Representations of Lie superalgebras, Hecke algebras and affine algebras

李超代数、赫克代数和仿射代数的表示

• 批准号: 1101268
• 负责人: Weiqiang Wang
• 金额: $ 18.02万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101268&HistoricalAwards=false
• 关键词: Representations; Lie; superalgebras; Hecke; algebras

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.

这些研究涵盖了特征为零的域上的李超代数、Hecke代数和仿射李代数的表示理论以及正特征域上的表示理论。在最近的工作中，PI和合作者已经获得了一个概念性的解决方案，在各种抛物范畴的经典李超代数的不可约特征标问题。PI提出了一种新的方法来攻击著名的公开问题的满范畴O的李超代数的不可约特征标。PI还建议开发Weyl群的不变量理论的自旋模拟和Hecke代数的自旋模拟。这些公式需要计算和比较自旋假度和自旋一般度，作为深层经典理论（与李型有限群密切相关）的惊人自旋模拟。PI进一步提议对正特征的无限维李代数的模表示进行系统的研究。用于描述自然界中对称性的数学语言通常涉及群、代数或其变体的概念。王的研究有助于解决李超代数（描述超对称性的语言的一部分）长期存在的老问题，也提出了许多令人兴奋的研究问题。PI的研究吸引了学生到群和代数的表示理论的活跃领域。他是夏季和冬季学校的热门演讲者，旨在培养研究生和年轻的数学家在他们的职业生涯开始。