We provide formulas for the Weyl-Kac denominator and superdenominator of a basic classical Lie superalgebra for a distinguished set of positive roots. Résumé. Nous donnons les formules pour les dénominateurs et super-dénominateurs de Weyl-Kac d’une superalgèbre de Lie basique classique pour un ensemble distingué de racines positives.

Let Am,n be the tensor product of Laurent polynomial algebra in m even variables and exterior n odd over complex field C, Witt superalgebra Wm,n Lie superderivations Am,n. In this paper, we classify simple weight modules with finite-dimensional spaces respect to standard Cartan subalgebra Wm,0. Every such module is either a quotient or highest type.

We analyse the Witten-Woronowicz's type deformations of the Lie su-peralgebra osp(2,2) and obtain a deformation parametrized by three independent parameters. For some of these algebras, finite dimensional representations are formulated in terms of a finite difference operator, providing operators that are relevant for the classification of quasi exactly solvable, finite difference equations. Si...

The five simple exceptional complex Lie superalgbras of vector fields are described. One of them is new; the other four are explicitely described for the first time. All of the exceptional Lie superalgebras are obtained with the help of the Cartan prolongation or a generalized prolongation. The description of several of the exceptional Lie superalgebras is associated with the Lie superalgebra a...

We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p > 2. A superalgebra generalization of the celebrated Kac-Weisfeiler Conjecture is formulated, which exhibits a mixture of p-power and 2-power divisibilities of dimensions of modules. We establish the Conjecture for basic classical Lie superalgebras.

1.1. In [PS1], I. Penkov and V. Serganova show that the category of representations of a basic classical Lie superalgebra g of type I with a fixed typical central character is equivalent to the category of representations of the even part g0 with a suitable central character. A similar result for type II was proven by I. Penkov in [P] for “generic” central characters. The aim of this paper is t...

We classify irreducible representations of finite W-algebra for the queer Lie superalgebra Q(n) associated with principal nilpotent coadjoint orbits. use this classification and our previous results to obtain a finite-dimensional super Yangian YQ(1).

0.1. Let g be a complex finite-dimensional contragredient Lie superalgebra. These algebras were classified by V. Kac in [K1] and the list (excluding Lie algebras) consists of four series: A(m|n), B(m|n), C(m), D(m|n) and the exceptional algebrasD(2, 1, a), F (4), G(3). The finite-dimensional contragredient Lie superalgebras with zero Killing form (or, equivalently, with dual Coxeter number equa...

In this paper we use the Etingof-Kazhdan quantization of Lie bisuperalgebras to investigate some interesting questions related to DrinfeldJimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie s...

A superconformal algebra is a simple complex Lie superalgebra g spanned by the coefficients of a finite family of pairwise local fields a(z) = ∑ n∈Z a(n)z , one of which is the Virasoro field L(z), [3, 8, 11]. Superconformal algebras play an important role in the string theory and conformal field theory. The Lie superalgebras K(N) of contact vector fields with Laurent polynomials as coefficient...