Self-injective algebras and the second Hochschild cohomology group

In this paper we study the second Hochschild cohomology group ${HH}^2(\mathrm{\Lambda })$ of a finite dimensional algebra $\mathrm{\Lambda }$. In particular, we determine ${HH}^2(\mathrm{\Lambda })$ where $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$; we give a basis for ${HH}^2(\mathrm{\Lambda })$ in the few cases where it is not zero.