Trigonometric Cherednik algebra at critical level and quantum many-body problems

TL;DR

This paper constructs a representation of the trigonometric Cherednik algebra at critical level for modules over affine Weyl groups, linking it to quantum integrable systems and deriving Bethe ansatz equations.

Contribution

It introduces a new representation of the algebra at critical level and connects it to quantum many-body problems, enabling spectral analysis via representation theory.

Findings

Representation of $A(k)$ at critical level constructed

Quantum conserved integrals derived from the algebra's center

Bethe ansatz equations expressed through normalized intertwiners

Abstract

For any module over the affine Weyl group we construct a representation of the associated trigonometric Cherednik algebra $A(k)$ at critical level in terms of Dunkl type operators. Under this representation the center of $A(k)$ produces quantum conserved integrals for root system generalizations of quantum spin-particle systems on the circle with delta function interactions. This enables us to translate the spectral problem of such a quantum spin-particle system to questions in the representation theory of $A(k)$. We use this approach to derive the associated Bethe ansatz equations. They are expressed in terms of the normalized intertwiners of $A(k)$.