The applications of probability groups on Hopf algebras

TL;DR

This paper explores the use of probability groups to analyze semisimple Hopf algebras, revealing structural properties of their centers and class sums, and classifies small probability groups.

Contribution

It introduces a novel application of probability groups to Hopf algebra theory and classifies all small 2-integral probability groups.

Findings

The dual probability group algebra equals the center of the Hopf algebra.

Product of class sums is an integral combination scaled by the inverse of the algebra's dimension.

Classifies all 2- and 3-element 2-integral probability groups.

Abstract

In this work, we use probability groups, introduced by Harrison in 1979, as a tool to study a semisimple Hopf algebra $H$ with a commutative character ring and prove that the algebra generalized by the dual probability group is the center $Z(H)$ of $H$ and the product of two class sums is an integral combination up to a factor of $\dim (H)^{-1}$ of the class sums of $H$. We classify all the 2-integral probability groups with 2 or 3 elements.