Analytic and combinatorial aspects of representation theory

表示论的分析和组合方面

• 批准号: RGPIN-2018-04044
• 负责人: Salmasian, Hadi
• 金额: $ 3.35万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759214
• 关键词: Analytic; combinatorial; aspects; representation; theory

Representation theory is an area of mathematics that interweaves ideas from several disciplines, including algebra, analysis, combinatorics, and mathematical physics, to study realizations (also known as representations) of abstract mathematical objects that parametrize symmetries. In many problems that emerge in dynamical systems and physics, one encounters continuous groups of symmetries, which are called Lie groups. Lie groups are objects that belong to differential geometry, but the study of their representations relies substantially on certain algebraic objects called Lie algebras. Representation theory of Lie groups and Lie algebras plays a distinguished role in harmonic analysis, number theory, algebraic combinatorics, and theoretical physics. In my research over the next five years, I aim to answer questions in representation theory of Lie groups and Lie algebras, and a generalization of Lie algebras which are called Lie superalgebras. These questions are closely related to the theory of symmetric polynomials. Symmetric polynomials, such as Jack and Macdonald polynomials and their deformations, occur frequently in representation theory. This has lead to remarkable interactions between representation theory and algebraic combinatorics, and the insight provided by each of these branches of mathematics has enriched the other one substantially. The goal of my proposed research is to shed more light on these interactions by focusing on the case of Lie superalgebras, and to establish new connections with other algebraic objects such as quantum groups. The passage from Lie algebras to Lie superalgebras results in many new challenges and technical difficulties, and tackling them requires novel ideas.I also plan to continue my study of unitary representations of Lie groups using tools from analysis. These representations are realized on Hilbert spaces, and typically the underlying Lie algebra does not act on the entire representation space. In particular, canonical dense subspaces which carry an action of the Lie algebra, such as the space of smooth vectors, play a crucial role in the theory. In my research I will study representations of both finite and infinite dimensional Lie groups. The finite dimensional case includes real and p-adic semisimple Lie groups, and therefore the impact of my research will be in the theory of automorphic forms. In the infinite dimensional case, the groups under investigation include loop groups and the Virasoro group. One of the most important classes of representations of the latter groups is the class of unitary representations of positive energy. Therefore in the infinite dimensional case the impact of my research will be in mathematical physics.

表示论是一个数学领域，它交织了几个学科的思想，包括代数、分析、组合学和数学物理，以研究将对称参数化的抽象数学对象的实现(也称为表示)。在动力系统和物理中出现的许多问题中，人们都会遇到连续的对称群，它们被称为李群。李群是属于微分几何的对象，但对其表示的研究在很大程度上依赖于称为李代数的某些代数对象。李群和李代数的表示理论在调和分析、数论、代数组合学和理论物理中有着重要的地位。在接下来的五年的研究中，我的目标是回答李群和李代数的表示论中的问题，以及李代数的推广，称为李超代数。这些问题与对称多项式理论密切相关。对称多项式，如Jack和Macdonald多项式及其变形，在表示理论中经常出现。这导致了表示理论和代数组合学之间的显著互动，而这些数学分支中的每一个提供的洞察力都大大丰富了另一个分支。我提出的研究的目的是通过关注李超代数的情况来更好地阐明这些相互作用，并与其他代数对象建立新的联系，例如量子群。从李代数到李超代数的过渡导致了许多新的挑战和技术困难，解决这些问题需要新的想法。我还计划使用分析中的工具继续研究李群的酉表示。这些表示是在希尔伯特空间上实现的，通常基础李代数并不作用于整个表示空间。特别是，具有李代数作用的典范稠密子空间，如光滑向量空间，在理论中起着至关重要的作用。在我的研究中，我将研究有限和无限维李群的表示。有限维的情形包括实半单李群和p-进半单李群，因此我的研究的影响将在自同构型理论中。在无限维情形下，所研究的群包括环群和Virasoro群。后一类群的最重要的表示类之一是正能量的酉表示类。因此，在无限维的情况下，我的研究的影响将是数学物理。