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. From the perspective of representation theory, Hopf algebras are in one-to-one correspondence with rigid monoidal categories. This fact may be “categorified”, passing from Hopf algebras to Hopf monads as defined by Bruguiéres and Virelizier [1]. Further, as studied by Aguiar and Chase [2], a Hopf monad may admit a comodule monad over it; this generalises the notion of a comodule algebra over a Hopf algebra, which representation theoretically expresses itself as a module category over the (rigid) monoidal base category.
In this talk, we will explore the classical theorem from this perspective, and extend it to co-module monads over Hopf monads. Hereto we construct the anti-Drinfeld double of a Hopf monad, which—analogously to the Hopf algebraic case—is a comodule over its double; the latter was studied in [3]. As it turns out the interplay between double and anti-double characterises when a rigid monoidal category is pivotal—i.e., the double dualising functor is (isomorphic to) the identity.
This talk is based on [4].

References

[1] A. Bruguières and A. Virelizier, Hopf monads, Adv. Math. 215, No. 2, 2007, 679–733.
[2] M. Aguiar and S. U. Chase, Generalized Hopf modules for bimonads, Theory Appl. Categ., Vol. 27,
2012, 263–326.
[3] A. Bruguiéres and A. Virelizier, Quantum double of Hopf monads and categorical centers,
Trans. Am. Math. Soc. 364, No. 3, 2012, 1225–1279.
[4] S. Halbig and T. Zorman, Pivotality, twisted centres, and the anti-double of a Hopf monad, Theory
Appl. Categ., Vol. 41, 2024, No. 4, 86–149