Abstract
Connection is established between one-dimensional Toda lattices, constructed on the basis of the systems of simple roots of classical and affine Lie algebras, and other integrable systems of interacting particles. That connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends. Some of the new lattices admit non-Abelian generalizations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Toda, M.: Wave propagation in anharmonic lattices. J. Phys. Soc. Jpn.23, 501–506 (1967)
Flaschka, H.: On the Toda lattice. I. Phys. Rev. B9, 1924–1925 (1974); II. Progr. Theor. Phys.51, 703–716 (1974)
Moser, J.: Finitely many mass points on the line under the influence of an exponential potential — an integrable system. Batelle Recontres-Lecture Notes in Physics, vol. 38, pp. 468–497. New York: Springer, Berlin, Heidelberg, New York: Springer 1974
Bogoyavlensky, O.I.: On perturbation of the periodic Toda lattices. Commun. Math. Phys.51, 201–209 (1976)
Olshanetsky, M.A., Perelomov, A.M.: Explicit solutions of classical generalized Toda models. Invent. Math.54, 261–269 (1979)
Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math.34, 195–338 (1979)
Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math.38, 267–317 (1980)
Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. Math.38, 318–379 (1980)
Reyman, A.G., Semenov-Tian-Schansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. I. Invent. Math.54, 81–100 (1979), II. Invent. Math.63, 423–432 (1981)
Adler, M., van Moerbeke, P.: Kowalewski's Asymptotic method, Kac-Moody Lie algebras and regularization. Commun. Math. Phys.83, 83–106 (1982)
Goodman, R., Wallach, N.R.: Classical and quantum mechanical systems of Toda lattice type. I. Commun. Math. Phys.83, 355–386 (1982); II. Solutions of the classical flows. Commun. Math. Phys.94, 177–217 (1984)
Symes, W.W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math.59, 13–52 (1980)
Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math.16, 197–220 (1975)
Olshanetsky, M.A., Perelomov, A.M.: Completely integrable systems connected with semisimple Lie algebras. Invent. Math.37, 93–108 (1976)
Inozemtsev, V.I., Meshcheryakov, D.V.: Extension of the class of integrable dynamical systems connected with semisimple Lie algebras. Lett. Math. Phys.9, 13–18 (1985)
Inozemtsev, V.I.: On a motion of classical integrable systems of interacting particles in the external field. Phys. Lett.98A, 316–318 (1983); New Completely integrable multiparticle systems. Phys. Scripta29, 518–521 (1984)
Reyman, A.G.: Integrable Hamiltonian systems related to affine Lie algebras. Zapiski LOMI95, 3–54 (1980) (in Russian)
Olshanetsky, M.A., Perelomov, A.M., Reyman, A.G., Semenov-Tian-Schansky, M.A.: Integrable systems-II, in contemporary problems in mathematics16, 86–226 (1987) (in Russian)
Inozemtsev, V.I.: Lax representation with the spectral parameter on a torus for integrable particle systems. JINR Preprint P2-88-219, Dubna, 1988
Ruijsenaars, S.N.M.: Complete integrability of relativistic Calogero-Moser systems and elliptic function identities. Commun. Math. Phys.110, 191–213 (1987)
Inozemtsev, V.I.: On relativistic two-particle Ruijsenaars-Schneider systems in an external field. JINR Preprint E2-88-219, Dubna, 1988
Author information
Authors and Affiliations
Additional information
Communicated by J. N. Mather
Rights and permissions
About this article
Cite this article
Inozemtsev, V.I. The finite Toda lattices. Commun.Math. Phys. 121, 629–638 (1989). https://doi.org/10.1007/BF01218159
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01218159