In the last lecture we have learned that the category of modules over a braided Hopf algebra H is a braided monoidal category. A braided Hopf algebra is a rather sophisticated algebraic object, it is not easy to give interesting nontrivial examples. In this text we develop a theory that will lead to a concrete recipe which produces a nontrivial braided Hopf algebra D(A) (called Drinfeld’s quant...

Results on the finiteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2-types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some finite crossed modules are given, using crossed complex methods.

Results on the niteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2-types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some nite crossed modules are given, using crossed complex methods.

Results on the niteness of induced crossed modules are proved both algebraically and topologically. Using the Van Kampen type theorem for the fundamental crossed module, applications are given to the 2-types of mapping cones of classifying spaces of groups. Calculations of the cohomology classes of some nite crossed modules are given, using crossed complex methods.

The notion of crossed product with a coquasi-Hopf algebra H is introduced and studied. The result of such a crossed product is an algebra in the monoidal category of right H-comodules. We give necessary and sufficient conditions for two crossed products to be equivalent. Then, two structure theorems for coquasi Hopf modules are given. First, these are relative Hopf modules over the crossed prod...

The goal of this article is to construct a crossed module representing the cocycle 〈[, ], 〉 generating H(g; C) for a simple complex Lie algebra g.

We introduce Yetter-Drinfeld modules over a weak Hopf algebra H, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of H-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak DoiHopf modules, and, a fortiori, a...

In this paper, we introduced the concept of crossed modules for Hom–Lie antialgebras. It is proved that category antialgebras and [Formula: see text]-Hom–Lie are equivalent to each other. The relationship between extension third cohomology group investigated.