Algebra Isomorphisms And Algebra Automorphisms Of U_q(osp(1,2,f))

As a generalization of quantum algebra Uq(osp(1,2)), we introduce the algebra Uq(osp(1,2, f)). We mainly study finite dimensional representations of Uq(osp(1,2,f)), and discuss the al-gebra isomorphisms, algebra automorphisms and super Hopf algebra isomorphisms of Uq (osp(1,2,f)) in the case q is not a root of unit.Let C be the complex field. Uq(osp(1,2,f)) is defined as an associative algebra generated by the four variables E, F, K, K-1with the relations:(R1) KK-1=K-1K=1;(R2) KEK-1=qE;(R3) KFK-1=q-1F(RA) EF+FE=f(K); whereIn this thesis, we only consider the case and m is odd. The thesis consists of five parts.In the first part, we introduce the definition of the algebra Uq(osp(1,2,f)) and give its super Hopf algebra structure.In the second part, we study the finite dimensional representations of the algebra Uq(osp(1,2, f)), and classify finite dimensional irreducible Uq(osp(1,2,f))-modules.In the third part, we mainly discuss the algebra isomorphisms of the algebra Uq(osp(1,2,f)), the main result is the following theorem:Theorem3.3Suppose0≠p∈C,0≠q∈C, and p≠±1, q≠±1,f(x)∈C[x,x-1], then Uq(osp(1,2,f)) and Up(osp(1,2,f)) are isomorphic as C-algebra if and only if p=q or p=q-1.In the fourth part, we mainly discuss the algebra automorphism of the algebra Uq (osp(1,2,f)), the main result is the following theorem:Theorem4.1Suppose q E C*is not a root of unit, α is a2m-th root of unity, r∈Z, λ∈C*,then (?)∈AutC(Uq(osp(1,2,f))) if and only if (?)(K)=αK,(?)(K-1)=α-1K-1,(?)(E)=λ-1EK-r,(?)(F)=αmλKrF.In the fifth part, we mainly discuss the super Hopf algebra isomorphism of Uq(osp(1,2,f)), the main result is the following theorem:Theorem5.3Suppose0≠p∈C,0≠q∈C, and p, q are not roots of unit, then (?):Uq(osp(1,2,f))â†'Up(osp(1,2,f)) is a super Hopf isomorphism if and only if p=q or p=q-1.