Quantum double of Hopf monads and categorical centers

Abstract

The center $\mathcal {Z}(\mathcal {C})$ of an autonomous category $\mathcal {C}$ is monadic over $\mathcal {C}$ (if certain coends exist in $\mathcal {C}$). The notion of a Hopf monad naturally arises if one tries to reconstruct the structure of $\mathcal {Z}(\mathcal {C})$ in terms of its monad $Z$: we show that $Z$ is a quasitriangular Hopf monad on $\mathcal {C}$ and $\mathcal {Z}(\mathcal {C})$ is isomorphic to the braided category $Z-\mathcal {C}$ of $Z$-modules. More generally, let $T$ be a Hopf monad on an autonomous category $\mathcal {C}$. We construct a Hopf monad $Z_T$ on $\mathcal {C}$, the centralizer of $T$, and a canonical distributive law $\Omega \colon TZ_T \to Z_T T$. By Beck’s theory, this has two consequences. On one hand, $D_T=Z_T \circ _\Omega T$ is a quasitriangular Hopf monad on $\mathcal {C}$, called the double of $T$, and $\mathcal {Z}(T-\mathcal {C}) \simeq D_T-\mathcal {C}$ as braided categories. As an illustration, we define the double $D(A)$ of a Hopf algebra $A$ in a braided autonomous category in such a way that the center of the category of $A$-modules is the braided category of $D(A)$-modules (generalizing the Drinfeld double). On the other hand, the canonical distributive law $\Omega$ also lifts $Z_T$ to a Hopf monad $\tilde {Z}_T^\Omega$ on $T-\mathcal {C}$, and $\tilde {Z}_T^\Omega (\mathbb {1}, T_0)$ is the coend of $T-\mathcal {C}$. For $T=Z$, this gives an explicit description of the Hopf algebra structure of the coend of $\mathcal {Z}(\mathcal {C})$ in terms of the structural morphisms of $\mathcal {C}$. Such a description is useful in quantum topology, especially when $\mathcal {C}$ is a spherical fusion category, as $\mathcal {Z}(\mathcal {C})$ is then modular.