆ(m|n).

Tensor product representations

In this paper, we study representations of a direct sum of Lie superalgebras. We deal with the question of whether the irreducible representations of a finite direct sum of Lie superalgebras are obtained in case representations of each constituent are available. We give an affirmative answer to this question for finite weight representations of a class of Lie superalgebras containing basic classical simple Lie superalgebras as well as affine Lie (super)algebras.

On Kostant root systems of Lie superalgebras

We study the eigenspace decomposition of a basic classical Lie superalgebra under the adjoint action of a toral subalgebra, thus extending results of Kostant. In recognition of Kostant's contribution we refer to the eigenfunctions appearing in the decomposition as Kostant roots. We then prove that Kostant root systems inherit the main properties of classical root systems. Our approach is combinatorial in nature and utilizes certain graphs naturally associated with Kostant root systems. In particular, we reprove Kostant's results without making use of the Killing form.

Typical irreducible characters of generalized quantum groups

We introduce the definition of the typical irreducible modules of the generalized quantum groups, and prove the Weyl–Kac-type formulas of their characters. As a by-product, we obtain the Weyl–Kac-type character formulas of the typical irreducible modules of the quantum superalgebras associated with the basic classical Lie superalgebras, which is explained in Introduction.

SIMPLE BOUNDED HIGHEST WEIGHT MODULES OF BASIC CLASSICAL LIE SUPERALGEBRAS

We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $$ \mathfrak{g} $$ . In particular, our result leads to the classification of the simple weight modules with finite weight multiplicities over all classical Lie superalgebras. We also obtain some character formulas of strongly typical bounded highest weight modules of $$ \mathfrak{g} $$ .

Conformal embeddings in affine vertex superalgebras

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra Vk(g) where g=g0¯⊕g1¯ is a basic classical simple Lie superalgebra. Let Vk(g0¯) be the subalgebra of Vk(g) generated by g0¯. We first classify all levels k for which the embedding Vk(g0¯) in Vk(g) is conformal. Next we prove that, for a large family of such conformal levels, Vk(g) is a completely reducible Vk(g0¯)–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of V−2(osp(2n+8|2n)) as a finite, non simple current extension of V−2(Dn+4)⊗V1(Cn). This decomposition uses our previous work [10] on the representation theory of V−2(Dn+4).We also study conformal embeddings gl(n|m)↪sl(n+1|m) and in most cases we obtain decomposition rules.

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.

Twisting functors and generalized Verma modules

Let $\mathfrak{g}$ be a reductive Lie algebra. We give a condition that ensures that the character of a generalized Verma module is well-behaved under a twisting functor. We show that a similar result holds for basic classical simple Lie superalgebras, and generalize a result from \cite{CM} about twisting Verma modules.

The Primitive Spectrum and Category for the Periplectic Lie Superalgebra

AbstractWe solve two problems in representation theory for the periplectic Lie superalgebra$\mathfrak{p}\mathfrak{e}(n)$, namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category ${\mathcal{O}}$into indecomposable blocks.To solve the first problem, we establish a new type of equivalence between category ${\mathcal{O}}$for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.

2-Local Automorphisms on Basic Classical Lie Superalgebras

Let G be a basic classical Lie superalgebra except A(n, n) and D(2, 1, α) over the complex number field ℂ. Using existence of a non-degenerate invariant bilinear form and root space decomposition, we prove that every 2-local automorphism on G is an automorphism. Furthermore, we give an example of a 2-local automorphism which is not an automorphism on a subalgebra of Lie superalgebra spl(3, 3).

Lie Superalgebras and Calogero–Moser–Sutherland Systems

We review recent results obtained at the intersection of the theory of quantum deformed Calogero–Moser–Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible deformations of root systems of basic classical Lie superalgebras. For classical series, we prove the existence of Lax pairs. Connections between infinite-dimensional Calogero–Moser–Sutherland systems, deformed quantum CMS systems, and representation theory of Lie superalgebras are discussed.

Jantzen filtration and strong linkage principle for modular Lie superalgebras

Abstract In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and obtain a sum formula in the corresponding Grothendieck groups. Consequently, we formulate a strong linkage principle for such categories.

The Core of an Extended Affine Lie Superalgebra (A Characterization)

We characterize a class of root graded Lie superalgebras containing a locally finite basic classical simple Lie superalgebra. As a by-product, we characterize the core and the core modulo the center of an extended affine Lie superalgebra whose root system is of finite type X \neq A(m,n), B(m,n), BC(m,n),C(m,n).