Representations of Lie Superalgebras in Prime Characteristic I

TL;DR

This paper develops the initial framework for understanding the representation theory of restricted Lie superalgebras over fields with characteristic p>2, including a new conjecture extending classical results.

Contribution

It formulates a superalgebra generalization of the Kac-Weisfeiler Conjecture and proves it for basic classical Lie superalgebras.

Findings

Established the superalgebra Kac-Weisfeiler Conjecture for basic classical Lie superalgebras.

Identified p-power and 2-power divisibility patterns in module dimensions.

Laid foundational work for further research in Lie superalgebra representations.

Abstract

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.