Hopf Monads And The Construction Of Ribbon Crossed Category

The aim of this dissertation is to carry out a series of further research on bimonads, entwining structures, weak Hom-Hopf algbebras, weak quasi-Turaev G-coalgebras and ribbon categories from the following aspects.First, we will introduce the notion of monoidal mixed datums and braided mixed datums, and discuss the braided structure on the category of mixed bimodules. We also give the necessary and sufficient conditions for the category of mixed bimodules to be a monoidal category and braided category. Further, as applications, we investigate the braided structure on the category of Hom-(co)modules of a (monoidal) Hom-bialgebra.Second, we will study the dualities in the category of entwined modules and define the notion of a rigid entwining datum. Also, we will describe the ribbon structure on the category of entwined modules of the entwining structure which is both a double quantum group and a rigid entwining datum. Further, as applications, we investigate the ribbon structure on the category of Yetter-Drinfeld modules and the category of Long-dimodules.Again, we will introduce the notion of a weak Hom-Hopf algebra and discuss the u-niqueness of the category of its representations. We show the tensor functor of a weak Hom-bialgebra is a weak bimonad. Further, the quasitriangular structure and ribbon structure of a weak Hom-bialgebra will be discussed.Finally, we will introduce the notion of a weak quasi-Turaev G-coalgebra, and discuss the cross structure over the category of its representations. Also we describe the quasitriangular structure and ribbon structure of a weak quasi-Turaev G-coalgebra, and give a sufficient and necessary condition of the category of its representations is a crossed ribbon G-category.