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，超表示理论被认为与通常的李代数的超表示理论有很大的不同，很大程度上是因为单李超代数的Weyl群不足以控制不可约表示的结构。PI试图在李超代数的表示理论和李代数的表示理论之间建立一种新的直接联系，称为超对偶。超对偶性证明了李超代数和李代数的Kazhdan-Lusztig多项式在适当的无穷极限上的某些等价和Kazhdan-Lusztig多项式的识别。一个主要的代数工具是Kazhdan-Lusztig理论的Fock空间公式。超对偶性有望为各种类型的单李超代数上的模范畴和单位根处的量子超群上的模范畴上的不可约特征标的Kazhdan-Lusztig型猜想提供一种新的途径。自然界中对称性的不同表现形式可以在圆、球面或五个正多面体之一等中找到。用来描述对称性的数学语言通常涉及群或它们的无穷小对应的概念，如李代数。表示论是通过用矩阵来表示群和李代数的一种研究方法。另一方面，不同的对称性可以相互关联。在寻求万物统一理论的过程中，物理学家提出了弦理论作为候选理论。超对称为这种考虑增加了另一个看不见的维度，而对李超代数的研究对于理解超对称是至关重要的。我们的超对偶项目可以被视为提供了一种将超对称性与通常意义上的对称性联系起来的精确和新的方法。这有助于提供令人信服的证据支持超对称性的想法，并可能应用于弦理论。