Yetter-Drinfeld modules over weak Hopf algebras and the center construction

TL;DR

This paper introduces Yetter-Drinfeld modules over weak Hopf algebras, showing their categorical equivalences and constructing the Drinfeld double, thereby extending the theory of quantum groups and braided categories.

Contribution

It defines Yetter-Drinfeld modules over weak Hopf algebras and establishes their categorical isomorphisms with the center and modules over the Drinfeld double, extending existing frameworks.

Findings

Yetter-Drinfeld modules form a braided monoidal category

Categories of different Yetter-Drinfeld modules are isomorphic

Yetter-Drinfeld modules are equivalent to modules over the Drinfeld double

Abstract

We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show that the category of Yetter-Drinfeld modules is isomorphic to the category of modules over the Drinfeld double.