Affine algebras, Lie superalgebras, Hecke algebras, and representations

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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.