Representation theory of W-algebras, quantum groups, symplectic reflection algebras and quantum Hamiltonian reductions

Northeastern University, Boston MA

Investigators

Abstract

This project studies the representation theory of several different yet related associative algebras: finite W-algebras, symplectic reflection algebras, quantum groups and quantum Hamiltonian reductions associated to quivers. The investigator plans to classify finite dimensional irreducible modules over W-algebras and cyclotomic rational Cherednik algebras. He is going to relate various categories of representations of the universal enveloping algebras of semisimple Lie algebras and of W-algebras and use this relation to compute the dimensions of irreducible W-algebra modules. Next, the investigator will study a connection between W-algebras and quantum groups at a root of unity. Another related topic is the study of Harish-Chandra bimodules over symplectic reflection algebras and quantum groups at roots of unity. The investigator also plans to work on a conjecture of Rouquier describing the multiplicities in the categories O and a conjecture of Etingof on counting finite dimensional irreducible modules over symplectic reflection algebras. The latter will be approached in a more general context of quantum Hamiltonian reductions corresponding to Nakajima quiver varieties. The area of this project is Representation theory. Roughly speaking, Representation theory deals with symmetry, in particular, coming from Quantum Physics. Symmetries are thought as algebraic structures such as groups or algebras. The main problem is therefore is to understand how a given algebraic structure can be represented as a symmetry of some other objects, usually vector spaces. The algebraic structures studied in this project are certain associative algebras mostly arising in Quantum Mechanics: finite W-algebras, symplectic reflection algebras or quantum groups. Mostly, the project concentrates on a fundamental representation-theoretic problem - understanding basic, so called "irreducible"representations that serve as building blocks for more general ones with an emphasis on finite dimensional representations. Problems to be studied include computing the number of such representations, classifying them, computing their dimensions or finer invariants, called characters.