Abstract
We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.V. Mikhailov (1979) Pisma v ZhETF 30 443
Ueno, K. and Takasaki, K.: Toda lattice hierarchy. Group representations and systems of differential equations (Tokyo, 1982), In: Adv. Stud. Pure Math., 4, North-Holland, Amsterdam, 1984, pp. 1–95.
M. Mineev-Weinstein P.B. Wiegmann A. Zabrodin (2000) Phys. Rev. Lett. 84 5106
Sklyanin, E. K.: Quantum variant of the method of the inverse scattering problem. Differential geometry, Lie groups and mechanics, III. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 95 (1980), 55–128, 161; translation in J. Soviet Math. 19 (5) (1982), 1546–1596 (in Russian).
Semenov-Tyan-Shanski, M. A.: What a classical r-matrix is, Funktsional. Anal. i Prilozhen. 17 (4) (1983), 17–33; translation in Functional Anal. Appl. 17 (4) (1983), 259–272.
W. Oevel O. Ragnisco (1989) ArticleTitlesd. R-matrices and higher Poisson brackets for integrable systems Phys. A 161 IssueID1 181–220
L.C. Li S. Parmentier (1989) ArticleTitleNonlinear Poisson structures and r-matrices Comm. Math. Phys. 125 IssueID4 545–563
Oevel, W.: Poisson brackets for integrable lattice systems. In: A. S. Fokas and I. M. Gelfand (eds), Algebraic aspects of integrable systems: in memory of Irene Dorfman, Progress in Non-linear differential equations and their applications, Vol. 26. 1997.
Kuperschmidt, B. A.: Discrete Lax equations and differential-difference calculus. Astérisque No. 123, (1985), 212 pp.
Pirozerski, A. L. and Semenov-Tian-Shansky, M. A.: Generalized q-deformed Gelfand-Dickey structures on the group of q-pseudodifference operators. In: L. D. Faddeev (ed.), Seminar on Mathematical Physics, M. A. Semenov-Tian-Shansky (ed.), Advances in the Mathematical Sciences, Vol. 201, AMS, 2000, 321 pp. Preprint math.QA/9811025.
Belavin, A. A. and Drinfel’d, V.G.: Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen. 16 (3) (1982), 1–29, 96; Translated in Functional Anal. Appl. 16 (3) (1982), 159–180 (in Russian).
O.I. Bogoyavlensky (1976) ArticleTitleOn perturbations of the periodic Toda lattice Commun. Math. Phys. 51 201–209
P. Moerbeke Particlevan D. Mumford (1979) ArticleTitleThe spectrum of difference operators and algebraic curves Acta Math. 143 93–154
M. Adler P. Moerbeke Particlevan (1980) ArticleTitleCompletely integrable systems, Euclidean Lie algebras, and curves Adv. Math. 38 267–317
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematical Subject Classifications (1991). 37K10.
Rights and permissions
About this article
Cite this article
Carlet, G. The Hamiltonian Structures of the Two-Dimensional Toda Lattice and R-Matrices. Lett Math Phys 71, 209–226 (2005). https://doi.org/10.1007/s11005-005-0629-y
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11005-005-0629-y