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On the adjoint representation of a hopf algebra

Part of: Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  11 November 2020

Stefan Kolb ,
Martin Lorenz ,
Bach Nguyen  and
Ramy Yammine
Stefan Kolb
Affiliation:
School of Mathematics, Statistics and Physics, Newcastle University, NewcastleNE1 7RU, UK
Martin Lorenz
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA19122, USA (lorenz@temple.edu)
Bach Nguyen
Affiliation:
Department of Mathematics, Xavier University of Louisiana, New Orleans, LA70125, USA
Ramy Yammine
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA19122, USA (lorenz@temple.edu)
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Abstract

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

Keywords

Infinite-dimensional Hopf algebracocommutative Hopf algebrapointed Hopf algebraHopf subalgebraadjoint representationlocally finite parttensor functorcoideal subalgebraΔ-methodsDietzmann's Lemma

MSC classification

Primary: 16T05: Hopf algebras and their applications
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 1092 - 1099
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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