Representations of Infinite Dimensional Lie Algebras and the McKay Correspondence

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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.