本项目就两个问题展开研究: 1.双参数量子群的构造及其几何实现；2.cluster代数范畴化及其应用。我们讨论flag variety上取值于域C(t)的函数空间，给出B型和D型双参数量子群的构造和几何实现，并给出Shur-Weyl对偶。在cluster代数范畴化方面，我们将Nakajima quiver variety的方法推广，范畴化更一般情形的cluster代数，并以Donaldson-Thomas不变量“层”为工具，给出cluster代数新的范畴化并讨论其在Donaldson-Thomas不变量计算方面的应用。

This project will focus on the two problems. One is the constructions of two parameter quantum groups and their geometric realizations. Another is the categorifications of cluster algebras and their applications.. We shall study the functions spaces valued in C(t) over the flag varieties. By this method, the constructions of quantum groups of type B and type D, Shur-Weyl duals and their geometric realizations are given. On the problem of categorifications of cluster algebras, we shall use two methods to study it. 1. Generalize the method of Nakajima quiver varieties to give categorifications of more general cluster algebras. 2. Study the problem of categorifications of cluster algebras by the tool of Donaldson-Thomas invariants “sheafies” and study the applications of cluster algebras in the calculations of Donaldson-Thomas invariants.