Pivotality, twisted centres and the anti-double of a Hopf monad

TL;DR

This paper extends the correspondence between pairs in involution and algebra isomorphisms from finite-dimensional Hopf algebras to general rigid monoidal categories, introducing a monadic perspective and studying the anti-Drinfeld double of a Hopf monad.

Contribution

It generalizes the known correspondence to broader categories and provides a monadic framework for the anti-Drinfeld double construction.

Findings

Established a monadic interpretation of the anti-Drinfeld double.

Extended the correspondence to rigid monoidal categories.

Connected pivotality of Drinfeld centres with the anti-Drinfeld double.

Abstract

Finite-dimensional Hopf algebras admit a correspondence between so-called pairs in involution, one-dimensional anti-Yetter--Drinfeld modules and algebra isomorphisms between the Drinfeld and anti-Drinfeld double. We extend it to general rigid monoidal categories and provide a monadic interpretation under the assumption that certain coends exist. Hereto we construct and study the anti-Drinfeld double of a Hopf monad. As an application the connection with the pivotality of Drinfeld centres and their underlying categories is discussed.