Reflective centers as categories of modules

TL;DR

This paper explores the structure of reflective centers in module categories over braided monoidal categories, generalizing the role of transmutation of Hopf algebras and providing new insights into their module categories.

Contribution

It generalizes the concept of transmutation of Hopf algebras within the context of reflective centers, extending to categories with internal reconstructed Hopf algebras.

Findings

Reflective centers can be described as module categories over certain Hopf algebras.

The role of Majid's transmutation is clarified through a broader generalization.

In some cases, the reflective center equals the category of modules over a Hopf algebra in B.

Abstract

In [LWY23] the authors construct the reflective center of a module category M over a braided monoidal category B. The reflective center is by construction a braided module category over B. In the case where B is the category of modules over a finite dimensional quasitriangular Hopf algebra H, acting on the category of modules over a comodule algebra, they construct a comodule algebra, the reflective algebra, whose modules are precisely the reflective center. In the construction, Majid's transmutation of H plays a crucial r{\^o}le. This note centers on the transmuted H, seeking to ''explain'' its appearance through a generalization in which the acting category is no longer a module category, but admits an internal reconstructed Hopf algebra; the transmutation is a special case of this notion. As a result, in certain cases, the reflective center is simply the category of modules in M over that Hopf algebra in B.