\(q\)-KP hierarchy, bispectrality and Calogero-Moser systems
In [\textit{E. Frenkel}, Int. Math. Res. Not. 2, 55-76 (1996; Zbl 0860.17033)] the author has proposed a \(q\)-deformation of the \(N\)th KdV hierarchy which is Hamiltonian with respect to the quantum Poisson algebra. A slightly different deformation on the KP hierarchy was proposed in [\textit{L. Haine} and \textit{P. Iliev}, J. Phys. A 30, 7217-7227 (1997; Zbl 0928.35148)]. It was shown that by making an appropriate shift in the arguments of the classical Schur polynomials one obtains rational solutions of the deformed hierarchy. In the article under review the author shows that there is a one-to-one correspondece between the \(q\)-tau functions of a \(q\)-deformation of the KP hierarchy and the planes in Sato Grassmannian. On the base of this correspondence some applications to \(q\)-deformation of the Calogero-Moser hierarchy are given.
scientific article; zbMATH DE number 1523514
scientific article