Jordan-Hölder theorem for finite dimensional Hopf algebras

Natale, Sonia

doi:10.1090/proc/12702

Jordan-Hölder theorem for finite dimensional Hopf algebras
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Proc. Amer. Math. Soc. 143 (2015), 5195-5211 Request permission

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 not be simple as Hopf algebras.

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