On Kac-Weisfeiler modules for general and special linear Lie superalgebras

TL;DR

This paper confirms the existence of Kac-Weisfeiler modules for general linear Lie superalgebras and extends results to special linear superalgebras under certain conditions, advancing understanding of their module structures.

Contribution

It verifies the existence of Kac-Weisfeiler modules for ig_{m|n} and extends the results to ig_{m|n} with specific restrictions, filling a gap in superalgebra representation theory.

Findings

Existence of Kac-Weisfeiler modules for ig_{m|n}

Extension of results to ig_{m|n} with restrictions on p

Implications for the structure of modules in superalgebras

Abstract

Let $\ggg:=\gl_{m|n}$ be a general linear Lie superalgebra over an algebraically closed field $\mathds{k}=\overline{\mathbb{F}}_p$ of characteristic $p>2$. A module of $\ggg$ is said to be of Kac-Weisfeiler if its dimension coincides with the dimensional lower bound in the super Kac-Weisfeiler property presented by Wang-Zhao in \cite{WZ}. In this paper, we verify the existence of the Kac-Weisfeiler modules for $\gl_{m|n}$. We also establish the corresponding consequence for the special linear Lie superalgebras $\mathfrak{sl}_{m|n}$ with restrictions $p>2$ and $p\nmid(m-n)$.