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.

Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra b gl∞|∞ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra b gl∞ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for the...

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuper-algebra. Then we classify the cuspidal Levi sub...

In [Zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l + 1) and the Lie superalgebra osp(1, 2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebrasà la Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie supera...

Soon after parastatistics has been introduced [1], it was discovered that it has a deep algebraic structure. It turned out that any n pairs of parafermion operators generate the simple Lie algebra so(2n + 1) [2, 3], and n pairs of paraboson creation and annihilation operators b 1 , . . . , bn generate a Lie superalgebra [4], isomorphic to one of the basic classical Lie superalgebras in the clas...

In this work, differential geometry of the Z3-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained. E-mail: [email protected]

Let H be a Heisenberg superalgebra. In this paper, the definition of biderivations and the properties of biderivations on Lie superalgebras are introduced. And some properties of biderivations on Heisenberg superalgebras are introduced by the definition of Heisenberg superalge-bras.

Using colored superanalysis and ε-Lie superalgebras, we build the minimal Poincaré superalgebra in the case of the Z n grading. We then build a representation of this algebra, and the corresponding Poincaré supergroup.