Categorical Hopf kernels and representations of semisimple Hopf algebras

TL;DR

This paper explores the properties of Hopf kernels in semisimple Hopf algebras, establishing their equivalence with representation kernels and analyzing the normality of kernels, including a self-duality property.

Contribution

It demonstrates the equivalence of Hopf kernels and representation kernels in semisimple Hopf algebras and introduces new results on the normality and self-duality of kernels.

Findings

Hopf kernels coincide with kernels of representation in semisimple Hopf algebras

Normality of kernels is a self-dual property

New results on the normality of kernels in Hopf algebras

Abstract

In the category of semisimple Hopf algebras the Hopf kernels introduced by Andruskiewitsch and Devoto in \cite{AD} coincide with kernels of representation as introduced in \cite{Bker}. Some new results concerning the normality of kernels are also presented. It is proven that the property for Hopf algebras to have all kernels normal Hopf subalgebras is a self dual property.