无限维代数及其表示与上同调

This project is to study the quantum groups associated with quivers of infinite representation type, and the algebras determined by (possibly infinite) quivers and relations..This research dealt with in general infinite-dimensional algebras, however it used the representation theory of finite-dimensional algebras as a main tool, and took the Ringel-Hall algebras as a bridge between the quantum groups and quivers. We studied the two mentioned objects from their behaviors in structure theory, representation theory, and cohomologies, not only restrictly in one respect. In particular, we classified all representations over any tame quiver such that they fall into the composition algebra of the quiver, which is a subalgebra of the Ringel-Hall algebra generated by all irreducible representations; gave a PBW basis of the composition algebra of the Kronecker algebras; described some properties of the Ringel-Hall algebras and the Green classes in the framework of twisted Hopf algebras and obtained new decompositions of their structures;.proved that the first Hochschild cohomology of a (possibly infinite) quiver vanish if and only if it is a tree. Under the support of this project, we have published up to now 19 papers such as in “Trans. Amer. Math. Soc.”, “J. reine angew. Math.”, “J. Algebra”, “Comm. Algebra”,.“Algebra Colloquium”, “Science in China” etc. These works were cited by colleagues, and were invited to present talks in international conferences and universities. They are certainly progress in the Hall algebra approach to the quantum groups, and also initial instudying relations between infinite quivers and cohomologies.

本项目研究两类具有重要背景的无限维代数：即无限型箭图相应的量子群和无限箭图及关系理想确定的代数。这种研究将表示作为量子群的元素而得到表示与结构之间的关系和分类；以图的组合性质研究代数的（上）同调群并得到相互的分类。这对于从结构、表示、上同调、组合和计算全方位整体地研究无限维代数具有基本的、内在的理论价值和意义。