Block algebras with \(HH^1\) a simple Lie algebra. (English) Zbl 1417.20001
Summary: 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 \(HH^1(B)\) 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 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.
MSC:
20C20 | Modular representations and characters |
20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |
20J05 | Homological methods in group theory |
16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
17B20 | Simple, semisimple, reductive (super)algebras |