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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2012

Quantum Double of Hopf monads and Categorical Centers

Résumé

The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category Z-C of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We construct a Hopf monad Z_T on C, the centralizer of T, and a canonical distributive law of T over Z_T. By Beck's theory, this has two consequences. On one hand, D_T=Z_T T is a quasitriangular Hopf monad on C, called the double of T, and Z(T-C)= D_T-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 also lifts Z_T to a Hopf monad on T-C which gives the coend of T-C. Hence, for T=Z, an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C, which is useful in quantum topology.

Dates et versions

hal-00463040 , version 1 (10-03-2010)

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Citer

Alain Bruguières, Alexis Virelizier. Quantum Double of Hopf monads and Categorical Centers. Transactions of the American Mathematical Society, 2012, 364 (3), pp.1225-1279. ⟨10.1090/S0002-9947-2011-05342-0⟩. ⟨hal-00463040⟩
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