Block algebras with HH1 a simple Lie algebra

TL;DR

This paper characterizes blocks of finite group algebras with a single simple module where the first Hochschild cohomology is a simple Lie algebra, showing it must be a Jacobson-Witt algebra associated with an elementary abelian defect group.

Contribution

It proves that such blocks have a nilpotent structure with elementary abelian defect groups and that their Hochschild cohomology is isomorphic to a Jacobson-Witt algebra, excluding other simple Lie algebras.

Findings

Blocks with a single simple module have Hochschild cohomology isomorphic to Jacobson-Witt algebra.

Such blocks are nilpotent with elementary abelian defect groups of order at least 3.

No other simple modular Lie algebras occur as Hochschild cohomology of these blocks.

Abstract

We show that if $B$ is a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ such that $\HH^1(B)$ is a simple Lie algebra and such that $B$ has a unique isomorphism class of simple modules, then $B$ is nilpotent with an elementary abelian defect group $P$ of order at least $3$, and $\HH^1(B)$ is in that case isomorphic to the Jacobson-Witt algebra $\HH^1(kP)$. In particular, no other simple modular Lie algebras arise as $\HH^1(B)$ of a block $B$ with a single isomorphism class of simple modules.