| It's well known that one-parameter Lie superalgebra Uq(osp(1, 2))and two-parameter Liesuperalgebra Ur,s(osp(1, 2)) can be regarded as quantum deformations of Lie superalgebraosp(1,2). In this paper, we construct a more general quantum deformation Uq(osp(1,2,f(K,H))),which is generated by six variables E,F,K,H,K-1,H-1 with the following relations:(R1) KH = HK;(R2) KK-1 = K-1K = 1,HH-1 = H-1H = 1;(R3) KEK-1 = qE, HEH-1 = q-1E;(R4) KFK-1 = q-1F, HFH-1 = qF;(R5) EF + FE = f(K,H).where f(K,H) =∑iN=-N∑jN=-N aijKiHj∈k[K,K-1,H,H-1].This paper is divided into the following three parts:In the first part, we discuss the algebraic properties of Uq(osp(1, 2,f(K,H))). We getthe relations among the generators, then we prove that the algebra Uq(osp(1, 2,f(K,H))) isNoetherian and has no zero divisors, and get its basis i.e. the set {EiFjKsHr}i,j∈N,s,r∈Z. Fur-thermore, we research the representations of Uq(osp(1, 2,f(K,H))) and define its Verma mod-ules.In the second part, we give a sufficient and necessary condition for the algebra Uq(osp(1, 2,f(K,H))) to be a Z2-graded super Hopf algebra. We have the following important theorem:Proposition 2.5 Assumef(K,H)∈k[K,K-1,H,H-1] is a Laurent polynomial. Thenthe algebra Uq(osp(1, 2,f(K,H))) is a Z2–graded super Hopf algebra such that K,K-1,H,H-1are group-like elements and E,F are skew primitives if and only if f(K,H) = a(KmHn -K-m′H-n′) for some 0=- a∈k,m,n,m′,n′∈Z,s-t = s′-t′,l-r = l′-r′,m = s-l′,n =t - r′,m′= s′- l,n′= t′- r.SetM = m - n = m′- n′= r + s - t - l.In the third part, we mainly construct the center of the algebra Uq(osp(1, 2,f(K,H))),using the Casimir element Cq and Harish-Chandra homomorphism, and prove that the centerhas a polynomial structure. The main result is following:Theorem 3.2.5 The center of the quantum group Uq(osp(1, 2,f(K,H))) is a polyno-mial algebra generated by KH and Cq. The restriction of Harish-Chandra homomorphismπto Z(U) is an isomorphism onto the subalgebra of k[K,K-1,H,H-1] generated by KH andqMK2mH2n + q9-M)K-2m′H-2n′.
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