Block algebras with HH1 a simple Lie algebra
Markus Linckelmann, Lleonard Rubio y Degrassi

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 is a block of a finite group algebra over an algebraically closed field of prime characteristic such that is a simple Lie algebra and such that has a unique isomorphism class of simple modules, then is nilpotent with an elementary abelian defect group of order at least , and is in that case isomorphic to the Jacobson-Witt algebra . In particular, no other simple modular Lie algebras arise as of a block with a single isomorphism class of simple modules.
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