| Quantum groups and their representations play an important role in many fields of mathematicis and physics. For the finite-dimensional Lie algebra sl2, the usual quantum enveloping algebra Uq(sl2) has been extensively studied. There are several generaliza-tions of Uq(sl2) have been studied.In this paper, we generalize Uq(osp(1,2)) by adding central generators c and c-1to get a new quantum superalgebras Uq(osp(1,2,c)). These new quantum superalge-bras share many similar properties with quantum superalgebras Uq(osp(1,2)). First, we give the definition of Uq(osp(1,2,c)) and obtain some properties of Uq(osp(1,2,c)). We also define the comultiplication and the counit on Uq(osp(1,2,c)) to make it into a Hopf supealgebra. Then we determine all finite-dimensional irreducible represen-tations and find that there exists finite-dimensional non-semisimple Uq(osp(1,2,c))-modules. Also, we discuss representations of Uq(osp(1,2,c)) in the case when q is a root of unity. Finally, we study the Harish-Chandra homomorphisim and the centre of Uq (osp(1,2, c)). We introduce the quantum Casimir element and the quantum Scasimir element. A generalization of classic results is obtained:1.(a) Let V be a finite-dimensional Uq(osp(1,2,c))-module generated by the highest weight vector v of weight (λ,α). Then(1) λ=εtnαr, where ε=±1and n is the integer defined by dim V=n+1;(2) Setting vp=1/[p]t!Fpv, v0=v,we have vp=0for p> n and the set{v0, v1,..., vn} is a basis of V;(3) Any other highest weight vector in V is a scalar multiple of v and is of weight (λ,α);(4) The module is simple.(b) Any simple finite-dimensional Uq(osp(1,2, c))-odule is generated by the high-est weight vector. Two finite-dimensional Uq (osp (1,2, c))-odules generated by high-est weight vector of the same weight are isomorphic.2. When q is not a root of unity, then centre Z(U) of U=Uq(osp(1,2,c)) is a polynomial algebra generated by elements Cq,c,c-1over the field K. The restriction of Harish-Chandra homomorphism to Z(U) is an isomorphism onto the subalgebra of K[K, K-1,c,c-1] generated by c,c-1, and qK2+q-1K-2c2r. |
