Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

Abstract

In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen-Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen-Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero-Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, providing explicit computations of some 'isospectral deformations' of the planar rational Calogero-Moser system in the case of the split rational potential.