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

TL;DR

This paper computes the structure of the first Hochschild cohomology for certain finite representation type cocommutative Hopf algebras, revealing a link between module complexity and Lie algebra rank.

Contribution

It provides the first explicit calculation of the restricted Lie algebra structure of Hochschild cohomology for these algebras, connecting cohomological and representation-theoretic properties.

Findings

Hochschild cohomology structure determined for finite type cases

Complexity of trivial module equals maximal toral rank of cohomology

Establishes a link between cohomology and module complexity

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 $\mathcal{L}$.