Character tables and normal left coideal subalgebras

TL;DR

This paper explores how generalized character tables of semisimple Hopf algebras encode normal left coideal subalgebras, extending classical group theory results to the Hopf algebra setting.

Contribution

It introduces the concept that character tables reflect normal left coideal subalgebras and proves several analogues of group theory theorems for Hopf algebras.

Findings

Columns of the character table are orthogonal.

Entries of the character table are algebraic integers.

Proved Hopf algebra analogues of Burnside-Brauer and related theorems.

Abstract

We continue studying properties of semisimple Hopf algebras $H$ over algebraically closed fields of characteristic 0 resulting from their generalized character tables. We show that the generalized character table of $H$ reflect normal left coideal subalgebras of $H.$ These are the Hopf analogues of normal subgroups in the sense that they arise from Hopf quotients. We apply these ideas to prove Hopf analogues of known results in group theory. Among the rest we prove that columns of the character table are orthogonal and that all entries are algebraic integers. We analyze `semi kernels' and their relations to the character table. We prove a full analogue of the Burnside-Brauer theorem for almost cocommutative $H.$ We also prove the Hopf algebras analogue of the following (Burnside) theorem: If G is a non-abelian simple group then $\{1\}$ is the only conjugacy class of G which has prime power order.