Hopf Algebra K _Q [x, X ^{-1 , Y] Algebra Automorphism Of I And U _Q (sl _{2 < / Sub>) - Module Coalgebra Structure}}

Quantum group, especially, the quantum enveloping algebra, is an important branch of algebra research in recent years. The quantum enveloping algebra of the finite-dimensional simple Lie algebra sl2is the basic content to study the general quantum enveloping algebras. Quantum polynomial algebra Oq=Kq[x1±1,…xr±1,xr+1,…,xp] plays an important role in non-commutative algebraic geometry, it is closely relative to quantum groups and is the important means to study quantum groups.It is an important work to investigate the automorphism group of various algebraic structure for algebras, there are lots of related research subjects and research results. The theory of the group actions and Hopf algebra actions on algebras is an important research topic, and many mathematicians have been working on the topic. With the development of the quantum group theory, the study of the quantum enveloping algebra acting on the quantum polynomial algebras aroused a lot of interest to mathematicians.In this thesis, we investigate the quantum polynomial algebra kg[x,x-1,y], which is a Hopf algebra, where q is not a root of unity. We mainly discuss the coalgebra automorphisms, the structure of the graded coalgebra automorphism group of kq[x,x-1,y],and the Uq(sl2)-module coalgebra structures of ka[x,x-1,y]. This thesis is arranged as follows. In Section1, we introduce some basic notions, such as graded coalgebras, the endomorphisms of a graded coalgebra, the module coalgebras over a Hopf algebra, and so on. In Section2, we first introduce the quantum polynomial algebra kq[x,x-1,y], its Hopf algebra structure and some related conclusions. It is proved that it is a graded Pointed Hopf algebra. Then, we discuss the coalgebra automorphisms and the graded coalgebra automorphism group of kq[x,x-1,y]. It is shown that the graded coalgebra automorphism group of kq[x,x-1,y] is isomorphic to the semidirect product group (k*)z x Z. In Section3, we investigate the Uq (sl2)-module coalgebra structures on kq[x,x-1,y]. Assume that kq[x,x-1,y] is a Uq(sl2)-module coalgebra. Then the action of K on kq[x,x-1,y] is a coalgebra automorphism. If it is a graded coalgebra automorphism, then we say that kg[x,x-1,y] is a normal Uq (sl2)-module coalgebra. In this section, we mainly consider the normal Uq (sl2)-module coalgebra structure of kq[x,x-1,y]. We first describe the general formulas for the actions of the generators of Uq(sl2) on the coradical and the primitive elements of kq[x,x-1,y]. These formulas are given in three cases according to the actions of K on the coradical, respectively. Then in the case that the action of K on the coradical is the identity map, we describe all the possible normal Uq(sl2)-module coalgebra structures of Kq[x, x-1,y].