2008/12/12 by Alain Bruguières, Bruguières, Alain, Alexis Virelizier +1
Mathematics · #16W30 #18C20 #18D10 #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA) #math.CT #math.QA #msc:16W30 #msc:18C20 #msc:18D10
paper · pdf · doi:10.48550/arxiv.0812.2443
arxiv created 2008/12/13 · arxiv updated 2009/12/01
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 ZT on C, the centralizer of T, and a canonical distributive law of T over ZT. By Beck's theory, this has two consequences. On one hand, DT=ZT T is a quasitriangular Hopf monad on C, called the double of T, and Z(T-C)= DT-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 ZT 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.