Demazure Flags, Hypergeometric Series, and Quantum Affine Algebras

李群和李代数的研究有着悠久而杰出的历史，可以追溯到19世纪中期。它的根源在于空间的几何形状是由它的一组对称性决定的。随着我们对空间理解的不断发展，李群和李代数（李论）的研究已成为现代数学的核心部分。这门学科也一直受益于与物理学的相互作用，这导致了上个世纪的两个主要发展；Kac-Moody代数和量子群的定义和研究。事实证明，这些物体与物理学、组合学和数论有很多联系。他们研究的早期成功之一是，由印度著名数学家Srinivasa Ramanujan（1887-1920）建立的许多经典恒等式可以用Kac-Moody代数的表示理论的语言来恢复和解释。量子群是Kac-Moody代数的变形，它们的研究导致了与结理论和数学物理的重要联系。这个项目的目标之一是利用Kac-Moody代数和它们的量子类似物之间的协同作用来研究表征理论中的问题，并将它们与拉马努金作品中出现的各种函数系列联系起来。本课题主要研究仿射Kac-Moody李代数中Demazure模与超几何级数、量子仿射代数、广义q系统和聚类代数理论之间的联系。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，一个固定水平的模块允许一个标志，其中连续商都是更高水平的Demazure模块。这些信息可以以生成级数的形式编码，项目的目标是将它们与q-超几何级数联系起来，并研究它们的模块化性质。Demazure模的研究在量子仿射Kac-Moody代数理论中也占有重要地位。本课题的另一个目标是建立量子仿射代数素数表示的特征公式。后者已知与簇代数有关，该提案将研究由这些许多连接引起的组合问题。

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.