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Transactions of the American Mathematical Society

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Quantum double of Hopf monads and categorical centers

Authors: Alain Bruguières and Alexis Virelizier
Journal: Trans. Amer. Math. Soc. 364 (2012), 1225-1279
MSC (2010): Primary 16W30, 18C20, 18D10
Published electronically: October 17, 2011
MathSciNet review: 2869176
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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 (\bbone ,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.

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Additional Information

Alain Bruguières
Affiliation: Département de Mathématiques, Université Montpellier II, Place Eugène Bataillan, 34095 Montpellier cedex 05, France

Alexis Virelizier
Affiliation: Département de Mathématiques, Université Montpellier II, Place Eugène Bataillan, 34095 Montpellier cedex 05, France

Received by editor(s): June 5, 2009
Received by editor(s) in revised form: March 4, 2010
Published electronically: October 17, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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