Affine algebras, Lie superalgebras, Hecke algebras, and representations

仿射代数、李超代数、赫克代数和表示

• 批准号: 0800280
• 负责人: Weiqiang Wang
• 金额: --
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800280&HistoricalAwards=false
• 关键词: Affine; algebras; Lie; superalgebras; Hecke

Wang's research proposal covers three very active areas of representation theory and aims to stretch them into new directions: (i) the Hecke algebras associated to double covers of the Weyl groups and their representations. He proposes to construct the quantum ``spin" Hecke algebras of finite, affine, and double affine types. Then he intends to develop the representation theory of these algebras at different levels of degeneration and connections to noncommutative geometry; (ii) modular representations of finite-dimensional (simple) Lie superalgebras over an algebraically closed field of prime characteristic. In particular, Wang proposes to establish a superalgebra analogue of the Kac-Weisfeiler conjecture and connections to finite W-superalgebras; and (iii) modular representation theory of affine Lie algebras over an algebraically closed field of prime characteristic. He proposes to study systematically Wakimoto modules, at the critical and non-critical levels, and affine W-algebras in the framework of modular vertex algebras.The mathematical language used to describe symmetries in nature and supersymmetry proposed by physicists often involves the concept of groups or algebras. Representation theory is a way of studying complicated groups and algebras by expressing them in matrix forms, sometimes in a deliberately simplified manner. One outcome of studying representations is to see how symmetries differ from one another and how seemingly different symmetries are related to each other. The study of groups and algebras has numerous applications to physics, chemistry, cryptography, and others. Wang's research will broaden the scope of the study of several central concepts in representation theory in the last three decades: Hecke algebras, Lie superalgebras, and affine Lie algebras.

Wang的研究建议涵盖了表示理论的三个非常活跃的领域，并试图将其扩展到新的方向：(I)与Weyl群的双重覆盖相关的Hecke代数及其表示。他建议构造有限、仿射和双仿射类型的量子“自旋”Hecke代数。然后，他打算发展这些代数在不同退化水平和与非交换几何的联系的表示理论；(Ii)有限维(单)李超代数在具有素数特征的代数闭域上的模表示。特别是，Wang建议建立Kac-Weisfeeller猜想的超代数模拟以及与有限W-超代数的联系；以及(Iii)具有素数特征的代数闭域上仿射李代数的模表示理论。他建议在模顶点代数的框架下，在临界和非临界水平上系统地研究Wakimoto模和仿射W-代数。物理学家提出的用于描述自然界的对称性和超对称性的数学语言通常涉及群或代数的概念。表示论是一种研究复杂群和代数的方法，它用矩阵的形式来表示它们，有时是故意简化的方式。研究表象的一个结果是看到对称性是如何彼此不同的，以及表面上不同的对称性是如何相互联系的。群和代数的研究在物理、化学、密码学和其他方面有许多应用。王的研究将拓宽过去三十年来表示论中几个核心概念的研究范围：Hecke代数、Lie超代数和仿射Lie代数。