On Lie Groups and Toda Lattices

TL;DR

This paper generalizes the construction of relativistic Toda chains as integrable systems on Lie groups beyond type A, including simply-laced and non-simply-laced cases, and explores their extensions to affine Lie algebras.

Contribution

It extends the relativistic Toda chain framework to arbitrary Lie groups, explicitly constructs integrals of motion, and introduces methods for non-simply-laced and affine cases.

Findings

Constructed integrals of motion for simply-laced Lie groups.

Extended Toda systems to non-simply-laced Lie groups using folding techniques.

Discussed potential extensions to affine Lie algebras.

Abstract

We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of A-series. For the simply-laced case this is just a direct generalization of the well-known relativistic Toda chains, and we construct explicitly the set of the Ad-invariant integrals of motion on symplectic leaves, which can be described by the Poisson quivers being just the blown up Dynkin diagrams. We also demonstrate how to get the set of "minimal" integrals of motion, using the co-multiplication rules for the corresponding Lie algebras. In the non simply-laced case the corresponding Bogoyavlensky-Coxeter-Toda systems are constructed using the Fock-Goncharov folding of the corresponding Poisson submanifolds. We discuss also how this procedure can be extended for the affine case beyond A-series, and consider explicitly an example from the affine D-series.