Infinite Dimensional Lie Algebras, Quantum Groups, and their Applications

RESHETIKHIN这个建议的主要主题是研究量子化的泛包络代数，它们的表示以及它们在低维拓扑和量子可积模型中的应用。该提议的具体目标包括：Q-W代数的量子化及其在量子仿射代数表示理论中的作用，Q-W代数的对偶方面，作为泊松李群上的球函数的tau函数及其量子化，量子与经典可积系统之间相互关系的各种方面以及表示理论。最近几十年的特点是数学和物理之间激烈而富有成效的相互作用。这一过程的很大一部分是在经典和量子可积系统的研究中开始的，并由此发展到与共形场理论有关的各种问题，以及弦理论的非常富有成效的发展。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由这种相互作用产生并受到其启发的代数结构的研究上。它们之所以引人注目，不仅是因为它们非常特殊，还因为它们具有从低维拓扑和几何到固态物理的重要应用。

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.