Categorification of Quantum Groups

量子群的分类

• 批准号: 0855713
• 负责人: Aaron Lauda
• 金额: $ 11.87万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0855713&HistoricalAwards=false
• 关键词: Categorification; Quantum; Groups

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Building off geometric constructions of Beilinson, Lusztig, and MacPherson, the PI in collaboration with Mikhail Khovanov categorified quantum sl(n), as well as the quantum deformation of the universal enveloping algebra of the ``lower-triangular'' subalgebra for an arbitrary Kac-Moody Lie algebra. While these categorifications are completely combinatorial, they utilize various diagrammatic calculi emphasizing a new interplay between topology and algebra. The proposal aims to further develop the theory of categorified quantum groups, study their Hopf structure (comultiplication and antipode), further develop their representation theory, categorify the entire quantum enveloping algebra for other Kac-Moody Lie algebras, and understand the relationship to geometric representation theory. Potential applications of categorified of quantum groups include the study of positivity properties for quantum enveloping algebras, a representation theoretic explanation of Khovanov homology and a categorification of the Witten-Reshetikhin-Turaev quantum 3-manifold invariants.Quantum groups are prevalent throughout mathematics and theoretical physics. These Hopf algebras provide the representation theoretic framework for understanding quantum link invariants such as the Jones polynomial, Kauffman and HOMFLY-PT polynomial, as well as the Witten-Reshetikhin-Turaev quantum 3-manifold invariants. They also relate to statistical mechanics, quantum field theory, and affine Lie algebras. Recent work suggests that the representation theory of quantum groups, and the quantum groups themselves, are shadows of a much richer algebraic structure known as categorified quantum groups. These structures were conjectured to exist by Crane and Frenkel and form the focal point of this proposal.

该奖项是根据2009年的《美国复苏与再现法案》（公法111-5）进行的。构建了贝林森，卢斯蒂格和麦克弗森的几何结构，与米哈伊尔·科瓦诺夫（Mikhail Khovanov对于任意的kac-moody lie代数。 尽管这些分类是完全组合的，但它们利用各种图解的计算强调了拓扑与代数之间的新相互作用。该提案旨在进一步发展分类量子组的理论，研究其HOPF结构（共同授课和反模式），进一步发展其代表理论，对其他Kac-Moody Lie代数的整个量子包围代数进行分类，并了解与几何表示理论的关系。量子组分类的潜在应用包括量子包围代数的阳性特性的研究，Khovanov同源性的代表理论解释以及对Witten-Reshetikhin-Turaev量子量子3- manifold variant群组的分类。这些HOPF代数提供了理论框架，以理解量子链接不变符，例如琼斯多项式，考夫曼和Homfly-pt-pt多项式，以及witten-Reshetikhin-turaev量子量子3个manifolds。 它们还与统计力学，量子场理论和仿射代数有关。 最近的工作表明，量子基团的表示理论以及量子群本身是一种更丰富的代数结构的阴影，称为分类量子基团。 这些结构由起重机和弗伦克尔（Crane and Frenkel）猜想，并构成了该提案的焦点。