On hochschild cohomology and modular representation theory

Abstract

The aim of this thesis is to study local and global invariants in representation theory of finite groups using the (restricted) Lie algebra structure of the first degree of Hochschild cohomology of a block algebra B as a main tool. This lead to two directions: In the first part we investigate the global approach. In particular, we prove the compatibility of the p-power map under stable equivalence of Morita type of subclasses of the first Hochschild cohomology represented by integrable derivations. Further results in this aspect include an example showing that the p-power map cannot generally be expressed in terms of the BV operator. We also study some properties of r-integrable derivations and we provide a family of examples given by the quantum complete intersections where all the derivations are r-integrable. In the second part our attention is focused on the local invariants. More precisely, we fully characterise blocks B with unique isomorphism class of simple modules such that the first degree Hochschild cohomology HH1(B) is a simple as Lie algebra. In this case we prove that B is a nilpotent block with an elementary abelian defect group P of order at least 3 and HH1(B) is isomorphic to the Witt algebra HH1(kP)