Self-injective algebras and the second Hochschild cohomology group

TL;DR

This paper investigates the second Hochschild cohomology group of finite dimensional self-injective algebras of finite representation type, showing it is usually zero and providing explicit bases when non-zero.

Contribution

It determines the second Hochschild cohomology group for a class of self-injective algebras and identifies cases where this group is non-zero, providing explicit bases.

Findings

${HH}^2( extstyle ext{Lambda})$ is zero for most self-injective algebras of finite type

Explicit bases are given for cases where ${HH}^2( extstyle ext{Lambda})$ is non-zero

The study advances understanding of deformation theory for these algebras

Abstract

In this paper we study the second Hochschild cohomology group ${HH}^2(\Lambda)$ of a finite dimensional algebra $\Lambda$. In particular, we determine ${HH}^2(\Lambda)$ where $\Lambda$ is a finite dimensional self-injective algebra of finite representation type over an algebraically closed field $K$ and show that this group is zero for most such $\Lambda$; we give a basis for ${HH}^2(\Lambda)$ in the few cases where it is not zero.