Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

Abstract: Hochschild cohomology $\mathrm{HH}^*(A)$ of an associative (unital) $k$-algebra $A$ (here $k$ is a field) has a rich structure. First Hochschild cohomology $\mathrm{HH}^1(A)$ is isomorphic to the quotient of the space of $k$-linear derivations of $A$ modulo its inner derivations. In the context of modular representation theory, if the field $k$ has characteristic $p$ and $G$ is a finite group, an indecomposable direct algebra factor $B$ of the group algebra $kG$ is called block algebra. Is $\mathrm{HH}^1(B)$ nontrivial for any block algebra $B$ with nontrivial defect group? This is a question launched by Markus Linckelmann at the ICRA 2016. We explain the basic facts needed to understand this question. We give methods to investigate the nontriviality of the first Hochschild cohomology of some twisted group algebras. As a consequence we show that for some block algebras, with nontrivial defect groups, the first Hochschild cohomology is nontrivial.

group theoryrepresentation theory

Audience: general audience

Mimar Sinan University Mathematics Seminars