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 TMF, 2015, Volume 183, Number 2, Pages 222–253 (Mi tmf8780)

New method for constructing semi-invariants and integrals of the full symmetric $\mathfrak{sl}_n$ Toda lattice

A. S. Sorinab, Yu. B. Chernyakovca

a Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia
b National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia
c Institute for Theoretical and Experimental Physics, Moscow, Russia

Abstract: We consider the full symmetric representation of the Lax operator matrix of the Toda lattice, which is known as the full symmetric Toda lattice. The phase space of this system is the generic orbit of the coadjoint action of the Borel subgroup $B^+_n$ of $SL_n(\mathbb R)$. This system is integrable. We propose a new method for constructing semi-invariants and integrals of the full symmetric Toda lattice. Using only the equations of motion for the Lax eigenvector matrix, we prove the existence of the semi-invariants that are Plücker coordinates in the corresponding projective spaces. We use these semi-invariants to construct the integrals. Our new approach provides simple exact formulas for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and also in terms of its eigenvector and eigenvalue matrices of the full symmetric Toda lattice without using the chopping and Kostant procedures. We describe the structure of the additional integrals of motion as functions on the flag space modulo the Toda flows and show how the Plücker coordinates of different projective spaces define different families of the additional integrals.

Keywords: Liouville integrability, integral of motion, semi-invariant, full symmetric Toda lattice, flag space, noncommutative integrability, Lax representation

 Funding Agency Grant Number Russian Foundation for Basic Research 11-02-01335_a13-02-91330-NNIO_à13-02-90602-Ar_a12-02-00594 Federal Agency for Science and Innovations of Russian Federation 14.740.11.0347

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/tmf8780

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English version:
Theoretical and Mathematical Physics, 2015, 183:2, 637–664

Bibliographic databases:

MSC: 37J35, 37K10, 70H06, 82B20

Citation: A. S. Sorin, Yu. B. Chernyakov, “New method for constructing semi-invariants and integrals of the full symmetric $\mathfrak{sl}_n$ Toda lattice”, TMF, 183:2 (2015), 222–253; Theoret. and Math. Phys., 183:2 (2015), 637–664

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf8780
• https://doi.org/10.4213/tmf8780
• http://mi.mathnet.ru/eng/tmf/v183/i2/p222

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Erratum

This publication is cited in the following articles:
1. A. S. Sorin, Yu. B. Chernyakov, G. I. Sharygin, “Phase portraits of the full symmetric Toda systems on rank-$2$ groups”, Theoret. and Math. Phys., 193:2 (2017), 1574–1592
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