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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Quantum double of Hopf monads and categorical centers
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by Alain Bruguières and Alexis Virelizier PDF
Trans. Amer. Math. Soc. 364 (2012), 1225-1279 Request permission

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.
References
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Additional Information
  • Alain Bruguières
  • Affiliation: Département de Mathématiques, Université Montpellier II, Place Eugène Bataillan, 34095 Montpellier cedex 05, France
  • Email: bruguier@math.univ-montp2.fr
  • Alexis Virelizier
  • Affiliation: Département de Mathématiques, Université Montpellier II, Place Eugène Bataillan, 34095 Montpellier cedex 05, France
  • Email: virelizi@math.univ-montp2.fr
  • Received by editor(s): June 5, 2009
  • Received by editor(s) in revised form: March 4, 2010
  • Published electronically: October 17, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 1225-1279
  • MSC (2010): Primary 16W30, 18C20, 18D10
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05342-0
  • MathSciNet review: 2869176