On Lie groups and Toda lattices

O. Kruglinskaya, A. Marshakov

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4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Article number125201
JournalJournal of Physics A: Mathematical and Theoretical
Volume48
Issue number12
DOIs
Publication statusPublished - 27 Mar 2015
Externally publishedYes

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