Integration of Modules I: Stability

TL;DR

This paper investigates the conditions under which modules over Lie algebras in positive characteristic can be extended to algebraic group actions, focusing on cohomological obstructions and integrability of certain modules.

Contribution

It introduces a cohomological framework to analyze stability and proves integrability of bricks for semisimple algebraic groups in positive characteristic.

Findings

Cohomological obstructions to algebraic group actions identified

Proved integrability of bricks for semisimple algebraic groups

Established criteria for stability under group twists

Abstract

We explore the integration of representations from a Lie algebra to its algebraic group in positive characteristic. An integrable module is stable under the twists by group elements. Our aim is to investigate cohomological obstructions for passing from stability to an algebraic group action. As an application, we prove integrability of bricks for a semisimple algebraic group.