Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

Keli Zheng,Yongzheng Zhang

doi:10.1515/math-2019-0117

Abstract

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

Highlights

Let F be an arbitrary eld of characteristic p >
In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of F
Let F be an arbitrary eld of characteristic p > and Z = { ̄, ̄} be the residue class ring mod

Summary

Introduction

In 1967, Rudakov and Shafarevich [1] described all the irreducible representations of sl( ) over an algebraically closed eld F of characteristic p >. They demonstrated that in addition to the p-representations known since 1930s, all of which possess a highest and lowest weight and are labeled by one integer, there are other representations that form a variety of dimension. They described the g-modules not possessing a p-structure for Lie algebras g with Cartan matrix. Since the classi cation of all the nite-dimensional simple complex Lie superalgebras was done by Kac [9], the problems of constructing a uni ed representation theory for all the types of simple Lie superalgebras

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Generalized Witt and special modular Lie superalgebras

Graded modules over modular Lie superalgebras