Representations of Infinite Dimensional Lie Algebras and the McKay Correspondence

研究无限维李代数的表示理论、有限群和几何学之间的联系。 受Nakajima，Grojnowski，Segal等人的影响，研究者在他最近的工作中建立了Heisenberg和仿射李代数的点在曲面上的Hilbert格式、圈积（orbifolds）和顶点表示之间的联系。 他建议研究这些主题之间更深层次的联系。 一种新的方法麦凯通信第一次观察到的调查已经制定了他的工作与他的合作者。 利用圈积构造了ADE型（量子）仿射和环面李代数的顶点表示。 调查人员建议实现其他（量子）顶点表示研究代数在文献中的花圈产品及其变化。 在另一个方向，他建议给一个组理论解释某些顶点代数的框架圈产品。调查研究对称性。 对称有不同的类型，离散的和连续的，有限的和无限的。 它们对于理解物理定律具有根本的重要性，并具有实际应用，例如编码理论。 事实证明，如果一个人把某些类别的离散和有限的对称在一起，以一种巧妙的方式，一个观察到连续和无限的对称。 研究者研究这些不同类型的对称性之间的相互作用，这带来了对这些主题的新见解，而这些新见解不能通过单独研究它们来获得。

The investigator studies the connections among representation theory of infinite dimensional Lie algebras, finite groups and geometry. Influenced by the works of Nakajima, Grojnowski, Segal and others, theinvestigator in his recent works established connections among Hilbert schemes of points on surfaces, wreath product (orbifolds) and vertex representations of Heisenberg and affine Lie algebras. He proposes to investigate the deeper connections among these subjects. A new approach to the McKay correspondence first observed by the investigator has been developed in his work with his collaborators. Here vertex representations of (quantum) affine and toroidal Lie algebras of ADE type are constructed by using wreath products. The investigator proposes to realize other (quantum) vertex representations studied algebraically in the literature by means of wreath products and their variations. In another direction, he proposes to give a group theoretic interpretation of certain vertex algebras in the framework of wreath products.The investigator studies symmetries. There are different types of symmetries, discrete and continuous, finite and infinite. They are of fundamental importance to understanding of the laws of physics and have practical applications such as to coding theory. It turns out that if one puts certain classes of discrete and finite symmetry together in a clever way one observes continuous and infinite symmetry. The investigator studies the interactions between these different types of symmetries which bring new insights of these subjects which can not be obtained by studying them separately.