Generalizations of Yetter-Drinfel'd modules and the center construction of monoidal categories

Abstract: This is joint work with Ryan Aziz. A Yetter-Drinfel'd module over a bialgebra $H$, is at the same time 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 as well as 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 $H$, $K$, together with an bimodule coalgebra $C$ and a bicomodule algebra $A$ over them. A generalized Yetter-Drinfel'd module in their sense, is an $A$-module that is at the same time a $C$-comodule satisfying a certain compatibility condition. Under finiteness conditions, they showed that these modules are exactly modules of a suitably constructed smash product build out of $A$ and $C$. The aim of this talk is to show how the category of these generalized Yetter-Drinfel'd 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 and we discuss the bicategorical structure that arises this way.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

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