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Commutative algebras of \(q\)-difference operators and Calogero-Moser hierarchy. (Algèbres commutatives d’opérateurs aux \(q\)-différences et systèmes de Calogero-Moser.) (French. Abridged English version) Zbl 0947.39010

Summary: We define a subspace \(Gr_q^{\text{ad}}\) of M. Sato and Y. Sato’s Grassmannian \(Gr\) [North-Holland Math. Stud. 81, 259-271 (1983; Zbl 0528.58020)], which is a \(q\)-deformation of G. Wilson’s adelic Grassmannian \(Gr^{\text{ad}}\) [J. Reine Angew. Math. 442, 177-204 (1993; Zbl 0781.34051)]. From each plane \(W\in Gr^{\text{ad}}_q\) we construct a bispectral commutative algebra \({\mathcal A}^q_W\) of \(q\)-difference operators. The common eigenfunction \(\Psi(x,z)\) for the operators from \({\mathcal A}^q_W\) is a \(q\)-wave (Baker-Akhiezer) function for a rational solution to a \(q\)-deformation of the KP hierarchy. The poles of these solutions are governed by a certain \(q\)-deformation of the Calogero-Moser hierarchy.

MSC:

39A12 Discrete version of topics in analysis
39A13 Difference equations, scaling (\(q\)-differences)
39A70 Difference operators
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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