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

这项研究的目标被放在一个如此高的水平，以至于我们无法在申请这项研究资助的情况下进行所有的项目。在我们对量子对称空间的研究过程中，我们认识到一个没有实现为量子对称空间的量子四维球，并且成功地在这样的空间上显式地构造了量子矢丛，这在量子规范理论的指导下是非常基础的，留下了许多有待于进一步研究的问题。然后，同时，我们面临着计算那些量子流形的循环上同调，所以我们决定使用一种特殊类型的计算机系统，这种计算机系统从未尝试过使用纯数学来解决这样的问题，因为手工计算的数量非常巨大。然而，计算机辅助计算的这个问题仍然与计算机系统本身的调谐一起留下。另一方面，一个处理所谓的局部紧致量子群的物体的一般框架最终完成，并已提交给法国的一家期刊。通过利用这个框架，我们现在准备好继续非常详细地研究明确给出的非紧致类型量子群的例子的方向。

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.