Block algebras with HH1 a simple Lie algebra

Abstract

The purpose of this note is to add to the evidence that the algebra structure of a p-block of a finite group is closely related to the Lie algebra structure of its first Hochschild cohomology group. 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 HH1(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 HH1(B) is in that case isomorphic to the Witt algebra HH1(kP)⁠. In particular, no other simple modular Lie algebras arise as HH1(B) of a block B with a single isomorphism class of simple modules.