Abstract
The full symmetric Toda system is a generalization of the open Toda chain, for which the Lax operator is a symmetric matrix of general form. This system is Liouville integrable and even superintegrable. Deift, Lee, Nando, and Tomei (DLNT) proposed the chopping method for constructing integrals of such a system. In the paper, a solution of Hamiltonian equations for the entire family of DLNT integrals is constructed by using the generalized QR factorization method. For this purpose, certain tensor operations on the space of Lax operators and special differential operators on the Lie algebra are introduced. Both tools can be interpreted in terms of the representation theory of the Lie algebra \(\mathfrak{sl}_n\) and are expected to generalize to arbitrary real semisimple Lie algebras. As is known, the full Toda system can be interpreted in terms of a compact Lie group and a flag space. Hopefully, the results on the trajectories of this system obtained in the paper will be useful in studying the geometry of flag spaces.
This is a preview of subscription content, log in via an institution to check access.
References
N. N. Nehorošev, “Action-angle variables, and their generalizations”, Tr. Mosk. Mat. Obs., MSU, Moscow, 1972, 181–198.
Yu. B. Chernyakov and A. S. Sorin, “Explicit semi-invariants and integrals of the full symmetric \(\mathfrak{sl}_n\) Toda lattice”, Lett. Math. Phys., 104:8 (2014), 1045–1052; arXiv: 1409.5332.
A. S. Sorin, Yu. B. Chernyakov, “New method for constructing semi-invariants and integrals of the full symmetric \(\mathfrak{sl}_n\) Toda lattice”, Teoret. Mat. Fiz., 183:2 (2015), 222–253; English transl.: Theoret. and Math. Phys., 183:2 (2015), 637–664; arXiv: 1312.4555.
P. Deift, L. C. Li, T. Nanda, and C. Tomei, “The Toda flow on a generic orbit is integrable”, Commun. Pure Appl. Math., 39 (1986), 183–232.
M. Gekhtman, M. Shapiro, “Noncommutative and commutative integrability of generic Toda flows in simple Lie algebras”, Comm. Pure Appl. Math., 52:1 (1999), 53–84.
D. V. Talalaev, “Quantum generalized Toda system”, TMF, 171:2 (2012), 312–320; English transl.: Theoret. and Math. Phys., 171:2 (2012), 691–699.
M. Adler, “On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg–de Vries equation”, Invent. Math., 50:3 (1979), 219–248.
B. Kostant, “The solution to a generalized Toda lattice and representation theory”, Adv. in Math., 34:3 (1979), 195–338.
W. W. Symes, “Systems of Toda type, inverse spectral problems, and representation theory”, Invent.Math., 59:1 (1980), 13–51.
N. M. Ercolani, H. Flaschka, S. Singer, “The geometry of the full Kostant–Toda lattice”, Integrable systems (Luminy, 1991), Progress in Mathematics, 115, Birkhäuser, Boston, 1993, 181–225.
S. V. Manakov, “Note on the integration of Euler’s equations of the dynamics of an \(n\)-dimensional rigid body”, Funktsional. Anal. i Prilozhen., 10:4 (1976), 93–94; English transl.: Funct. Anal. Appl., 10:4 (1976), 328–329.
A. S. Mishchenko, A. T. Fomenko, “Euler equations on finite-dimensional Lie groups”, Izv. Akad. Nauk SSSR Ser. Mat., 42:2 (1978), 396–415; English transl.: Math. USSR-Izv., 12:2 (1978), 371–389.
G. Schrader, “Integrable systems from the classical reflection equation”, Int. Math. Res. Not., 2016:1 (2016), 1-23.
B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Grad. Texts in Math., 222 Springer, Cham, 2015.
Yu. B. Chernyakov, G. I. Sharygin, and A. S. Sorin, On the invariants of the full symmetric Toda system, arXiv: 1910.04789v1.
P. A. Damianou and F. Magri, “A gentle (without chopping) approach to the full Kostant– Toda lattice”, SIGMA, 1 (2005).
I. Bobrova, V. Retakh, V. Rubtsov, and G. Sharygin, “A fully noncommutative analog of the Painlevè”, J. Phys. A, 55:47 (2022).
Yu. B. Chernyakov, G. I. Sharygin, and A. S. Sorin, “Bruhat order in general Toda system”, Comm. Math. Phys., 330:1 (2014), 367–399.
Y. Xie, On the full Kostant–Toda lattice and the flag varieties. I. The singular solutions, arXiv: 2212.03679.
Y. Kodama and Y. Xie, On the full Kostant-Toda hierarchy and its \(\ell\)-banded reductions for the Lie algebras of type A, B and G, arXiv: 2303.07331.
Funding
The work on Sections 1, 2.1, 2.3, 3.1, 2.2, and Appendix was performed by Yu. B. Chernyakov, the work on Sections 2.4, 3.3, 4.1, and 4.2 was performed by D. V. Talalaev, and the work on Sections 3.2, 4.1, and 5 was performed by G. I. Sharygin. The work of G. I. Sharygin was supported by the Russian Science Foundation under grant no. 22-11-00272, https://rscf.ru/project/22-11-00272/, and carried out at Lomonosov Moscow State University, National Research Centre “Kurchatov Institute,” and Moscow Institute of Physics and Technology (National Research University). The work of D. V. Talalaev was carried out within the framework of a development program for Regional Scientific and Educational Mathematical Center of P. G. Demidov Yaroslavl State University under the support of the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-886 on the provision of subsidy from the federal budget) and was partially supported by the Russian Foundation for Basic Research under grant no. 20-01-00157.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that he has no conflicts of interest.
Additional information
Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 100–122 https://doi.org/10.4213/faa4105.
To I. M. Gel’fand, whose influence on mathematics cannot be overestimated
Translated by O. V. Sipacheva
Publisher’s note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Talalaev, D.V., Chernyakov, Y.B. & Sharygin, G.I. Full Symmetric Toda System: Solution via QR-Decomposition. Funct Anal Its Appl 57, 346–363 (2023). https://doi.org/10.1134/S0016266323040081
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0016266323040081