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

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 $A(k)$.

31 pages

References in corpus (6)

• Exchange Operator Formalism for Integrable Systems of Particles
• Quasiclassical asymptotics of solutions to the KZ equations
• Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation
• Periodic integrable systems with delta-potentials
• Many Body Problems with "Spin"-Related Contact Interactions
• Bethe Equations for a g_2 Model

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• Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions
• Traces of Integrability in Relaxation of One-Dimensional Two-Mass Mixtures
• Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra
• Discrete harmonic analysis on a Weyl alcove
• A new take on spherical, Whittaker and Bessel functions (Spherical and Whittaker functions via DAHA I,II)
• Creating Entanglement Using Integrals of Motion
• Extended trigonometric Cherednik algebras and nonstationary Schrödinger equations with delta-potentials
• On the basic representation of the double affine Hecke algebra at critical level
• Dirac induction for rational Cherednik algebras
• Bispectral quantum Knizhnik-Zamolodchikov equations for arbitrary root systems
• Completeness of the Bethe ansatz on Weyl alcoves