Some Algebraic Properties About Weak Hopf Superalgebra Wosp(1,2)

In 1998, Ian M. Musson and Zou Yiming showed that quantum enveloping algebrasUq(osp(1,2n)) of contragredient classical Lie superalgebras osp(1,2n) have Hopf superalge-bra structure. They discussed crystal bases[14] and integrable representations of Uq(osp(1, 2n))[21].When n = 1, the structure of Uq(osp(1, 2)) is similar to that of Uq(sl(2)). Li libin[12] constructedthe center of Uq(osp(1, 2)) using the method used to give the center of Uq(sl(2)). In 1998, LiFang defined weak Hopf algebras[10]. In this paper, we define the weak Hopf super-algebrawosp(1,2) using the method of constructing wslq(2)[9]. The contents of this paper are dividedinto three chapters to research the algebraic properties of wosp(1, 2) .In chapter 1, we first recall the basic results of Hopf super algebra Uq(osp(1, 2)) and givethe definition of weak Hopf algebra wosp(1, 2). Next, we prove that wosp(1, 2) is Noetherianand the seis the base of wosp(1, 2), usingweak Ore expansion. Finally, we prove that it has a weak Hopf super-algebra structure, andgive that the set of the group-like elements of wosp(1, 2), denoted as G(wosp(1, 2)), that isIn the beginning of the chapter 2, we study the integrable representations of wosp(1, 2).Since K, Kˉare not reversible, the discussion of weightλof wosp(1, 2)-module V is classifiedtwo cases: 1)λ= 0; 2)λ= 0. Next, we find that for any finite dimensional irreduciblewosp(1,2)-module V , either V～= Vε,n or V～= W(0). Finally, we get an important conclusion inthis paper that is the quantum Clebsch Gordan decomposition formula of finite-dimensionalintegrable highest weight Z2-graded wosp(1, 2)-modules.In the last chapter, set W = wosp(1, 2)J and Y = wosp(1, 2)(1 J). As ideals ofwosp(1,2), we have wosp(1,2) = W⊕Y , and the isomorphism of Hopf super-algebrasW～= Uq(osp(1,2)). It's well known that the center of Uq(osp(1,2)) is a polynomial algebragenerated by Cq. So, we have the center of W and the center of wosp(1, 2) is a direct sum ofthe center of W and the center of Y . Finally, we give the necessary and sufficient conditions forwospq(1,2)～= wospp(1,2) to be a weak Hopf super-algebra is that p = q.