Infinite Dimensional Lie Algebras, Quantum Groups, and their Applications

列谢季欣 这个建议的主题是研究量子化的泛包络代数，它们的表示及其在低维拓扑和量子可积模型中的应用。 该提案的具体目标包括：q-W代数，它们的量子化以及它们在量子表示论中的作用。 量子仿射代数，对偶方面的q-W代数，tau函数作为球面函数的泊松李群和他们的量化，量子和经典可积系统和表示理论之间的相互关系的各个方面。 近几十年的特点是数学和物理学的密切和富有成效的相互作用。这一过程的很大一部分是在研究经典和量子可积系统时开始的，并从那里发展到与共形场论有关的各种问题，以及弦理论中非常富有成效的发展。 在这种相互作用的数学方面是代数，几何和拓扑学的各个方面。 我的研究主要集中在代数的研究上 这种相互作用产生的结构， 它们之所以引人注目，不仅是因为它们非常特殊，还因为它们具有 重要的应用范围从低维拓扑和几何固态物理。

RESHETIKHIN The main theme of this proposal is the study of quantized universal enveloping algebras, their representations and their applications to low dimensional topology and to quantum integrable models. Among specific goals of the proposal are: q-W algebras their quantization and their role in the representation theory of quantum affine algebras, the duality aspects for q-W algebras, tau- functions as spherical functions on Poisson Lie groups and their quantization, various aspects of interrelations between quantum and classical integrable systems and representation theory. Recent decades were characterized by intense and productive interaction of mathematics and physics. Large part of this process was initiated in the study of classical and quantum integrable systems and developed from there to various questions related to conformal field theory and to a very productive developments in the string theory. On the mathematical side of this interaction are various aspects of algebra, geometry and topology. My research is focused mostly on the study of algebraic structures resulted from this interaction and inspired by it. They are remarkable not only because they are quite exceptional but also because they have important applications ranging from low dimensional topology and geometry to solid state physics.