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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.

Publication Type:	Article
Additional Information:	This is a pre-copyedited, author-produced version of an article accepted for publication in Quarterly Journal of Mathematics following peer review. The version of record Linckelmann, M. and Degrassi, L. R. Y. (2018). Block algebras with HH1 a simple Lie algebra. Quarterly Journal of Mathematics, 69(4), pp. 1123-1128 is available online at: https://doi.org/10.1093/qmath/hay017.
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics
SWORD Depositor:	Symplectic Administrator

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