QUANTUM SUPER SPHERES AND THEIR TRANSFORMATION GROUPS, REPRESENTATIONS AND LITTLE t-JACOBI POLYNOMIALS

Quantum super 2-shpheres and the corresponding quantum super transformation group are introduced in analogy to the well-known quantum 2-shpheres and quantum SL(2), connection between little t-Jacobi polynomials and the finite dimensional representations of the quantum super group is formulated, and the Peter-Weyl theorem is obtained. The quantum group SUq(2) introduced by Woronowicz and the quantum 2spheres introduced by Podles̀ play basic roles in quantum group and quantum homogeneous space theory. In terms of “quantum homogeneous spaces”, the Podles̀ spheres can be regarded as quantum homogeneous spaces with SUq(2) as their transformation group. The quantum group SUq(2) (or SLq(2)) and quantum 2spheres have been studied intensively (there is a comprehensive bibliography in [KS]). The results obtained in these studies revealed connections with other fields of mathematics and form the foundation of more general studies in quantum group and quantum homogeneous spaces. For a contragredient Lie superalgebra, besides sl(2), there are two other basic building blocks, namely sl(1, 1) and osp(1, 2). The structure of the corresponding quantized enveloping algebra of sl(1, 1) is relatively simple because of the condition f = e = 0 on the generators. Though there have been many publications devoted to the quantized enveloping algebra Uq(osp(1, 2n)) of osp(1, 2n) (especially when n = 1) and its representations, there does not seem to be a detail study of the corresponding quantum group that is comparable to the existing SUq(2) theory, perhaps due partially to the fact that in general, the theory of osp(1, 2n) is similar to that of so(2n + 1), and for the quantum osp(1, 2) the theory is similar to that of SUq(2). But the similarity deserves further explanation. Over the field of complex numbers, the first nontrivial representation of the Lie superalgebra osp(1, 2) is 3-dimensional, and all nontrivial finite dimensional representations have odd dimensions. In contrary, the finite dimensional representations of the deformation Uq(osp(1, 2)) assume both even and odd dimensions (see [MZ]). This makes the representation theory of the Uq(osp(1, 2)) even closer to that of Uq(sl(2)) in this aspect. On the other hand, there is a difference between the nontrivial even dimensional representations of these two quantum algebras: the ones for Uq(sl(2)) are self-dual, but not for Uq(osp(1, 2)). When the duals of these modules are used to 1991 Mathematics Subject Classification. 17B35, 17B60, 17B70, 22E70. Typeset by AMS-TEX 1