Degenerate Quantum General Linear Groups

In this thesis,we introduce a new class of Hopf algebras,and develop their structure and representation theory.We will call these new Hopf algebras degen-erate quantum groups because of their close connections with the Drinfeld-Jimbo quantum groups.We shall focus on the type A case in this thesis.The main idea of this work originated from Zacho s algebra and the quantum correspondences be-tween quantum supergroups and quantum groups.We will employ Hopf algebraic and Lie theoretical techniques to investigate the degenerate quantum groups.We first study the degenerate quantum group of Uq(sl3)as a motivating example,then generalise the investigation to the degenerate quantum group of Uq(gln)for arbi-trary n.This thesis consists of two chapters,which are as follows:In the first chapter:we introduce a degenerate version Uq(sl2,1)of the quantum group Uq(sl3),the Hopf algebra structure of Uq(sl2,1)is explicitly described,and a PBW theorem for its negative part is established.We classify the finite dimension-al irreducible modules for Uq(sl2,1).and construct an explicit basis for each such module.The main conclusions are as follows:Lemma 1.2.2 For all i=1,2,the Uq(sl2,1)has the structure of a Hopf alge-bra:co-multiplication Δ:Uq(sl2,1)→Uq(sl2,1)(?)Uq(sl2,1),Δ(ei)=ei(?)ki+1(?)ei,Δ(fi)=fi(?)1+ki-1(?)fi,Δ(ki)=ki(?)ki.co-unit ε:Uq(sl2,1)→C(q),ε(ei)=ε(fi)=0,ε(ki)=1.antipode S:Uq(sl2,1)→Uq(sl2,1),S(ei)=-eiki-1,S(fi)=-kifi,S(ki)=Ki-1.Theorem 1.2.4 Let J be the two-sided ideal in Uq(sl2.1)generated by the ele-ments S12(+),S12(-),S2(+)and S2(-)Then J is a Hopf ideal,and Uq(sl2,1)=Uq(sl2,1)/J is a Hopf algebra with the induced comultiplication,counit and antipode.Lemma 1.4.1 The simple Uq(sl2,1)module L(A)with highest weight λ=(λ1,λ2)is finite dimensional if and only if λi=±q(?),(?)∈Z+.Theorem 1.4.2 Let λ=(λ1,λ2)with λ1=±ql,l∈Z+.Then the simple Uq(sl2,1)-module L(λ)has dimension 4(l+1)if and only if(qλiλ2—q-1λ1-1λ2-1)(λ2-λ2-2)≠0.In this case,vectors f1kuλ,f2f1kuλ,Ff1kuλ,Ff21kuλ,k=0,1,...,l forms a basis of L(λ).Theorem 1.4.3 Assume that λ1=±ql,l∈Z+,and assume that the atyp-icality condition(qλ1λ2-q-1λ1-1λ2-1)(λ2-λ2-1)=0 holds.Then λ2 belongs to one of the following mutually exclusive c:ases:(a)λ2 =±1,or(b)λ2=±q-1λ1-1.Ifλ2=±1,L(λ)is 2l＋1 dimensional with a basis{fiuλ,Ff1kuλ|0≤j≤l,0≤k≤l-1},If λ2=±g-1λ1-1,L(λ)is 2(l＋1)+1 dimensional with a basis{f1kuλ,f2f1kuλ,Ff1luλ|0≤k≤l}.In the second chapter,we generalise the definition of the degenerate quantum group Ua(sl2,1)given in Chapter 1 to construct a degenerate quantum general lin-ear group Uq(glm.n),which includes the degenerate quantum special linear group Ug(slm,n)as a subalgebra.The Hopf algebraic structure of Ug(glm,n),and Uq(slm,n)as a Hopf subalgebra,is developed.The finite dimensional simple modules for Uq(glm.n)are classified in terms of highest weights,and we give its tensor represen-tation.The main results are as follows:Lemma 2.2.1 For all a ∈I’,b∈I,Uq(glm,n)has the structure of a Hopf algebra:co-multiplicatjion Δ:Uq(glm,n)→Uq(glm,n)(?)Uq(glm,n),Δ(ea)=ea(?)ka+1(?)ea,A(fa)=fa(?)1+ka-1(?)fa,Δ(Kb)=Kb(?)Kb.co-unit ε:Uq(glm,n)→(g),ε(ea)=ε(fa)=0,ε(Kb)=1.antipode S:Uq(glm,n)→Uq(glm,n),S(ea)=-eaka-1,S(fa)=-kafa,S(Kb)=Kb-1.Theorem 2.2.3 Let J be the two-sided ideal in Uq(glm,n)generated by the elements Q-Q-.Then J is a Hopf ideal,and U,(glm,n)=Ug(glm,n)/J is a Hopf algebra with the induced co-multiplication,counit and antipode.lemma 2.4.1 The finite dimensional Uq(glm,n)-modules are all the highest weight modules.Theorem 2.4.2 The simple Uq(glm,n)-module L(A)is finite dimensional if and only if its highest weight A satisfies the condition λa/λa+1=ωaqala,where la∈Z+,ωa=±1,(?)a≠m.Lemma 2.5.1 There is a Uq(glm,n)-action on V defined,for all a∈I’,b,c∈I,by eauc=δa+l,cua,fauc=δacua+1,:Kb±1uc=qb±δbcuc.The corresponding representation v:Uq(glm,n)→Endc(q)(V)is given by v(ea)=ea,a+1,v(fa)=ea+1,a,v(Kb)=1+(qb-1)ebb.By using the Hopf algebraic structure of Ug(glm,n),we can turn the tensor prod-ucts of any Uq(glm,n)-modviles into a Uq(glm,n)-module.In paxticulax,we have the Uq(glm,n)-modules V(?)r(?)(V*)(?)s for r,s=1,2,...,We will call them tensor modules.Note that Ug(glm,n)acts on these modules through the iterated comultiplication,Δ(r+s-1)=(Δ(?)id(?)(r+s-2))(Δ(?)id(?)(r+s-3))…(Δ?id)Δ.