On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation type

Abstract

Let $\mathscr{B}_0(\mathcal{G})\subseteq k\mathcal{G}$ be the principal block algebra of the group algebra $k\mathcal{G}$ of an infinitesimal group scheme $\mathcal{G}$ over an algebraically closed field $k$ of characteristic ${\rm char}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology $\mathcal{L}:={\rm H}^1(\mathscr{B}_0(\mathcal{G}),\mathscr{B}_0(\mathcal{G}))$ whenever $\mathscr{B}_0(\mathcal{G})$ has finite representation type. As a consequence, we prove that the complexity of the trivial $\mathcal{G}$-module $k$ coincides with the maximal toral rank of $\mathcal{L}$.