G(2)-Calogero-Moser Lax operators from reduction

TL;DR

This paper develops a Lax operator for the G(2)-Calogero-Moser model through a two-step reduction process involving the A6 and B3 models, revealing algebraic structures linked to Lie algebra degrees.

Contribution

It introduces a novel reduction-based method to construct G(2)-Calogero-Moser Lax operators from known models, connecting Lie algebra embeddings with integrable systems.

Findings

Successfully constructed the G(2)-Lax operator via reduction

Identified the degrees of conserved charges match Lie algebra degrees

Demonstrated the embedding relationships between root systems

Abstract

We construct a Lax operator for the $G_2$-Calogero-Moser model by means of a double reduction procedure. In the first reduction step we reduce the $A_6$-model to a $B_3$-model with the help of an embedding of the $B_3$-root system into the $A_6$-root system together with the specification of certain coupling constants. The $G_2$-Lax operator is obtained thereafter by means of an additional reduction by exploiting the embedding of the $G_2$-system into the $B_3$-system. The degree of algebraically independent and non-vanishing charges is found to be equal to the degrees of the corresponding Lie algebra.