On the Adjoint Representation of a Hopf Algebra

Martin Lorenz, Bach Nguyen, Ramy Yammine

Published 2019-05-08Version 1

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 cocommutative $H$, 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 pointed Hopf algebra. If $H$ is generated by grouplike and by skew-primitive elements, then $H_{\text{adfin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.