无限维李代数

申请人近五年在数学中心领域之仿射型量子群，Kac-Moody代数，代数组合论及等相关领域进行了一系列工作：和合作者将McKay对应的新形式推广到双参数的量子环面代数上，并首次给出A型量子双参数访射代数的费米表示。得到了Jack多项式的顶点代数实现的一类一般形式，用广义拉普拉斯微分算子得到了Jack多项式的递推公式。本项目主要目标是，进一步对仿射量子代数和双参数的量子仿射代数及其推广代数、顶点代数、对称函数方面的三个重要问题展开研究。具体将解决量子圈代数顶点算子表示；构造特征p－顶点算子代数；研究代参数的KP－Hierarchy方程及其李代数方法；研究Mcdonald对称函数的顶点算子和W-代数方法。这些均是当前该领域中急待解决和具有挑战性的研究课题，这些研究将开辟该领域新的增长点，同时有助带动华南地区在这些中心数学研究领域。申请者目前在广州组织大量学术活动，指导十数名研究生从事相关研究。

The applicant has been working on important aspects of affine quantum algebras, Kac-Moody algebras, and algebraic combinatorics during the past five years. He and collaborators formulated a new form of McKay correspondence for two-parameter quantum toroidal algebras, and derived the Fermionic representations of the quantum toroidal Lie algebra of type A for the first time. He also obtained a general form of the vertex operator realization of the Jack polynomials, and used the Laplace operator to give a new characterization of Jack symmetric functions. The main goal of the current project is to develop more tools to study quantum affine algebras and two-parameter quantum affine algebras, vertex algebras, and Macdonald symmetric functions. The project will give new constructions of the quantum loop algebras; p-vertex operator algebras; and deformation of KP-the Hierarchy equations via Lie algebraic method. These challenging problems will open up a new area in the field and help promote mathematical research in China. The grant also devotes a significant effort to education in southern China, where he supervises 10 students year round, and organize numerous workshops and summer schools to cultivate new working forces in the field.