量子群和无限维李代数的表示

Lie theory has been at the center of the mathematical stage since the late 1960s. Quantum groups and infinite dimensional Lie algebras, the most important two generalizations of finite dimensional simple Lie algebras, have diverse applications in areas of soliton equations, integrable systems, combinatorics, number theory. Lie theory also profoundly influences several areas in theoretical physics, such as string theory, quantum field theory and conformal field theory, etc. ..The goal of this proposed research is to study the representation theory of quantum groups and infinite dimensional Lie algebras. In recent years, as new methods and theories have been developed in the field of Lie theory, great progresses have been made for the representation theory of quantum groups and Lie (super)algebras. Especially, some important problems, that had been open for a long time, were solved successfully. This attracts more attentions from mathematicians in Lie theory of various interests and provides motivations for new developments. Based on the known results obtained by other mathematicians and the research works we have done so far, the present project will make deeper investigations into the research of representation theory of quantum groups and infinite dimensional Lie algebras. Some key problems related to quantum groups in this project include the construction and classification of irreducible representations of quantum groups under some conditions. We will define modular Yangians and study their finite dimensional irreducible representations. For infinite dimensional Lie algebras, we will focus on studying the representations of affine Kac-Moody Lie algebras, of the Virasoro algebra, of root graded Lie (super)algebras, and also of the modular Lie superalgebras.

二十世纪六十年代以来李理论一直是数学研究的热点之一，其中李理论代数方向的两个主要研究对象为：量子群和无限维李代数，它们是有限维半单李代数概念的自然推广，它们在数学领域里，除了代数学以外，还丰富了孤立子理论、可积系统、奇性理论、数论和组合等方向的研究；物理学中，它在弦论、共形场论、量子场论等分支中都有着十分重要的应用。..近年来，随着李代数表示理论新方法的发展和建立，量子群和无限维李代数的表示理论也进一步丰富，尤其是一些重要的公开问题的解决对该领域的发展提供了新的动力。本项目将依托项目组多年相关研究积累的坚实基础，继续深入研究量子群、Yangian和一些无限维李（超）代数的表示理论。主要拟解决的核心问题有：构造并分类量子群的不可约表示；给出模Yangian的定义并研究有限维不可约表示；同时我们将进一步研究仿射李代数、Virasoro代数、根系分次李代数、模李超代数的结构和表示理论。

二十世纪六十年代以来李理论一直是数学研究的热点之一，其中李理论代数方向的两个主要研究对象为：量子群和无限维李代数，它们是有限维半单李代数概念的自然推广，它们在数学领域里，除了代数学以外，还丰富了孤立子理论、可积系统、奇性理论、数论和组合等方向的研究；物理学中，它在弦论、共形场论、量子场论等分支中都有着十分重要的应用。本项目主要侧重于量子群（Yangian）、无限维李代数表示理论的研究，特别是构造或分类了量子群和一些重要的无穷维李代数的表示。主要研究结果包括：完全分类了A1型量子群的秩1自由表示，并研究了这类模与有限维不可约张量积得到了类似于经典Clebsch-Gordan公式的张量积分解公式；更进一步完全分类了有限型量子群的秩1自由表示，得到了大量不可约非权表示；研究了Rueda代数上秩1自由模和一些广义Weyl代数的Whittaker模的分类；研究了可对称化Kac-Moody代数的抛物范畴；还研究了三类有重要物理背景的Virasoro型李代数的表示：Heisenberg-Virasoro代数，镜像Heisenberg-Virasoro代数和平面Galilean共形代数。这些研究成果有利于量子群和Yangian以及无穷维李代数相应研究工作的深入开展，有很重要的理论价值。经过四年的努力，本项目接收发表含有项目标注SCI论文7篇，主要代表性文章有4篇发表在著名代数期刊Journal of Algebra（2篇）和Journal of Pure and Applied Algebra（2篇）上，另外有两篇分别发表在著名综合期刊Communications in Mathematical Physics和Forum Mathematicum上。联合培养毕业博士2人，独立指导毕业博士1人，硕士2人，出站博士后工作人员1人。