Computing the Frobenius-Schur indicator for abelian extensions of Hopf algebras

TL;DR

This paper provides a method to compute the Frobenius-Schur indicator for a class of abelian extension Hopf algebras, including the Drinfeld double of a group algebra, and explores conditions for its positivity.

Contribution

It introduces a general formula for the Frobenius-Schur indicator for abelian extension Hopf algebras and analyzes its positivity in specific cases.

Findings

The indicator is computable for abelian extension Hopf algebras.

The indicator is always positive for the Drinfeld double of the symmetric group.

A general formula for the indicator is established.

Abstract

In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.