The doubles of a braided Hopf algebra

TL;DR

This paper generalizes the concept of the Drinfeld double for Hopf algebras in braided categories, introducing a Hopf monad construction that extends the classical framework to broader categorical contexts.

Contribution

It constructs a quasitriangular Hopf monad d_A on the center of B, generalizing the double of a Hopf algebra without requiring the existence of a coend.

Findings

The Hopf monad d_A may not be representable by a Hopf algebra.

When B admits a coend C, the double D(A) is the cross product of d_A by C.

The category of modules over D(A) is isomorphic to the center of A-modules as a braided category.

Abstract

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.