| In this article,we will introduce a set of new Hopf algebra,it is closely related to the cor-responding quantum groups and quantum super groups,we call these new Hopf algebras the degenerate quantum groups.Our study of the degenerate quantum groups of type C are based on the degenerate quantum groups of type A research.This thesis consists of two chapters.In the first chapter,the main results are as follows:Theorem 1.3.1 Let Uq(sp2l,2m)be the unital associative algebra over C(q)generated by the set of elements {ea,fa,Ka,Ka-1|a∈I} subject to the relations from(1.19)to(1.26)only.Then Uq(sp2l,2m)has the structure of a Hopf algebra with co-multiplication△:Uq(sp2l,2m)→Uq(sp2l,2m)(?)Uq(sp2l,2m),△(ea)=ea(?)ka+1(?)ea,△(fa)=fa(?)1+ka-1(?)fa,△(Ka)=Ka(?)Ka,co-unit ε:Uq(sp2l,2m)→C(q),ε(ea)=ε(fa)=0,ε(Ka)=1,and antipode S:Uq(sp2l,2m)→Uq(sp2l,2m),S(ea)=-eaka-1,S(fa)=-kafa,S(Ka)=Ka-1,for all a ∈I.Theorem 1.3.3 Let J be the two-sided ideal generated by Q+ and Q-in Uq(sp2l,2m)Then J is a Hopf ideal,and Uq(sp2l,2m)=Uq(sp2l,2rn.)/J,which is a Hopf algebra with co-multiplication,co-unit and antipode induced by the corre-sponding structure maps of Uq(sp2l,2m)In the second chapter,we will focus on research of the exchange relations among elements of Uq(sp2l,2m).Lemma 2.1.1 The algebra Uq(sp2l,2m)admits the following triangular decomposition Uq(sp2l,2m)=U-U0U+,that is,every element of Uq(sp2l,2m)can be expressed as a linear combination of elements of the form u_u0u+ with u_∈ U-,u0∈U0 and u+∈U+.Further-more,Uq(b)=U0U+.Theorem 2.2.1 F(θ)KE(θ)can spans Uq(sp2l,2m)linearly. |
