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.

表示理论是数学的一个领域，它将几个学科的思想交织在一起，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）。在动力系统和物理学中出现的许多问题中，人们会遇到连续的对称群，这些对称群被称为李群。李群是属于微分几何的对象，但是对它们表示的研究主要依赖于某些称为李代数的代数对象。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作用。******在我未来五年的研究中，我的目标是回答李群和李代数的表示理论中的问题，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，如杰克多项式和麦克唐纳多项式及其变形，是表征理论中经常出现的问题。这导致了表示理论和代数组合学之间显著的相互作用，这些数学分支中的每一个分支所提供的洞察力都极大地丰富了另一个分支。我提出的研究目标是通过关注李超代数的情况来阐明这些相互作用，并与其他代数对象（如量子群）建立新的联系。从李代数到李超代数的过渡带来了许多新的挑战和技术难题，解决它们需要新颖的思想。******我还计划继续学习李群的酉表示，使用分析中的工具。这些表示是在希尔伯特空间上实现的，通常底层的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正则密子空间，如光滑向量空间，在李代数理论中起着至关重要的作用。在我的研究中，我将研究有限维和无限维李群的表示。有限维的情况包括实李群和p进半单李群，因此我的研究将在自同构形式理论方面产生影响。在无限维的情况下，所研究的群包括环群和Virasoro群。后一类表示的最重要的一类是正能量的幺正表示。因此，在无限维的情况下，我的研究将影响到数学物理。