Quantum double of Hopf monads and categorical centers
By Alain Bruguières, Alexis Virelizier. 2012
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The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of a 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 Ω: TZT → ZTT . By Beck’s theory, this has two consequences. On one hand, DT = ZT ◦Ω 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 Ω also lifts ZT to a Hopf monad Z Ω T on T−C, and ZΩ T ( , T0) is the coend of T−C. For T = Z, this gives an explicit description of the Hopf algebra structure of the coend of Z(C) in terms of the structural morphisms of C. Such a description is useful in quantum topology, especially when C is a spherical fusion category, as Z(C) is then modular.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/cfc5b32fa63ce2ec4031639ad4d90770f5b73e92
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- Non-degeneracy conditions for braided finite tensor categories (2016, background): “C(V ; 1 ) for V ∈ C. Then the triple (ZC,µ,η) is a monad on C, which is in fact a quasitriangular Hopf monad on C (we omit the description of other structure morphisms since we will not use them; see [BV12] for details). Definition 2.5. We call ZC the central Hopf monad on C. Given an object (V,c) ∈ ”