Quantum integrable systems and representations of Lie algebras

Abstract

In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain homomorphism from the center of U(gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=-N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many-body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky-Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. We also give a new expression for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of gl(N).

17 pages, no figures

Cited by in corpus (13)

• Duality in Integrable Systems and Gauge Theories
• Elliptic Calogero-Moser system from two dimensional current algebra
• From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models
• Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism
• Surface defects in gauge theory and KZ equation
• Degenerate integrability of quantum spin Calogero--Moser systems
• On a Duality in Calogero-Moser-Sutherland Systems
• From Vertex Operators to Calogero-Sutherland Models
• A note on the action-angle variables for the rational Calogero-Moser system
• Quantization of classical integrable systems. Part III: systems in n-dimensional Euclidean space
• Traces of intertwiners for quantum affine sl_2 and Felder-Varchenko functions
• Jack polynomials for the $B{C}_n$ root system and generalized spherical functions
• Etingof Trace, Path Hypergeometric Functions and Integrable Systems