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Jordan-Hölder theorem for finite dimensional Hopf algebras


Author: Sonia Natale
Journal: Proc. Amer. Math. Soc. 143 (2015), 5195-5211
MSC (2010): Primary 16T05; Secondary 17B37
DOI: https://doi.org/10.1090/proc/12702
Published electronically: April 14, 2015
MathSciNet review: 3411137
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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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Sonia Natale
Affiliation: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM – CONICET, (5000) Ciudad Universitaria, Córdoba, Argentina
Email: natale@famaf.unc.edu.ar

DOI: https://doi.org/10.1090/proc/12702
Keywords: Hopf algebra, adjoint action, normal Hopf subalgebra, composition series, principal series, isomorphism theorems, Jordan-H\"older theorem, composition factor
Received by editor(s): July 10, 2014
Received by editor(s) in revised form: November 6, 2014
Published electronically: April 14, 2015
Additional Notes: This work was partially supported by CONICET, Secyt (UNC) and the Alexander von Humboldt Foundation
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2015 American Mathematical Society

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