Vol. 10, No. 1, 2016

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ISSN: 1944-7833 (e-only)
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On tensor factorizations of Hopf algebras

Marc Keilberg and Peter Schauenburg

Vol. 10 (2016), No. 1, 61–87
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