Irreducible unitary representation of non compact quantum group SUq(1,1) and its quantum symmetric space

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The target of this research had been put in such a high level so that we were not able to carry out all the projects on the occasion that we asked for this research grant. On the process of our research concerning the quantum symmetric space, we recognized a quantum 4-dimensional spheres which are not realized as the quantum symmetric space, and we also succeeded to have an explicit construction of a quantum vector bundle on such spaces which are regarded to be very basic in the direction of quantum gauge theories leaving lots of problems to be studied in the future. Then, at the same time, we faced to compute the cyclic cohomologies of those quantum manifolds so that we decided to use a special type of computer system which had never tried to use for such a problem coming from pure mathematics due to the extremely huge amount of computations by hands. However, this problem of computer aided computation is still left together with the tunung of the computer system itself. On the other hand, a general framework to deal with the objects which are so called the locally compact quantum groups was eventually finished and already submitted to a journal in France. By making use of this framework, we are now ready to proceed in the direction of the very detailed study of the explicitly given examples of the quantum groups of the non compact type.