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: $ 1.68万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679150
• 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.

表示理论是数学领域，它交织了包括代数，分析，组合和数学物理学在内的几个学科的思想，用于研究参数化对称性的抽象数学对象的实现（也称为表示）。在动态系统和物理中出现的许多问题中，一个遇到连续的对称组，称为谎言组。谎言组是属于差异几何形状的对象，但是对其表示的研究基本依赖于某些称为Lie代数的代数对象。谎言群体和谎言代数的表示理论在谐波分析，数量理论，代数组合和理论物理学中起着杰出的作用。 ******在未来五年的研究中，我的目标是回答谎言群体和谎言代数的代表理论中的问题，以及被称为Lie Superalgebras的Lie代数的概括。这些问题与对称多项式的理论密切相关。代表理论中经常出现对称多项式（例如Jack和MacDonald多项式及其变形）。这导致了表示理论与代数组合学之间的显着相互作用，而这些数学分支所提供的洞察力实质上富含了另一种。我拟议的研究的目的是通过关注谎言超级甲虫的情况，并与其他代数对象（例如量子群）建立新的联系，从而更多地阐明了这些相互作用。从谎言代数来撒谎的通道导致许多新的挑战和技术困难，并解决它们需要新颖的想法。这些表示形式在希尔伯特的空间上实现，通常基本的谎言代数不会在整个表示空间上作用。特别是，具有谎言代数的作用的规范密集子空间，例如平滑矢量的空间，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尺寸谎言组的表示。有限维度的情况包括真实和p-adic的半神经谎言组，因此我的研究的影响将是在自动形式的理论中。在无限的尺寸案例中，所研究的组包括循环组和Virasoro组。后一组最重要的表示形式之一是正能量的单一表示类别。因此，在无限的尺寸情况下，我的研究的影响将在数学物理学中。