On factorized Lax pairs for classical many-body integrable systems

TL;DR

This paper explores factorization formulas for Lax matrices in classical integrable many-body systems, linking IRF-Vertex relations and vector bundle modifications to derive explicit M-matrix expressions.

Contribution

It introduces new factorization formulas for Lax matrices of Ruijsenaars-Schneider and Calogero-Moser models based on two different theoretical approaches.

Findings

Derived explicit M-matrix formulas from IRF-Vertex relations.

Linked factorization formulas to vector bundle modifications.

Proposed formulas for special cases of Calogero-Moser models.

Abstract

In this paper we study factorization formulae for the Lax matrices of the classical Ruijsenaars-Schneider and Calogero-Moser models. We review the already known results and discuss their possible origins. The first origin comes from the IRF-Vertex relations and the properties of the intertwining matrices. The second origin is based on the Schlesinger transformations generated by modifications of underlying vector bundles. We show that both approaches provide explicit formulae for $M$-matrices of the integrable systems in terms of the intertwining matrices (and/or modification matrices). In the end we discuss the Calogero-Moser models related to classical root systems. The factorization formulae are proposed for a number of special cases.