权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Demazure Flags, Hypergeometric Series, and Quantum Affine Algebras

Demazure 标志、超几何级数和量子仿射代数

基本信息

批准号：
1719357
负责人：
Vyjayanthi Chari
金额：
$ 15.5万
依托单位：
University of California-Riverside
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719357&HistoricalAwards=false
关键词：
Demazure Flags Hypergeometric Series Quantum

项目摘要

The study of Lie groups and Lie algebras has a long and distinguished history going back to the mid-nineteenth century. It had its roots in the idea that the geometry of space is determined by its group of symmetries. As our understanding of space has evolved, the study of Lie groups and Lie algebras (Lie theory) has become a central part of modern mathematics. The subject has also always benefited by interactions with physics and these led to two of the major developments in the last century; the definition and study of Kac-Moody algebras and quantum groups. These objects turned out to have many connections to physics, combinatorics and number theory. One of the early successes of their study was the fact that many classical identities established by the famous Indian mathematician Srinivasa Ramanujan (1887-1920) could be recovered and interpreted in the language of the representation theory of Kac-Moody algebras. Quantum groups are deformations of Kac-Moody algebras and their study has led to important connections with knot theory and mathematical physics. One of the goals of this project it to exploit the synergy between the study of Kac-Moody algebras and their quantum analogs to study questions in representation theory and to relate them to various series of functions which appear in the work of Ramanujan.This project focuses on the study of the connections between Demazure modules in an affine Kac-Moody Lie algebra and the theory of hypergeometric series, quantum affine algebras, generalized Q-systems and cluster algebras. In the simplest cases, Demazure modules are indexed by a pair of positive integers one of which is called the level of the module. It can be shown that a module of a fixed level admits a flag where the successive quotients are all higher level Demazure modules. This information can be encoded in the form of a generating series and a goal of the project is to relate them to q-hypergeometric series and to study their modularity properties. The study of Demazure modules is also important in the theory of quantum affine Kac-Moody algebras. Another goal of this project is to establish character formulae for prime representations of quantum affine algebras. The latter are known to be related to cluster algebras and the proposal will study the combinatorial problems arising from these many connections.

李群和李代数的研究有着悠久而杰出的历史，可以追溯到世纪中期。它的根源在于空间几何是由它的对称群决定的。随着我们对空间的理解不断发展，李群和李代数（李理论）的研究已经成为现代数学的核心部分。这个问题也一直受益于相互作用与物理和这些导致了两个重大发展，在上个世纪;的定义和研究卡茨穆迪代数和量子群。这些对象原来有许多连接到物理学，组合数学和数论。他们研究的早期成功之一是，著名印度数学家斯里尼瓦萨·拉马努金（Srinivasa Ramanujan，1887-1920）建立的许多经典等式都可以用卡茨-穆迪代数表示论的语言来恢复和解释。量子群是Kac-Moody代数的变形，它们的研究与纽结理论和数学物理有着重要的联系。该项目的目标之一是利用Kac-Moody代数及其量子类似物的研究之间的协同作用来研究表示论中的问题，并将它们与Ramanujan工作中出现的各种函数系列联系起来。该项目重点研究仿射Kac-Moody李代数中Demazure模与超几何级数理论，量子仿射代数，广义Q-系统和簇代数。在最简单的情况下，Demazure模块由一对正整数索引，其中一个称为模块的级别。可以证明，一个固定级别的模块允许一个标志，其中连续的后继者都是更高级别的Demazure模块。这些信息可以以生成序列的形式进行编码，该项目的目标是将它们与q-超几何序列联系起来，并研究它们的模块性属性。Demazure模的研究在量子仿射Kac-Moody代数理论中也是重要的。这个项目的另一个目标是建立量子仿射代数素表示的特征公式。后者是已知的有关集群代数和建议将研究组合问题所产生的这些许多连接。