GENERALIZATIONS OF YETTER-DRINFEL'D MODULES AND THE CENTER CONSTRUCTION OF MONOIDAL CATEGORIES

A Yetter-Drinfel'd module over a bialgebra H is both a module and a comodule over H satisfying a particular compatibility condition. It is well known that the category of Yetter-Drinfel'd modules (say, over a finite-dimensional Hopf algebra H) is equivalent to the center of the monoidal category of H-(co)modules and to the category of modules over the Drinfel'd double of H. Caenepeel, Militaru and Zhu introduced a generalized version of Yetter-Drinfeld modules. More precisely, they consider two bialgebras H and K together with a bimodule coalgebra C and a bicomodule algebra A over them. A generalized Yetter-Drinfel'd module, in their sense, is an A-module C-comodule satisfying a certain compatibility condition. Under finiteness conditions, they showed that these modules are exactly modules of a suitably constructed smash product built out of A and C. The aim of this talk is to show how the category of these generalized Yetter-Drinfel'd modules can be obtained as a relative center of the category of A-modules viewed as a bi-actegory over the categories of H-modules and K-modules. Moreover, we also show how other variations of Yetter-Drinfel'd modules, such as anti-Yetter-Drinfel'd modules, arise as a particular case. (Joint work with Ryan Aziz.)