Calogero-Moser Lax Pairs with Spectral Parameter for General Lie Algebras

TL;DR

This paper constructs spectral parameter Lax pairs for elliptic Calogero-Moser systems linked to all finite-dimensional Lie algebras, including new integrable systems for twisted affine types, extending previous results.

Contribution

It provides explicit Lax pairs with spectral parameters for Calogero-Moser systems associated with all finite Lie algebras, including classical, exceptional, and twisted affine types, filling a gap in integrable systems theory.

Findings

Lax pairs with spectral parameter for classical and exceptional Lie algebras

Reduction to known Lax pairs at special spectral parameter values

Introduction of new integrable systems for twisted affine Lie algebras

Abstract

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of $G_2$).