Yetter-Drinfeld modules over weak multiplier bialgebras

Abstract

We continue the study of the representation theory of a regular weak multiplier bialgebra with full comultiplication, started in arXiv:1306.1466, arXiv:1311.2730. Yetter-Drinfeld modules are defined as modules and comodules, with compatibility conditions that are equivalent to a canonical object being (weakly) central in the category of modules, and equivalent also to another canonical object being (weakly) central in the category of comodules. Yetter-Drinfeld modules are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. Finite dimensional Yetter-Drinfeld modules over a regular weak multiplier Hopf algebra with full comultiplication are shown to possess duals in this monoidal category.