Jordan-Hölder theorem for finite dimensional Hopf algebras

2014/07/03 by Sonia Natale, Natale, Sonia
Mathematics · #16T05 #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.QA #math.RA #msc:16T05

paper · pdf · doi:10.48550/arxiv.1407.0931

15 pages, amslatex

arxiv created 2014/07/03 · arxiv updated 2014/07/04

Abstract

We show that a Jordan-Hölder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus' butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Hölder theorem holds as well for lower and upper composition series, even though the factors of such series may be not simple as Hopf algebras.

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