On the Hochschild Cohomology for Frobenius Kernels

TL;DR

This paper analyzes the Hochschild cohomology structure of Frobenius kernels, especially for G=SL_2, using spectral sequences to provide a detailed algebraic description.

Contribution

It introduces a method to compute Hochschild cohomology of Frobenius kernels via spectral sequences, with explicit results for G=SL_2.

Findings

Complete description of the Hochschild cohomology algebra for G_1 where G=SL_2

Development of spectral sequence techniques for cohomology calculations

Insights into the adjoint action on restricted enveloping algebras

Abstract

In this paper the authors investigate the structure of the Hochschild cohomology for Frobenius kernels. The authors first establish some fundamental constructions to compute Hochschild cohomology by using spectral sequences. This enables us to provide a complete description of the $G$-algebra structure of the Hochschild cohomology for the first Frobenius kernel $G_{1}$ where $G=SL_{2}$. This computation heavily relies on the calculation of the adjoint action on the restricted enveloping algebra.