Quantised algebras and quiver representation theory

量化代数和箭袋表示理论

• 批准号: 1787346
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1787346
• 关键词: Quantised; algebras; quiver; representation; theory

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.

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的连接到量子代数和李代数。在1990年初的林格尔定义了霍尔代数的类别表示的箭，并表明这样一个代数给出了实现的积极部分quantised包络代数，即量子群，延长加布里埃尔和卡茨的结果连接的箭和李代数。林格尔的卓越工作给出了意义上的乘法量子包络代数，并介绍了一个强大的工具到量子世界。它激发了一系列杰出的作品在quantised代数，使用霍尔代数的方法。同样在20世纪90年代，Lusztig与他的合作者Beilinson和MacPherson给出了q-Schur代数的几何构造。他还建造了整个quantised包络代数，使用反常层。2011年，Bridgeland给出了一个新的量子化包络代数的构造，比Lusztig的构造更初级。这个项目将特别探索仿射量子包络代数的新构造，它们与量子代数的关系，它们在零处的退化版本以及量子表示理论和李理论之间的新联系。