量子群、Groebner-Shirshov 基理论及相关问题研究

结合代数的Groebner-Shirshov 基理论（下面简称G-S基理论）在代数结构的研究中有很重要的作用。用量子群的G-S基理论来研究其结构将对研究量子群提供一个新方法。目前用G-S基理论来研究量子群结构方面的工作，至少据我所知，非常少。本项目主要是用量子群（或者其正部分）的G-S基来研究量子群的以下几个结构方面的性质：一、Verma 模的投射分解；二、Gelfand-Krillov 维数；三、整体为数；四、Hilbert序列；五、Koszul 性质；六、其它一些相关代数的可以用G-S基来研究的性质及构造几种相关代数的 G-S 基。我们的研究方案是用我们已得到的量子群的G-S基以及其它有关G-S基理论在非交换代数上的应用方面的结果来在量子群上讨论以上问题并得到较好的结果。我们认为用G-S基理论来讨论以上问题是可行的，并且预期的结果将对更好地了解量子群的结构有非常重要的意义。

Groebner-Shirshov bases theory (G-S bases theory for short in the following) of associative algebras have many important roles in the study of structures of corresponding associative algebras.Study quantum groups through its G-S bases will provide a new way to understand the structure of quantum groups. Now, at least up to our knowledge, there are very few work on the study of structural properties of quantum groups through their G-S bases. The aim of this project is to study the following structural properties of quantum groups using their G-S bases:1. the projective resolutions of Verma modules of quantum groups; 2. Gelfand-Krillov dimensions of quantum groups; 3. the global dimensions of quantum groups; 4. Hilbert series of quantum groups; 5. Koszulity of quantum groups; 6. some properties of related algebras which can be studied through their G-S bases theory and construct G-S basis for several relevant algebras. Our philosophy is to use the known G-S bases of quantum groups which we obtained and the relevant results about the applications of G-S bases theory on non-commutative algebras, and will get some nice results. We believe that the study of above problems using G-S bases theory is feasible and the expected results will be very important for the study of structures of quantum groups.