On tensor factorizations of Hopf algebras

Keilberg, Marc; Schauenburg, Peter

doi:10.2140/ant.2016.10.61

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On tensor factorizations of Hopf algebras

Marc Keilberg and Peter Schauenburg

Vol. 10 (2016), No. 1, 61–87

DOI: 10.2140/ant.2016.10.61

Abstract

We prove a variety of 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.)

Keywords

Hopf algebra, factorization, Fitting's lemma, Krull–Remak–Schmidt

Mathematical Subject Classification 2010

Primary: 16T05

Secondary: 18D35, 18D10, 20D99, 81R05

Milestones

Received: 22 October 2014

Accepted: 9 September 2015

Published: 14 February 2016

Authors

Marc Keilberg
Institut de Mathématiques de Bourgogne, UMR5584 CNRS Université Bourgogne Franche-Comté F-21000 Dijon France

Peter Schauenburg
Institut de Mathématiques de Bourgogne, UMR5584 CNRS Université Bourgogne Franche-Comté F-21000 Dijon France

Vol. 10, No. 1, 2016