Kernels of representations and coideal subalgebras of Hopf algebras

TL;DR

This paper introduces the concepts of left and right kernels of Hopf algebra representations, generalizing classical kernel notions and Brauer's theorem from group algebras to Hopf algebras.

Contribution

It defines and compares kernels of representations in Hopf algebras, establishing their equivalence with categorical Hopf kernels and extending classical theorems.

Findings

Left and right kernels coincide in group algebras

Kernels match categorical Hopf kernels in general case

Brauer's theorem is generalized to Hopf algebras

Abstract

We define left and right kernels of representations of Hopf algebras. In the case of group algebras, left and right kernels coincide and they are the usual kernels of modules. In the general case we show that these kernels coincide with the categorical left and right Hopf kernels of morphisms of Hopf algebras defined in \cite{AD}. Brauer's theorem for kernels over group algebras is generalized to Hopf algebras.