The double of a Hopf monad

📝 Abstract

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.

💡 Analysis

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.

📄 Content

The center Z(C) of an autonomous category C, introduced by Drinfeld, is a braided autonomous category. This construction establishes a bridge between the non-braided world and the braided world. It is useful, in particular, for comparing quantum invariants of 3 manifolds. Indeed, the center Z(C) of spherical fusion category C is modular (see [Mü03]) and it is conjectured that the Turaev-Viro invariant TV C (as revisited in [BW96]) is equal to the Reshetikhin-Turaev invariant RT Z(C) (see [Tur94]).

Date: October 27, 2018. 2000 Mathematics Subject Classification. 16W30,18C20,18D10.

Let C be an autonomous category. If the coend:

exists for all object X of C, then Day and Street [DS07] showed that Z is a monad on C and the center Z(C) is isomorphic to the category Z-C of Z-modules in C (also called Z-algebras). By Tannaka reconstruction, we endow the monad Z with a quasitriangular Hopf monad structure which reflects the braided autonomous structure of Z(C) in the sense that Z(C) ≃ Z-C as braided categories. The notion of Hopf monad, which generalizes Hopf algebras to the non-braided (and non-linear) setting, was introduced in [BV07] for this very purpose.

The Reshetikhin-Turaev invariant can be expressed in terms of the simple objects of the category (as in Reshetikhin and Turaev’s original construction) or in terms of the coend of the category (following Lyubashenko, see [Lyu95,BV05]). In order to compute RT Z(C) , the first approach is not practicable for lack of a workable description of the simple objects of the center. And so we need to provide an explicit description of the coend of Z(C) and its algebraic structure. To fulfill this objective, we extend the previous construction of Z to a more general situation. Let T be a Hopf monad on an autonomous category C. We denote T -C the category of T -modules (also called T -algebras), which is autonomous. Assume T is centralizable, meaning that the coend:

exists for every object X of C. We endow Z T with a structure of a Hopf monad on C and call Z T the centralizer of T . In particular, Z 1C = Z as Hopf monads. Note that the coend of C is Z(½) = Z 1C (½), and so the coend of

Using adjunction and exactness properties of Hopf monads, we show that 1 T -C is centralizable and:

Note that this implies that, as an object of C, the coend of the category T -C is Z T (½) and, in particular, the coend of Z(C) = Z-C is Z Z (½). Now U T Z 1 T -C = Z T U T means in fact that the Hopf monad Z 1 T -C is a lift to T -C of the Hopf monad Z T . Extending Beck’s theory of distributive laws to Hopf monads, we show that such a lift is encoded by an invertible comonoidal distributive law Ω : T Z T → Z T T , called the canonical distributive law of T . The coend of T -C is therefore (Z T (½), Z T (T 0 )Ω ½ ). When T is quasitriangular, this coend has a structure of a Hopf algebra in the braided autonomous category T -C, which we elucidate in terms of T . Hence, for T = Z, an explicit description of the coend of Z(C). The case of fusion categories is treated in detail.

The canonical distributive law Ω also endows the composition of Z T and T with a Hopf monad structure, denoted D T = Z T • Ω T and called the double of T . We prove that D T is quasitriangular and give a braided isomorphism:

This construction, which holds for any centralizable Hopf monad on an autonomous category, generalizes the Drinfeld double in an non-braided setting. As an illustration, we apply this to Hopf monads associated with Hopf algebras. This leads to the double D(A) of a Hopf algebra A in a braided autonomous category B. More precisely, the endofunctor ? ⊗ A is a Hopf monad on B. Assuming B admits a coend C, the Hopf monad ? ⊗ A is centralizable, and its centralizer is of the form ? ⊗ Z(A), where Z(A) = ∨ A ⊗ C is a Hopf algebra in B. The canonical distributive law of ?⊗A is of the form id 1B ⊗Ω, where Ω : Z(A)⊗A → A⊗Z(A) is a distributive law of Hopf algebras. Then D(A) = A ⊗ Ω Z(A) is a quasitriangular Hopf algebra in B, such that: Z(B A ) ≃ B D(A) ≃ D(A) B ≃ Z( A B). as braided categories, where A B and B A denote the categories of left and right modules over A. Note that a Hopf algebra B in B is quasitriangular when it is endowed with a R-matrix. In this context, we define R-matrices to be morphisms r : C ⊗ C → B ⊗ B which encode braidings on B B (or equivalently B B). When B is the category of finite-dimensional vector spaces over a field k, we recover the usual definition of R-matrices and the Drinfeld double of a Hopf algebra H. Indeed, in that case: C = k, Z(H) = H * cop , and D(H) = H ⊗ Ω H * cop .

The canonical distributive law of a Hopf monad is in fact naturally defined in a more general setting. Let T be a Hopf monad on an autonomous category C and Q be a Hopf monad on T -C. Their cross product Q ⋊ T = U T QF T is a Hopf monad on C. If Q ⋊ T is centralizable, then so is Q and the Hopf monad Z Q is a lift to T -C of the Hopf monad Z Q⋊T :

Hence a canonical distributive law Ω : T Z Q⋊T

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