Solvability for semisimple Hopf algebras via integrals

TL;DR

This paper develops harmonic analysis techniques for semisimple Hopf algebras using integrals, introduces a new definition of solvability for these algebras, and proves an analogue of Burnside's theorem for certain dimensions.

Contribution

It introduces an intrinsic definition of solvability for semisimple Hopf algebras and extends classical group solvability concepts to this algebraic setting.

Findings

Embedded the space of functionals on coideal subalgebras into the dual of the Hopf algebra.

Provided explicit formulas for induced characters and their embeddings.

Proved that certain quasitriangular Hopf algebras are solvable.

Abstract

We use integrals of left coideal subalgebras to develop Harmonic analysis for semisimple Hopf algebras. We show how $N^*,$ the space of functional on $N,$ is embedded in $H^*.$ We define a bilinear form on $N^*$ and show that irreducible $N$-characters are orthogonal with respect to that form. We then give an explicit formula for induced characters of $N$ and show how the induced characters are embedded in $R(H).$ In the second part we give an intrinsic definition for solvable semisimple Hopf algebras via left coideal subalgebras and their integrals. We show how this definition generalizes solvability for finite groups. In particular, commutative and nilpotent Hopf algebras are solvable. We finally prove an analogue of Burnside theorem: A semisimple quasitriangular Hopf algebras of dimension $p^aq^b$ is solvable.