On Doi-Hopf modules and Yetter-Drinfeld modules in symmetric monoidal categories. (Q2448501)

Let \(\mathcal C\) be a monoidal category. A (right) entwining structure is a triple \((A,C,\psi)\), where \(A\) is an algebra in \(\mathcal C\), \(C\) is a coalgebra in \(\mathcal C\), and \(\psi\colon C\otimes A\to A\otimes C\) is a morphism in \(\mathcal C\) satisfying four appropriate compatibility conditions. Now assume that \(\mathcal C\) is braided and that \(H\) is a bialgebra in \(\mathcal C\). It is possible to define Doi-Koppinen structures and Yetter-Drinfeld structures over \(H\), and, moreover, a Yetter-Drinfeld structure gives rise to a Doi-Koppinen structure, and Doi-Koppinen structures (over a Hopf algebra) give rise to entwining structures. Moreover, the category of representations of a Doi-Koppinen structure (also called the category of Doi-Hopf modules) is isomorphic to the category of entwined modules over the corresponding entwining structures. A similar result holds for Yetter-Drinfeld structures. Now the authors present the following generalization of these results. Let \(B\) be an algebra and a coalgebra in \(\mathcal C\), but not necessarily a bialgebra. Two different versions of Doi-Koppinen structures are introduced, and it is shown that they give rise to entwining structures. Then 16 different versions of lax Yetter-Drinfeld structures are introduced, depending on 4 choices of the braiding or its inverse. If \(\mathcal C\) is symmetric, then they all coincide. Then lax Hopf algebras are introduced. For a symmetric monoidal category \(\mathcal C\), lax Yetter-Drinfeld structures over a lax Hopf algebra induce Doi-Koppinen structures, and, a fortiori, entwining structures. Let \(\mathcal D\) be a \(\mathcal C\)-category (also called a module category). Entwined modules can be defined in \(\mathcal D\), and similar properties hold for Doi-Hopf modules and Yetter-Drinfeld modules. The main result is now the following: the category of Yetter-Drinfeld modules in \(\mathcal D\) over a lax Yetter-Drinfeld structure over a lax Hopf algebra in a symmetric monoidal category \(\mathcal C\) is isomorphic to the category of entwined modules over the corresponding entwining structure.