QUANTUM GROUP SU (1,1)INFINITY-Z2 AND SUPER-TENSOR PRODUCTS

A quantum analogue of the group SU(1, 1)x with connected to right of it Z2 - the normalizer of SU(1, 1) in SL2(C) - is introduced and studied. Although there is no correctly defined tensor product in the category of *-representations of the quantum algebra C[SU(1, 1)]q of regular functions, some categories of *-representations of C [SU(1, 1)x with bar connected to right of it Z2]q turn out to be endowed with a certain Z2 -graded structure which can be considered as a ''super''-generalization of the monoidal category structure. This ''quantum effect'' may be considered as a step to understanding the concept of quantum topological locally compact group. In fact, there seems to be a family of quantum groups SU(1, 1)x with bar connected to right of it Z2 parameterized by unitary characters beta is-an-element-of T1 of the fundamental group of the two-dimensional symplectic leaf of SU(1, 1)/T, where T is the subgroup of diagonal matrices. It is shown that the quasi-classical analogues of the results of the paper are connected with the decomposition of Schubert cells of the flag manifold SL 2(C)R/B (where B is the Borel subgroup of upper-triangular matrices) into symplectic leaves.