Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Intermediate Toda systems

Intermediate Toda systems 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 I in a Borel subalgebra b+ of an arbitrary simple Lie algebra g. The classical Kostant-Toda lattice corresponds to the case of I = [n+, n+], where n+ is the unipotent ideal of b+, while the full Kostant-Toda lattice corresponds to I = {0}. We mainly focus on the case I = [[n+, n+], 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Regular and Chaotic Dynamics Springer Journals

Loading next page...
 
/lp/springer-journals/intermediate-toda-systems-Gc19t5BEY7

References (36)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Pleiades Publishing, Ltd.
Subject
Mathematics; Dynamical Systems and Ergodic Theory
ISSN
1560-3547
eISSN
1468-4845
DOI
10.1134/S1560354715030053
Publisher site
See Article on Publisher Site

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 I in a Borel subalgebra b+ of an arbitrary simple Lie algebra g. The classical Kostant-Toda lattice corresponds to the case of I = [n+, n+], where n+ is the unipotent ideal of b+, while the full Kostant-Toda lattice corresponds to I = {0}. We mainly focus on the case I = [[n+, n+], 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.

Journal

Regular and Chaotic DynamicsSpringer Journals

Published: Jun 17, 2015

There are no references for this article.