Commutative Subalgebras and Bethe Ansatz for Quantum Affine and Toroidal Algebras via the Shuffle Approach

Yale University, New Haven CT

Investigators

Abstract

This research project lies in the intersection of three fields of mathematics: algebraic representation theory, integrable systems, and geometric representation theory. The former two branches of mathematics originate from physics, while the last deals with applications of purely algebraic concepts to geometry. Representation theory concerns the study of symmetries of a vector space such as our three-dimensional space (more generally, an infinite dimensional space) with additional structures. These symmetries can be often thought of as algebraic structures. The following two cases are of particular interest: (1) the case of pair-wise commuting symmetries, when sufficiently many exist, is of central importance in the study of integrable systems; (2) the case when the underlying vector space arises from geometric objects is of central importance in geometric representation theory. In this project the principal investigator plans to explore these concepts in the particular cases of algebras known as quantum toroidal algebras and affine Yangians. These associative algebras can be viewed as deformations of Lie algebras and provide generalizations of the classical quantum affine algebras and Yangians that have been studied extensively in recent decades. This project is devoted to the study of quantum toroidal algebras and affine Yangians. The PI's plan is as follows: (1) Develop shuffle realizations of all quantum toroidal/affine algebras of ADE type. (2) Unify all known different constructions of their representations and provide a wider class of shuffle type modules. (3) Study the maximal commutative subalgebras of quantum toroidal algebras via the shuffle realization, and develop a new (shuffle) approach to the well-known Bethe ansatz problem, concerning diagonalization of such maximal commutative subalgebras in interesting classes of representations. (4) Relate the aforementioned maximal commutative subalgebras to the study of quantum cohomology and quantum K-theory of Nakajima quiver varieties and affine Laumon spaces. (5) Generalize all the above to the additive case of affine Yangians. (6) Study quantizations of the shift of the argument algebras corresponding to all vertices of the "degeneration" rhombus and describe the relation of these algebras to the right hand sides of the KZ equation and Casimir equation rhombi.