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

TL;DR

This paper investigates the structure of planar quasi-invariant algebras, classifies their rank one Cohen-Macaulay modules, and applies these findings to spectral modules and explicit solutions of Calogero-Moser systems.

Contribution

It provides a classification of Cohen-Macaulay modules over these algebras and links them to spectral modules of Calogero-Moser systems, including a new explicit example.

Findings

Classified all rank one Cohen-Macaulay modules

Determined the Picard groups of the algebras

Computed a new explicit example of a deformed Calogero-Moser system

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, computing a new unexpected explicit example of a deformed Calogero-Moser system.