Abstract
The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Cherednik operators, which reduces to the generating function of Dell commuting Hamiltonians on the space of symmetric functions. Although the double elliptic Cherednik operators do not commute, they can be used for construction of the N → ∞ limit.
Highlights
Introduction and summaryWe discuss the double elliptic (Dell) integrable model being a generalization of the CalogeroRuijsenaars family of many-body systems [25,26,27,28,29, 45, 46] to elliptic dependence on the particles momenta
We mostly study the degeneration p → 0 of (1.2), which is the system similar to the model dual to elliptic Ruijsenaars-Schneider one, so that it is elliptic in momenta and trigonometric in coordinates (for simplicity, we will most of the time refer to this Dell (p = 0) case as just-model)
One can consider the special subalgebra - spherical Double Affine Hecke Algebras (DAHA), which preserve the subspace of symmetric polynomials ΛN inside of C[x1, . . . , xN ]
Summary
We discuss the double elliptic (Dell) integrable model being a generalization of the CalogeroRuijsenaars family of many-body systems [25,26,27,28,29, 45, 46] to elliptic dependence on the particles momenta. The generating function of quantum Hamiltonians in this version are given by a relatively simple expression, where both modular parameters (for elliptic dependence on momenta and coordinate) are free constants Another feature of the Koroteev-Shakirov formulation is that it admits some algebraic constructions, which are widely known for the Calogero-Ruijsenaars family of integrable systems. It provides the generating function of the (ell, trig)-model Hamiltonians in the following way: IN (u) = DN (ut)DN (u)−1 = 1 + Pθω(uU )Pθω(uZ)−1E.
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