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

Abstract

Let ${\mathcal{B}}_0(\mathcal{G})\subseteq k\mathcal{G}$ 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 $\mathrm{char}(k)=:p\ge 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology $\mathcal{L}:={\mathrm{H}}^1({\mathcal{B}}_0(\mathcal{G}),{\mathcal{B}}_0(\mathcal{G}))$ whenever ${\mathcal{B}}_0(\mathcal{G})$ 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 $\mathcal{L}$.

Cited by in corpus (1)

• Stable invariance of the restricted Lie algebra structure of Hochschild cohomology