Quantised algebras and quiver representation theory

箭图表示理论是一个丰富的研究课题，与数学中的各个学科有着广泛的联系。这个项目的重点是它与量子化代数和李代数的联系。20世纪90年代初，S·林格尔在箭图表示范畴上定义了一个霍尔代数，并证明了这种代数实现了量子化包络代数的正部分，即量子群，推广了Gabriel和Kacs关于箭图与李代数联系的结果。Ringel的杰出工作给出了量子包络代数中乘法的含义，并将一个强大的工具引入了量子世界。它启发了量子代数中的一系列杰出工作，使用了霍尔代数方法。同样在20世纪90年代，Lusztig和他的合作者Beilinson和MacPherson给出了Q-Schur代数的几何构造。他还利用倒数组构造了整个量子化包络代数。2011年，Bridgeland给出了整个量子包络代数的一个新的构造，它比Lusztig的构造更初级。这个项目将特别探索仿射量化包络代数的新构造，它们与量子化代数的关系，它们在零点的退化形式，以及箭图表示理论和李理论之间的新联系。

Quiver representation theory is a rich research subject and has broad connections to various subjects in mathematics. This project focuses on its connections to quantised algebras and Lie algebras. In early 1990's Ringel defined a Hall algebra on the categories of representations of quivers and showed such an algebra gives a realisation of the positive part of quantised enveloping algebras, i.e. quantum groups, extending Gabriel and Kacs results on the connections of quivers and Lie algebras. Ringel's remarkable work gives a meaning of the multiplication in quantised enveloping algebras and introduces a powerful tool into the quantum world. It has inspired a series of distinguished works in quantised algebras, using the Hall algebra approach. Also in 1990s, Lusztig gave a geometric construction of q-Schur algebras with his collaborators Beilinson and MacPherson. He also constructed the whole quantised enveloping algebras, using perverse sheaves. In 2011, Bridgeland gave a new construction of the whole quantised enveloping algebra, on a more elementary level than Lusztig's construction. This project will in particular explore new constructions of affine quantised enveloping algebras, their relations to quantised algebras, their degenerate version at zero and new connections between quiver representation theory and Lie theory.