Infinite Dimensional Lie Algebras, Quantum Groups, and their Applications

无限维李代数、量子群及其应用

• 批准号: 0070931
• 负责人: Nicolai Reshetikhin
• 金额: $ 15万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070931&HistoricalAwards=false
• 关键词: Infinite; Dimensional; Lie; Algebras; Quantum

Among the specific problems, which this proposal addresses, is the description of finite dimensional representations of quantized affine Lie algebras in terms of the geometry of the symplectic leaves of the corresponding Poisson Lie group. When completed, this project should substantially clarify the structure of most known integrable systems. Another problem is the investigation of Weyl-type dualities for quantum affine algebras in the limit when the number of factors in the tensor product goes to infinity. Reshetikhin wants to investigate new infinite dimensional algebras that will appear in this limit and how they are related to the thermodynamics of the corresponding integrable quantum field theory. Some of the other problems he is planning to investigate are the asymptotic expansion of Witten-Reshetikhin-Turaev invariants for large levels, the deformation quantization of generic integrable systems (not necessarly regular, for which the Lagrangian fibration is a fiber bundle). He also intends to investigate characteristic classical and quantum systems and classical and quantum integrable systems related to Kac-Moody Lie algebras and Lie groups.Quantum groups appeared as an algebraic object (Hopf algebras) describing the symmetries of a wide class of quantum integrable systems. In the last decade there has been a fascinating development in structural theory of quantum groups and their representation theory. This development was stimulated by (and stimulated) various applications to integrable systems topology and geometry. Their representation theory is far more sophisticated than the representation theory of Lie groups. Quantum groups and their representation theory were instrumental in several path-breaking results: the invariants of links in 3 manifolds, integrable systems, crystal bases, and just recently, some of the ideas from the borderline between quantum groups and non-commutative geometry were used in string theory. One may say that the conceptual goal of this direction is to understand most sophisticated symmetries which may appear (and some of them appear) as symmetries of interactions of elementary particles (or strings, if they are really there instead of particles).

在这个建议所涉及的具体问题中，量化仿射李代数的有限维表示是根据相应Poisson李群的辛叶的几何来描述的。当这个项目完成后，应该会实质上阐明大多数已知的可积系统的结构。另一个问题是当张量积的因子数达到无穷大时，量子仿射代数的Weyl型对偶在极限上的研究。Reshetikhin想要研究在这个极限中出现的新的无限维代数，以及它们如何与相应的可积量子场论的热力学相关。他计划研究的其他一些问题包括大能级的Witten-Reshetikhin-Turaev不变量的渐近展开，一般可积系统的形变量子化(不一定是正则的，其中拉格朗日纤维是纤维丛)。他还打算研究与Kac-Moody李代数和李群有关的特征经典和量子系统以及经典和量子可积系统。量子群以代数对象(Hopf代数)的形式出现，描述了一大类量子可积系统的对称性。在过去的十年里，量子群的结构理论及其表示理论有了令人着迷的发展。这一发展受到了可积系统、拓扑学和几何学的各种应用的刺激。他们的表象理论比李群的表象理论复杂得多。量子群及其表示理论在几个开创性的结果中发挥了重要作用：3个流形、可积系统、晶基中的链接的不变量，最近，一些来自量子群和非对易几何之间的分界线的思想被用于弦理论。有人可能会说，这个方向的概念目标是理解大多数复杂的对称性，这些对称性可能看起来(其中一些看起来)是基本粒子(或弦，如果它们真的存在而不是粒子)相互作用的对称性。