# Higher Deformations of Lie Algebra Representations II

## TL;DR

This paper extends Steinberg's tensor product theorem to deformed algebraic structures called higher reduced enveloping algebras, enabling new insights into their modules and representation theory.

## Contribution

It demonstrates that Steinberg decomposition can be deformed for higher Frobenius kernels, linking their representation theory to that of reduced enveloping algebras.

## Key findings

- Deformation of Steinberg decomposition established.
- Structural results obtained for modules over deformed algebras.
- Results hold without reductivity assumptions.

## Abstract

Steinberg's tensor product theorem shows that for semisimple algebraic groups the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper we prove that Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.