The Lie algebra structure of $\mathrm{HH}^1(B)$ for blocks with a normal defect group

Abstract

It is an open question whether the Lie algebra $\mathrm{HH}^1(B)$ of a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ is simple only if the block is nilpotent with an elementary abelian defect group of order at least $3$. We show this to be the case if $B$ has a normal abelian defect group. Another open question is whether $\mathrm{HH}^1(kS_n)$ is a solvable Lie algebra, where $S_n$ denotes a symmetric group on $n$ letters. We give some general solvability criteria for $\mathrm{HH}^1(B)$ of blocks $B$ with a normal defect group, and, making use of results of Chuang-Kessar and Chuang-Rouquier, we deduce that if $B$ is a block of a symmetric group algebra with an abelian defect group, then $\mathrm{HH}^1(B)$ is solvable.