Higher Deformations of Lie Algebra Representations I

TL;DR

This paper extends the study of deformations of distribution algebras from the first Frobenius kernel to higher kernels, providing new insights into Lie algebra representations in positive characteristic.

Contribution

It develops an analogue of deformed distribution algebras for higher Frobenius kernels, addressing a longstanding open question in the field.

Findings

Constructed deformed distribution algebras for higher Frobenius kernels

Analyzed their representation theory for the special linear group

Provided new tools for understanding Lie algebra representations in positive characteristic

Abstract

In the late 1980s, Friedlander and Parshall studied the representations of a family of algebras which were obtained as deformations of the distribution algebra of the first Frobenius kernel of an algebraic group. The representation theory of these algebras tells us much about the representation theory of Lie algebras in positive characteristic. We develop an analogue of this family of algebras for the distribution algebras of the higher Frobenius kernels, answering a 30 year old question posed by Friedlander and Parshall. We also examine their representation theory in the case of the special linear group.