Symplectic Reflection Algebras and Noncommutative Geometry

University Of California-Riverside, Riverside CA

Investigators

Abstract

This project is devoted to the study of symplectic reflection algebras and their variants. Symplectic reflection algebras are relatively new objects in Lie theory and we are only at the beginning stages of its representation theory. The main focus of the proposed project will be on symplectic reflection algebras of wreath product types, which include as a special case the rational Cherednik algebras of type A. The symplectic reflection algebras of wreath product types are closely related to Kleinian singularities and their resolutions. In particular, representations of the symplectic reflection algebra are expected to be closely related to sheaves on Hilbert scheme of points on the minimal resolution of the associated Kleinian singularity. The symplectic reflection algebra also provides quantization of the Hilbert scheme of points. A first goal of the project is to develop systematically the representation theory of the symplectic reflection algebras of wreath product type from a geometric point of view. The second goal of the project is to explore applications of similar algebras in noncommutative geometry. The PI hopes to unify and extend some of the existing methods from rational Cherednik algebras and deformed preprojective algebras of quivers to the wreath product case. The PI also hopes that the methods developed for symplectic reflection algebras can be extended to other algebras that arise from noncommutative deformations of surfaces. This will lead to a more coherent theory that links representation theory to noncommutative geometry. Symplectic reflection algebras have already provide us with several applications to combinatorics and noncommutative geometry. This project will require techniques from representation theory and algebraic geometry. It will hopefully bring all these branches in mathematics closer together and enrich each other with new constructions and techniques. In particular, wreath product generalizations appear in all these subjects, and symplectic reflection algebras is a central tool to formalize and exploit this connection.