Iliev, Plamen 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 C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 10, 877-882 (1999). 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. Cited in 4 Documents 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.) Keywords:\(q\)-wave function; Baker-Akhiezer function; \(q\)-difference operator; Grassmannian; bispectral commutative algebra; eigenfunction; KP hierarchy; Calogero-Moser hierarchy Citations:Zbl 0528.58020; Zbl 0781.34051 PDFBibTeX XMLCite \textit{P. Iliev}, C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 10, 877--882 (1999; Zbl 0947.39010) Full Text: DOI