Duality between representations of Lie superalgebras and Lie algebras via Kazhdan-Lusztig theory

通过 Kazhdan-Lusztig 理论研究李超代数和李代数表示之间的对偶性

• 批准号: 0500374
• 负责人: Weiqiang Wang
• 金额: $ 10万
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500374&HistoricalAwards=false
• 关键词: Duality; between; representations; Lie; superalgebras

The super representation theory is traditionally regarded as fairly different from the usual one for Lie algebras largely because the Weyl group for a simple Lie superalgebra does not suffice to control the structures of the irreducible representations. The PI intends to formulate and establish a new direct link, termed as super duality, between the representation theories of Lie superalgebras and of Lie algebras. The super duality asserts certain equivalences of module categories in a suitable infinite limit and the identification of the Kazhdan-Lusztig polynomials for Lie superalgebras and Liealgebras. A main algebraic tool is a Fock space formulation of the Kazhdan-Lusztig theory. The super duality is expected to provide a new approach toward the Kazhdan-Lusztig type conjectures on irreducible characters for module categories over simple Lie superalgebras of various types and for module categories over quantum supergroups at roots of unity.There are different manifestations of symmetries in nature, which one can find in, for example, a circle, a sphere, or one of the five regular polyhedra, and others. The mathematical language used to describe symmetries often involves the concept of groups or their infinitesimal counterparts such as Lie algebras. Representation Theory is a way of studying the groups and Lie algebras by expressing them in terms of matrices. On the other hand, different symmetries can be related to each other. In search for a unified theory of everything, physicists have proposedString Theory as a candidate theory. Supersymmetry adds another invisible dimension to such considerations and the study of Lie superalgebras is crucial to understanding the supersymmetry. Our project on Super Duality can be regarded as providing a precise and new way of relating supersymmetry to symmetry in the usual sense. This helps to provide a convincing evidence supporting the idea of supersymmetry and may have applications to String Theory.

传统上，超级代表理论与谎言代数的常规代数相当不同，这很大程度上是因为简单的谎言superalgebra群不足以控制不可约说明的结构。 PI打算在Lie Superalgebras和Lie代数的代表理论之间建立和建立一个新的直接链接，称为超双重性。超级双重性在适当的无限限制中主张了模块类别的某些等效性，并鉴定了kazhdan-lusztig多项式用于谎言超级甲壳虫和liealgebras。一个主要的代数工具是Kazhdan-Lusztig理论的Fock空间公式。预计超级二元性能为模块类别的Kazhdan-lusztig类型提供一种新方法，而不是各种类型的简单谎言超级代码，以及在Unity根部的量子超级组上的模块类别。用于描述对称性的数学语言通常涉及群体的概念或它们的无限同行，例如Lie代数。表示理论是一种研究群体并通过矩阵来表达代数的方式。另一方面，不同的对称性可以彼此相关。为了寻找一切的统一理论，物理学家提出了拟议的理论作为候选理论。超对称性增加了此类考虑的另一个不可见维度，而谎言超级乳房的研究对于理解超对称性至关重要。我们关于超级双重性的项目可以被视为提供了一种与通常意义上的对称性相关联的精确和新的方式。这有助于提供令人信服的证据，以支持超对称的思想，并可能对弦理论有应用。