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.
Publication Source (Journal or Book title)
Pacific Journal of Mathematics
Negron, C., & Ng, S. (2017). Gauge invariants from the powers of antipodes. Pacific Journal of Mathematics, 291 (2), 439-460. https://doi.org/10.2140/pjm.2017.291.439