Weak Hopf Algebras And Quantum Groups

The notion of a Hopf algebra was developed by algebraic topologists abstracting the work of H. Hopf on manifolds in 1941. Starting with issue of the article "On the structure of Hopf algebras", written by J. Milnor and J. Moore in 1965, Hopf algebras became a subject of study from a strictly algebraic point of view. By the end of the 1990s, research in this field was given a strong boost by the appearance of quantum groups and the development of the action theory of Hopf algebras which unifies earlier results known for group actions, actions of Lie algebras, and graded rings. Since the 1990s, there appear many Hopf algebraic constructions, such as weak Hopf algebra, quasi-Hopf algebra, Hopf π-coalgebra and so on. The aim of this Ph.D. dissertation is to study the properties of weak Hopf algebra and quantum groups from the following aspects.In Part One we discuss the conditions for the weak antipode to be an involution and study the actions of weak Hopf algebras. We give a criterion for the smash product A#H to be separable over A, prove the Maschke type Theorem for the L-R smash products and investigate the properties of the Drinfeld double, which is a special case of the diagonal crossed products.In Part Two we study the theory of entwining structures and that of factorization structures. We prove the Fundamental Theorem of the entwined modules and give a necessary and sufficient condition for the R-twisted product of two bialgebras to be a bialgebra.In Part Three we study the properties of quantum algebras u_q(osp(1, 2, f)) and u_q(f(K, H)). We first construct a new quantum algebra u_q(osp(1, 2, f)) and discuss its center structure, moreover, we prove the Isomorphism Theorem of u_q(osp(1,2)). Then we study the adjoint action of the quantum algebra u_q(f(K, H)) and discuss the structure of its locally finite subalgebra.