# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.01762