Gauge invariants from the powers of antipodes

TL;DR

This paper proves that certain trace functions of the antipode in Hopf algebras are gauge invariants, leading to invariance of the antipode's order under Drinfeld twists, with implications for tensor categories.

Contribution

It establishes gauge invariance of the trace of antipode powers and confirms the invariance of the antipode's order for Hopf algebras with the Chevalley property.

Findings

Trace of nth power of antipode is gauge invariant.

Order of antipode is invariant under Drinfeld twists.

Confirmed Shimizu's question for specific tensor categories.

Abstract

We prove that the trace of the $n$th power of the antipode of a Hopf algebra with the Chevalley property is a gauge invariant, for each integer $n$. As a consequence, the order of the antipode, and its square, are invariant under Drinfeld twists. The invariance of the order of the antipode is closely related to a question of Shimizu on the pivotal covers of finite tensor categories, which we affirmatively answer for representation categories of Hopf algebras with the Chevalley property.