Abstract
We show how to construct semi-invariants and integrals of the full symmetric \({\mathfrak{sl}_n}\) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. The existence of additional integrals which constitute a full set of independent non-involutive integrals was known but the chopping and Kostant procedures have crucial computational complexities already for low-rank Lax matrices and are practically not applicable for higher ranks. Our new approach solves this problem and results in simple explicit formulae for the full set of independent semi-invariants and integrals expressed in terms of the Lax matrix and its eigenvectors, and of eigenvalue matrices for the full symmetric \({\mathfrak{sl}_n}\) Toda lattice.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arhangelskii A.A.: Completely integrable hamiltonian systems on a group of triangular matrices. Math. USSR-Sbornik 36(1), 127–134 (1980)
Adler M.: On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg–de Vries equation. Invent. Math. 50, 219–248 (1979)
Kostant B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338 (1979)
Symes W.W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math. 59(1), 13–51 (1980)
Deift P., Li L.C., Nanda T., Tomei C.: The Toda flow on a generic orbit is integrable. CPAM 39, 183–232 (1986)
Ercolani, N., Flaschka, H., Singer, S.: The geometry of the full Kostant-Toda lattice. In: Integrable Systems. Progress in Mathematics, vol. 115, pp. 181–226. Birkhauser, Basel (1993)
Shipman B.A.: The geometry of the full Kostant-Toda lattice of sl(4;C). J. Geom. Phys. 33, 295–325 (2000)
Talalaev, D.: Quantum generic Toda system. arXiv:1012.3296
Nehoroshev N.N.: Action-angle variables and their generalization. Tr. Mosk. Mat. O.-va. 26, 181–198 (1972)
Bloch A.M., Gekhtman M.: Hamiltonian and gradient structures in the Toda flows. J. Geom. Phys. 27, 230–248 (1998)
Gekhtman M., Shapiro M.: Noncommutative and commutative integrability of generic Toda flows in simple Lie algebras. Comm. Pure Appl. Math. 52, 53–84 (1999)
Bloch A.M., Gekhtman M.: Lie algebraic aspects of the finite nonperiodic Toda flows. J. Comp. Appl. Math. 202, 3–25 (2007)
Fre, P., Sorin, A.S.: The arrow of time and the Weyl group: all supergravity billiards are integrable. Nucl. Phys. B 815, 430 (2009). arXiv:0710.1059 [hep-th]
Fre, P., Sorin, A.S., Trigiante, M.: Integrability of supergravity black holes and new tensor classifiers of regular and nilpotent orbits. JHEP 1204, 015 (2012). arXiv:1103.0848 [hep-th]
Fulton W.: Young Tableaux. Cambridge University Press, Cambridge (1977)
Chernyakov, Y.B., Sharygin, G.I., Sorin, A.S.: Bruhat order in full symmetric toda system. Commun. Math. Phys. (2014, in press). arXiv:1212.4803 [nlin.SI]
Chernyakov, Y.B., Sorin, A.S.: Semi-invariants and Integrals of the Full Symmetric \({\mathfrak{sl}_n}\) Toda Lattice. arXiv:1312.4555 [nlin.SI]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chernyakov, Y.B., Sorin, A.S. Explicit Semi-invariants and Integrals of the Full Symmetric \({\mathfrak{sl}_n}\) Toda Lattice. Lett Math Phys 104, 1045–1052 (2014). https://doi.org/10.1007/s11005-014-0698-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-014-0698-x