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

Abstract

Let B0(G) ⊆ kG be the principal block algebra of the group algebra kG of an infinitesimal group scheme G over an algebraically closed field k of characteristic char(k) =: p ≥ 3. We calculate the restricted Lie algebra structure of the first Hochschild cohomology L := H 1 (B0(G), B0(G)) whenever B0(G) has finite representation type. As a consequence, we prove that the complexity of the trivial G-module k coincides with the maximal toral rank of L.

Stable invariance of the restricted Lie algebra structure of Hochschild cohomology

“…By combining Corollary 1 with the the idea that a precise knowledge of the Lie algebra structure of HH 1 (A) should contain significant information about A ( [26,8,27,10,37,4] to name a few), the authors, in joint work with Manuel Saorín, are trying to uncover new information in the stable equivalence class of finite dimensional algebras.…”