Jordan-H\"older theorem for finite dimensional Hopf algebras

TL;DR

This paper establishes a Jordan-H"older theorem for finite dimensional Hopf algebras, extending classical group theory results and answering an open question in the field.

Contribution

It proves a Jordan-H"older theorem for finite dimensional Hopf algebras and develops analogues of Noether isomorphism theorems under certain conditions.

Findings

Jordan-H"older theorem holds for finite dimensional Hopf algebras

Analogues of Noether isomorphism theorems established for Hopf algebras

An analogue of Zassenhaus' butterfly lemma proved for these algebras

Abstract

We show that a Jordan-H\"older 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-H\"older theorem holds as well for lower and upper composition series, even though the factors of such series may be not simple as Hopf algebras.