Affine algebras, Lie superalgebras, Hecke algebras, and representations

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。他建议系统地研究Wakimoto模块，在关键和非关键的水平，仿射W-代数的框架内的模块顶点代数。数学语言用来描述对称性的性质和超对称性提出的物理学家往往涉及的概念，集团或代数。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。

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.