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

这项研究的目标被置于如此高的水平，以至于我们无法在我们要求这项研究资助的时候执行所有的项目。在我们对量子对称空间的研究过程中，我们认识到了一个量子四维球体，它并没有被实现为量子对称空间，我们也成功地在这种空间上明确地构造了一个量子向量束，这在量子规范理论的方向上被认为是非常基础的，但在未来还有很多问题需要研究。与此同时，我们还面临着计算这些量子流形的循环上同调的问题，因此我们决定使用一种特殊的计算机系统，这种系统从未尝试过用于纯数学中的这种问题，因为手工计算的量非常大。然而，计算机辅助计算的问题仍然与计算机系统本身的调试问题一起遗留下来。另一方面，一个处理所谓局部紧量子群的一般框架最终完成，并已提交给法国的一家杂志。通过利用这一框架，我们现在准备对非紧型量子群的明确给出的例子进行非常详细的研究。

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.