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Article Dans Une Revue Contemporary mathematics Année : 2013

The doubles of a braided Hopf algebra

Résumé

Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix for a Hopf algebra in B involves the coend C of B). In general, i.e. when B does not necessarily admit a coend, we construct a quasitriangular Hopf monad d_A on the center Z(B) of B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. We prove that the Hopf monad d_A may not be representable by a Hopf algebra. If B has a coend C, then D(A) is the cross product of the Hopf monad d_A by C. Equivalently, the Hopf monad d_A is the cross quotient of the Hopf algebra D(A) by the Hopf algebra C.

Dates et versions

hal-00807274 , version 1 (03-04-2013)

Identifiants

Citer

Alain Bruguières, Alexis Virelizier. The doubles of a braided Hopf algebra. Contemporary mathematics, 2013, 585, pp.175-198. ⟨10.1090/conm/585⟩. ⟨hal-00807274⟩
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