Braided Drinfeld and Heisenberg Doubles

TL;DR

This paper introduces a unified framework for braided Drinfeld and Heisenberg doubles, generalizing their structures within monoidal categories and establishing their intrinsic definitions and relationships.

Contribution

It generalizes the Drinfeld center to mixed centers, introduces a Heisenberg analogue, and describes their interaction in the context of braided monoidal categories.

Findings

Unified description of Drinfeld and Heisenberg doubles

Intrinsic definitions via braided reconstruction theory

Heisenberg double as a 2-cocycle twist of the Drinfeld double

Abstract

In this paper, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result in [Lu94] that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras.