Block algebras with HH1 a simple Lie algebra

Abstract

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 Jacobson-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.2010 Mathematics Subject Classification. 16E40, 16G30, 16D90.

On the Lie algebra structure of 𝐻𝐻¹(𝐴) of a finite-dimensional algebra 𝐴

“…Now f preserves the submodules AjAi in this sum, thanks to Lemma 2.5. The result follows.The following observations are variations of the statements in[6, Proposition 3.5].Proposition 2.7. Let A be a split finite-dimensional k-algebra, and let E be a separable subalgebra of A such that A = E ⊕ J(A).…”

Maximal tori in $HH^1$ and the fundamental group

“…sl 2 , Witt, Jacobson-Witt, and solvable Lie algebras [10]), we can deduce from Theorem 5.1 that the number of nonisomorphic simple u(HH 1 (A), χ)-modules is bounded by β(Q A ). These families of Lie algebras have been studied in various articles, for example [27,42,43,50]. In particular, if A is a semimonomial algebra such that HH 1 (A) is a solvable Lie algebra, then the projective cover of the trivial irreducible module of HH 1 (A) is induced from the one dimensional trivial module of a maximal torus (see [31]) and by Theorem 5.1 we have p β(QA) = p mt-rank HH 1 (A) .…”

“…The first Hochschild cohomology Lie algebra HH 1 (A), along with all of the Lie theoretic invariants which may derived from it, is an important object attached to any finite dimensional algebra A, and this structure has been studied intensely in recent years; see for example [2,12,21,27,42,50,53,54]. In particular, authors have been interested in properties such as solvability and nilpotence of HH 1 (A), for specific classes of algebras.…”

On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation type

“…For the reduced group scheme, the first Hochschild cohomology of the block algebra has been studied in [18]. They showed that the only restricted simple Lie algebra occurs as H 1 (B, B) of some block algebra B of a finite group is the Jocobson-Witt algebra.…”

“…We show in Section 2 that the Lie algebra H 1 (kU, kU) associated with a unipotent group scheme is a simple Lie algebra if and only if U is elementary abelian. This can be thought of as a generalization of [18,Proposition 2.5]. Section 3 is concerned with the restricted Lie algebra structure of H 1 (B 0 (G), B 0 (G)) for the infinitesimal group scheme of finite representation type.…”