Representations of Lie superalgebras in prime characteristic III

TL;DR

This paper extends the deformation method to Lie superalgebras in prime characteristic, proving the Super Kac-Weisfeiler conjecture with improved assumptions and establishing a semisimplicity criterion for reduced enveloping superalgebras.

Contribution

It generalizes the deformation method to Lie superalgebras, providing a new proof of the Super Kac-Weisfeiler conjecture and a semisimplicity criterion for reduced enveloping superalgebras.

Findings

Proof of the Super Kac-Weisfeiler conjecture for basic classical Lie superalgebras.

Optimal improvement of the assumption on characteristic p.

Semisimplicity criterion for reduced enveloping superalgebras.

Abstract

For a restricted Lie superalgebra g over an algebraically closed field of characteristic p > 2, we generalize the deformation method of Premet and Skryabin to obtain results on the p-power and 2-power divisibility of dimensions of g-modules. In particular, we give a new proof of the Super Kac-Weisfeiler conjecture for basic classical Lie superalgebras. The new proof allows us to improve optimally the assumption on p. We also establish a semisimplicity criterion for the reduced enveloping superalgebras associated with semisimple p-characters for all basic classical Lie superalgebras using the technique of odd reflections.