On Lie Groups and Toda Lattices

Understand

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.