结合代数的表示与导出范畴

结合代数的表示理论是数学研究中一个十分活跃的领域，它与数学、物理领域的许多分支相互渗透，具有广泛而深刻的应用。Grothendieck和Verdier在代数几何中引入的导出范畴(三角范畴)，通过Happel、Rickard、Keller等著名数学家开创性的工作，现已成为当今代数几何和代数表示论等研究领域中不可缺少的关键方法之一。本项目将利用导出范畴、同调代数等一系列工具，并借鉴组合与几何方法，对无限倾斜模理论、代数的导出等价和Morita型稳定等价、量子齐次空间等热点、前沿问题展开深层次的系统性研究，并将代数表示论的方法应用于高阶代数K-理论和量子群等核心数学问题的讨论。这些问题的研究对代数表示论、代数K-理论、量子群和相关学科的深入发展有十分重要的意义。

The representation theory of associative algebras is one of the most active and attractive research fields in mathematics. It is of great importance and has wide applications in many other branches in mathematics and physics. Derived categories were introduced by Grothendieck and Verdier in algebraic geometry. Because of the works of Happel, Rickard, Keller and many other international leading mathematicians, derived categories become one of the most important methods in representation theory and geometry. We shall use derived categories, homological algebra, together with methods from combinatorics and geometry, to investigate collaboratively and systematically the several important topics: infinite tilting theory, derived equivalences and stable equivalences of Morita type, quantum homogeneous spaces; and use methods from representation theory to study higher algebraic K-theory, quantum groups and related topics. This project will lead to a significant benefit to the developement of representation theory, algebraic K-theory, quantum algebras and related topics.