Infinite Dimensional Lie Algebras, Quantum Groups, and their Applications

在具体的问题，这一建议解决，是描述有限维表示的量化仿射李代数在相应的泊松李群的辛叶的几何。当完成时，这个项目应该从本质上阐明大多数已知的可积系统的结构。另一个问题是研究当张量积的因子数趋于无穷时，量子仿射代数的weyl型对偶性。Reshetikhin想要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。他计划研究的其他一些问题是大水平Witten-Reshetikhin-Turaev不变量的渐近展开，一般可积系统的变形量化（不一定是规则的，拉格朗日纤维是一个纤维束）。他还打算研究与Kac-Moody李代数和李群相关的特征经典和量子系统以及经典和量子可积系统。量子群是作为一种代数对象（霍普夫代数）出现的，它描述了一类广泛的量子可积系统的对称性。近十年来，量子群的结构理论及其表示理论取得了令人瞩目的发展。可积系统拓扑和几何的各种应用刺激了这一发展。他们的表征理论比李群的表征理论要复杂得多。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流形中链路的不变量，可积系统，晶体基，以及最近，量子群和非交换几何之间的边界的一些思想被用于弦理论。有人可能会说，这个方向的概念目标是理解最复杂的对称性，这些对称性可能表现为（其中一些表现为）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).