Gr(?)bner-Shirshov Basis Of Quantum Group And Quantum Supergroup

The Grobner basis theory for commutative algebras was introduced by Buchberger and provides a solution to the reduction problem for commutative algebras. Bergman and Shirshov developed the Grobner basis theory for associative algebras and Lie algebras respectively, then Bokut proved that Shirshov's method works for associative algebras as well, therefor, Shirshov's theory for Lie algebras and their universal enveloping algebras is called the Grobner-Shirshov basis theory. Bokut and Malcolmson developed the theory of Grobner-Shirshov basis for the quantum enveloping algebras,or the so-called quantum groups and explicitly constructed the basis for the quantum groups of type An for (q8≠1). In this dissertation, we follow this idea and using the calculation of composition as a basic tool,we prove that the set of given relations is closed under composition, and consequently get a Grobner-Shirshov basis for Z/3Z-quantum group and the quantum groups of type B2.Yamane introduced a new quantum group U, that is the Z/3Z-quantum group, and it can be viewed as a Z/3Z- graded version of a quantum superalgebra. As a Hopf algebra, this new quantum group is not isomorphic to any of quantum algebras and quantum superalgebras. In the first part of this dissertation,we give a Grobner-Shirshov basis of this new quantum group.For constructing a PBW type basis for quantum groups,Ringel constructed a gen-erating sequence for Ringel-Hall algebras and some skew commutator relations for these generators using the Auslander-Reiten theory. In the second part of this dissertation, as an attempt to construct a bridge between Grobner-Shirshov basis theory and the representation theory of finite dimension algebra, we give a Grobner-Shirshov basis for quantum group of type B2 by using the relations constructed by Ringel and the canonical isomorphism between quantum groups and the Ringel-Hall algebra. This work is very meaningful,because in the process of finding the Grobner-Shirshov basis for quantum group we found that it is very difficult to construct the Grobner-Shirshov basis for quan-tum group using the method of Shirshov. So we used the Auslander-Reiten theory in the representation theory of finite dimension algebra and Ringel-Hall algebra to get some skew-commutator relations for the generators, then prove that the set of these relations is closed under composition.Finally, get the Grobner-Shirshov basis for quantum group of type B2.We hope that this new approach will enable us to construct a Grobner-Shirshov basis for quantum group of type Bn.