Some Sub - Algebras Of U_q And Its Related Structures

Quantum enveloping algebra is an important topic in algebra. During the past nearly thirty years, many mathematicians have done a lot of researches on Uq and made great progress. It is found that quantum polynomial algebra which is a significant tool to research the quantum group is closely related with the non-commutative algebraic geometry. In algebra, various algebraic structures together with their automorphism groups attract numerous mathematicians’ attention. The reason is that the automorphism group is an important invariant, which reflects the symmetry of the algebra structure and plays an important role in the classification of the algebraic structures. On the other hand, the action theory of Hopf algebras also play an important role in the study of algebraic structures, which may be applied to construct a new algebra through extension.Based on the previous studies, we will study some subcoalgebras and their related structures. The automorphism groups of these subcoalgebras as well as the Uq-module coalgebra structures on them will be investigated. This paper is organized as follows. In Section 1, as preliminaries, some related concepts are introduced, such as coalgebra, coalgebra automorphisms, Hopf algebras and quantum enveloping algebra, module coalgebra and so on. In Section 2, a coalgebras (C,Δ,s) is firstly constructed, which has a k-basis {gn,hn|n∈Z}, where gn is group like element and hn is semi-primitive element. It is shown that C is isomorphic to some subcoalgebras of Uq. Then, the set of the group like elements of C is determined. It is shown that G(C)={gn|n ∈ Z}. Finally, the coalgebra automorphisms of C are examined, and the coalgebra automorphism group of C is described. In Section 3, the actions on C of the quantum enveloping algebra Uq is investigated. All the actions on C of Uq such that C becomes a Uq-module coalgebra are determined, which display basically all the Uq-module coalgebra structures on C.