The Lie algebra structure of the $HH^1$ of the blocks of the sporadic Mathieu groups

Abstract

Let G be a sporadic Mathieu group and k an algebraically closed field of prime characteristic p, dividing the order of G. In this paper we describe some of the Lie algebra structure of the first Hochschild cohomology groups of the p-blocks of kG. In particular, letting B denote a p-block of kG, we calculate the dimension of HH 1 (B) and in the majority of cases we determine whether HH 1 (B) is a solvable Lie algebra.

The nonvanishing first Hochschild cohomology of twisted finite simple group algebras

“…Our motivation to study the more general problem of the nonvanishing of HH 1 (B) comes in part from a wider investigation into the links between the Lie algebra structure of HH 1 (B) and the k-algebra structure of B, examining how each influences the other. In particular, the Lie algebra structure is also expected to provide information useful to the Auslander-Reiten conjecture (see [2,4,5,7,14,15,16,17] for more examples of this).…”

Hochschild cohomology of symmetric groups and generating functions,II

“…It is an open question [6,Question 7.4] whether for G a finite group and B a block of kG, if the defect groups of B are non-trivial, then HH 1 (B) is non-zero. This question has been shown to have a positive answer in some cases in [8] and [7], for instance. We prove that this question has an affirmative answer if G is a symmetric group S n on n letters.…”