Self-injective algebras and the second Hochschild cohomology group

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.