On tensor factorizations of Hopf algebras

TL;DR

This paper investigates tensor product factorizations of finite dimensional Hopf algebras, establishing analogs of classical group decomposition results and analyzing automorphisms, with applications to Drinfeld doubles of finite groups.

Contribution

It extends classical decomposition theorems to Hopf algebras, analyzing endomorphisms and automorphisms, and computes automorphism groups of Drinfeld doubles with abelian factors.

Findings

Proves tensor product factorization results for Hopf algebras.

Analyzes normal and conormal Hopf algebra endomorphisms.

Computes automorphism groups of Drinfeld doubles with abelian factors.

Abstract

We prove a variety results on tensor product factorizations of finite dimensional Hopf algebras (more generally Hopf algebras satisfying chain conditions in suitable braided categories). The results are analogs of well-known results on direct product factorizations of finite groups (or groups with chain conditions) such as Fitting's Lemma and the uniqueness of the Krull-Remak-Schmidt factorization. We analyze the notion of normal (and conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and automorphisms of tensor products. The results are then applied to compute the automorphism group of the Drinfeld double of a finite group in the case where the group contains an abelian factor. (If it doesn't, the group can be calculated by results of the first author.)