Yetter-Drinfel'd Modules And Generalized Weak Group-Like Algebras

The main research objects of this Ph.D. dissertation are weak Hopf algebras and generalized weak group-like algebras. We carry out a series of further research from the following aspects.Firstly, we introduce the notions of a symmetric pair and u-condition in the Yetter-Drinfel'd module category HHYD over a weak Hopf algebra H Then we prove that some properties of H are determined by some symmetric pairs in the category HHyD. Furthermore, a symmetric subcategory of the category HHyD is constructed.Secondly, we propose the notion of a pseudosymmetric weak Hopf algebra and give suffi-cient and necessary conditions for a quasitriangular weak Hopf algebra to be pseudotriangular from two different perspectives. In order to obtain more examples of pseudosymmetric weak Hopf algebras, on the one hand, we introduce the concept of an almost-triangular weak Hopf algebra which is a special class of pseudosymmetric weak Hopf algebras, on the other hand, we show that the pseudosymmetry of the category HHyD is equivalent to the commutativity and cocommutativity of H.Again, for a right weak H-module algebra B, we introduce a new cocyclic object which is isomorphic to the cocyclic object studied in [16]. Moreover, we establish a generalized trace map for this new equivariant cyclic cohomology and define an equivariant K-theory for B.Finally, we introduce the notion of a generalized weak group-like algebra and construct a list of examples. Then we show that a generalized weak group-like algebra can be regarded as a generalized weak bi-Frobenius algebra. Moreover, we discuss when a generalized weak group-like algebra is a groupoid algebra and give the classifications of 2-dimensional and 3-dimensional generalized weak group-like algebras.