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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2015, Volume 20, Issue 3, Pages 277–292 (Mi rcd43)

Intermediate Toda Systems

Pantelis A. Damianoua, Hervé Sabourinb, Pol Vanhaeckeb

a Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
b Laboratoire de Mathématiques, UMR 7348 du CNRS, Université de Poitiers, 86962 Futuroscope Chasseneuil Cedex, France

Abstract: We construct a large family of Hamiltonian systems which interpolate between the classical Kostant–Toda lattice and the full Kostant–Toda lattice and we discuss their integrability. There is one such system for every nilpotent ideal $\mathcal{I}$ in a Borel subalgebra $\mathfrak{b}_+$ of an arbitrary simple Lie algebra $\mathfrak{g}$. The classical Kostant–Toda lattice corresponds to the case of $\mathcal{I}=[\mathfrak{n}_+,\mathfrak{n}_+]$, where $\mathfrak{n}_+$ is the unipotent ideal of $\mathfrak{b}_+$, while the full Kostant–Toda lattice corresponds to $\mathcal{I}=\{0\}$. We mainly focus on the case $\mathcal{I}=[[\mathfrak{n}_+,\mathfrak{n}_+],\mathfrak{n}_+]$. In this case, using the theory of root systems of simple Lie algebras, we compute the rank of the underlying Poisson manifolds and construct a set of (rational) functions in involution, large enough to ensure Liouville integrability. These functions are restrictions of well-chosen integrals of the full Kostant–Toda lattice, except for the case of the Lie algebras of type $C$ and $D$ where a different function (Casimir) is needed. The latter fact, and other ones listed in the paper, point at the Liouville integrability of all the systems we construct, but also at the nontriviality of obtaining the result in full generality.

Keywords: Toda lattices, integrable systems

DOI: https://doi.org/10.1134/S1560354715030053

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Bibliographic databases:

MSC: 37J35, 17B20, 17B22, 70H06
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Citation: Pantelis A. Damianou, Hervé Sabourin, Pol Vanhaecke, “Intermediate Toda Systems”, Regul. Chaotic Dyn., 20:3 (2015), 277–292

Citation in format AMSBIB
\Bibitem{DamSabVan15} \by Pantelis A. Damianou, Herv\'e Sabourin, Pol Vanhaecke \paper Intermediate Toda Systems \jour Regul. Chaotic Dyn. \yr 2015 \vol 20 \issue 3 \pages 277--292 \mathnet{http://mi.mathnet.ru/rcd43} \crossref{https://doi.org/10.1134/S1560354715030053} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3357276} \zmath{https://zbmath.org/?q=an:06488657} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015RCD....20..277D} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84934964767}