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的研究计划涵盖了表示论的三个非常活跃的领域，并旨在将它们扩展到新的方向：（i）与Weyl群及其表示的双覆盖相关的Hecke代数。他建议构建量子"自旋”赫克代数有限，仿射，双仿射类型。然后，他打算发展的代表性理论，这些代数在不同层次的退化和连接到非交换几何;（ii）模块化表示有限维（简单）李超代数在代数封闭领域的主要特点。特别是，王建议建立一个超代数类似的卡茨-Weisfeiler猜想和连接有限W-超代数;和（iii）模表示理论的仿射李代数在一个代数闭域的主要特征。他提出在临界和非临界水平上系统地研究Wakimoto模，并在模顶点代数的框架下研究仿射W-代数。物理学家提出的用于描述自然界对称性和超对称性的数学语言经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，通过矩阵形式来表达它们，有时以故意简化的方式。研究表征的一个结果是看到对称性如何彼此不同，以及看似不同的对称性如何彼此相关。群和代数的研究在物理学、化学、密码学和其他领域有许多应用。王的研究将拓宽在过去三十年中表示论的几个中心概念的研究范围：Hecke代数，李超代数和仿射李代数。