Matrix Coefficients of Covering Groups, Quantum Groups, and Lie Superalgebras

覆盖群、量子群和李超代数的矩阵系数

• 批准号: 1801527
• 负责人: Benjamin Brubaker
• 金额: $ 15万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801527&HistoricalAwards=false
• 关键词: Matrix; Coefficients; Covering; Groups; Quantum

The study of symmetry is of fundamental scientific importance. In particular, many physical theories of the universe and of elementary particles are described by collections of symmetries called Lie groups. Understanding the spaces on which these symmetries may act provides important insights into such physical theories. This project will study surprising connections between number theory, quantum groups, and mathematical physics that provide a new way of understanding such spaces.More specifically, the investigator and his students and collaborators will explore matrix coefficients of representations of p-adic groups and their arithmetic covers, known as metaplectic groups. Matrix coefficients allow one to extract numerical invariants from representations. They play a key role in both the construction of automorphic L-functions and the proofs of their analytic properties, and also in the determination of scattering amplitudes in string theory. The project will develop new relations between matrix coefficients, quantum groups, and statistical mechanics, and new methods to study matrix coefficients using Hecke algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对称性的研究具有基本的科学重要性。特别是，许多关于宇宙和基本粒子的物理理论都是由称为李群的对称性集合描述的。了解这些对称性可能作用的空间，可以提供对这些物理理论的重要见解。该项目将研究数论、量子群和数学物理之间令人惊讶的联系，为理解此类空间提供一种新的方法。更具体地说，研究者及其学生和合作者将探索p-adic群及其算术覆盖的表示的矩阵系数，称为元群。矩阵系数允许人们从表示中提取数值不变量。它们在自守L-函数的构造及其解析性质的证明中起着关键作用，也在弦理论中散射振幅的确定中起着关键作用。该项目将开发矩阵系数、量子群和统计力学之间的新关系，以及使用Hecke代数研究矩阵系数的新方法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Colored five-vertex models and Demazure atoms

• DOI: 10.1016/j.jcta.2020.105354
• 发表时间: 2019-02
• 期刊: J. Comb. Theory A

A Yang–Baxter equation for metaplectic ice

• DOI: 10.4310/cntp.2019.v13.n1.a4
• 发表时间: 2016-04
• 期刊: Communications in Number Theory and Physics
• 影响因子: 1.9
• 作者: Ben Brubaker;Valentin Buciumas;D. Bump

Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker Functions

• DOI: 10.1007/s00220-020-03842-w
• 发表时间: 2018-06
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Ben Brubaker;Valentin Buciumas;D. Bump;H. Gustafsson