Representations of Lie superalgebras, Hecke algebras and affine algebras

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

The proposed research covers the representation theory of Lie superalgebras, Hecke algebras and affine Lie algebras over fields of characteristic zero as well as over fields with positive characteristics. In recent work, the PI and collaborators have obtained a conceptual solution to the irreducible character problem in various parabolic categories of classical Lie superalgebras. The PI proposes a new approach to attack the well-known open problem on the irreducible characters for the FULL category O for Lie superalgebras. The PI also proposes to develop a spin analogue of the invariant theory for Weyl groups and a spin analogue of Hecke algebras. These formulations call for the computation and comparison of spin fake degrees and spin generic degrees, as a striking spin analogue of a deep classical theory (which is intimately related to finite groups of Lie type). The PI further proposes to initiate a systematic study of the modular representations of infinite-dimensional Lie algebras in positive characteristic. The mathematical language used to describe symmetries in nature often involves the concepts of groups, algebras, or their variants. Wang's research has helped to solve long standing old problems for Lie superalgebras (part of the language used in describing the supersymmetry), and also to raise many exciting research problems. The PI's research has attracted students to the active area of representation theory of groups and algebras. He is a popular speaker for summer and winter schools aiming at training graduate students and young mathematicians at the beginning of their careers.