ALGB Seminar: Lleonard Rubio y Degrassi (University of Leicester)
Title: On the Lie structure of $HH^1$ of blocks with one simple module and of tame blocks
Every group algebra $kG$ of a finite group $G$ is decomposed as a direct sum of two-sided ideals called blocks. In modular representation theory to each block we assign a $p$-subgroup of $G$ called defect group which measures how far the block is from being semisimple. The block decomposition allow us also to study blocks depending on the number of simple $kG$-modules. However, even in the case of blocks with one simple module, the situation is poorly understood.
Blocks are usually classified by Morita or derived equivalence or by stable equivalence of Morita type. Stable equivalences of Morita type are the most general and frequent in modular representation theory but also the least understood. The main reason is that we do not know if many of the invariants for Morita and derived equivalences are still invariants for stable equivalences of Morita type. In this context, Hochschild cohomology records crucial information about a block $B$: its first degree component, denoted by $HH^1(B)$ is a Lie algebra and it is invariant under stable equivalences of Morita type.
In this talk I will show how the Lie structure of $HH^1$ helps to understand blocks with one simple module up to stable equivalences of Morita type. More precisely, I will prove that for a block $B$ with one simple module, $HH^1(B)$ is a simple Lie algebra if and only if $B$ is nilpotent (Morita equivalent to the group algebra of its defect group) with an elementary abelian defect group $P$ of order at least $3$. In addition, I will show that if $B$ is a block with tame representation type, then $HH^1(B)$ is a solvable Lie algebra. The latter result provides a positive answer for tame blocks of a recent conjecture of Schroll and Solotar.