量子群与量子代数的模及其相关理论

本项目将主要研究量子群与量子代数的表示理论中几个重要的关键问题，考虑量子环面代数的表示的张量积模的分解；完善扭型的量子仿射代数的McKay-Slodowy对应，利用McKay-Slodowy对应，研究扭型仿射李代数在张量代数中庞加莱级数；以及量子群的有限维模和余模的关系等课题。对这些问题的研究一方面有助于丰富和完善量子群和量子代数的表示理论，促进量子群和量子代数的表示理论相关问题的解决，另外一方面，本项目中部分课题是用群表示理论去实现扭型(量子)仿射李代数，对这些课题的研究还能建立量子代数与群表示等方向的交叉研究课题，促进其他相关问题的解决，具有重要的理论意义和研究价值。

This project will mainly focus on several important key problems of the representation theories of quantum groups and quantum algebras. We consider the decomposition of the tensor product modules of quantum toroidal algebras; we will complete the McKay-Slodowy correspondence of quantum twisted affine algebras, and using the McKay-Slodowy correspondence, we study the Poincare series in tensor algebra of twisted affine algebras; and study the relations between of the finite-dimensional modules and the comudules of quantum groups and others related problems. On the one hand, the research on these problems will helpful to enrich and improve the representation theory of quantum groups and quantum algebras, to promote to solve the related problems of the representation theory of quantum groups and quantum algebras. On the other hand, some problems in this project are to use the representation theory of group to realize twisted affine algebras, studying these problems will establish the cross topic problems of quantum algebras and group representation, it will promote the solution of other related problems, so this project has important theoretical significance and research value.