On Lie groups and Toda lattices

We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of the A-series. For the simply laced case this is just a direct generalization of the well-known relativistic Toda chains procedure, and we construct explicitly the set of Ad-invariant integrals of motion on symplectic leaves, which can be described using Poisson quivers, which are just 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 the A-series, and consider explicitly an example from the affine D-series.