Infinite Dimensional Lie Algebras, Quantum Groups, and their Applications

其中的具体问题，这一建议的地址，是有限维表示的量化仿射李代数的几何形状的辛叶相应的泊松李群的描述。 当完成后，这个项目应该基本上澄清了大多数已知的可积系统的结构。另一个问题是当张量积中因子的个数趋于无穷大时，量子仿射代数在极限中的Weyl型对偶的研究。 Reshetikhin希望研究在这个极限下出现的新的无限维代数，以及它们与相应的可积量子场论的热力学之间的关系。 其他一些问题，他正计划调查的渐近展开维滕-Reshetikhin-Turaev不变量的大水平，变形量子化的一般可积系统（不一定经常，其中拉格朗日纤维化是一个纤维丛）。 他还打算研究特征经典和量子系统以及与Kac-Moody李代数和李群相关的经典和量子可积系统。量子群作为一个代数对象（Hopf代数）出现，描述了广泛的一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。 这一发展是刺激（和刺激）各种应用可积系统拓扑和几何。 他们的表示理论比李群的表示理论复杂得多。 量子群和它们的表示论在几个开创性的结果中发挥了重要作用：三维流形中的链的不变量、可积系统、晶体基，以及最近，量子群和非对易几何之间的边界的一些想法被用于弦理论。 人们可以说，这个方向的概念目标是理解最复杂的对称性，这些对称性可能（其中一些可能）表现为基本粒子（或弦，如果它们真的存在而不是粒子）相互作用的对称性。

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).