On the adjoint representation of a hopf algebra. Issue 4 (11th November 2020) - Kolb, Stefan; Lorenz, Martin; Nguyen, Bach; Yammine, Ramy
Abstract: We consider the adjoint representation of a Hopf algebra H focusing on the locally finite part, H_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_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_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.