# On the Adjoint Representation of a Hopf Algebra

## TL;DR

This paper investigates the structure of the adjoint representation of Hopf algebras, showing that the locally finite part forms a Hopf subalgebra in certain cases and a coideal subalgebra generally, with new theoretical insights and a version of Dietzmann's Lemma.

## Contribution

It establishes conditions under which the locally finite part of a Hopf algebra forms a Hopf subalgebra or coideal subalgebra, and introduces a Hopf algebra version of Dietzmann's Lemma.

## Key findings

- $H_{adfin}$ is a Hopf subalgebra for virtually cocommutative $H$.
- $H_{adfin}$ is a left coideal subalgebra for general $H$.
- A Hopf algebra analogue of Dietzmann's Lemma is proved.

## Abstract

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{\text{adfin}}$, 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 $H_{\text{adfin}}$ 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, $H_{\text{adfin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.