Abstract
The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.
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Avan, J., Talon, M.: Rational and trigonometric constant, non-antisymmetricR-matrices. Phys. Lett. B241, 77–82 (1990)
Babelon, O., Viallet, C.-M.: Hamiltonian structures and Lax equations. Phys. Lett. B237, 441–416 (1990)
Burstall, F.E., Ferus, D., Pedit, F., Pinkall, U.: Harmonic Tori in symmetric spaces and communting Hamiltonian systems on loop algebras. University of Bath/Technische Universität Berlin/Emory University Atlanta, Preprint 1991
Vega, H.J. de, Eichenherr, H., Maillet, J.-M.: Classical and quantum algebras of non-local charges in sigma models. Commun. Math. Phys.92, 507–524 (1984)
Eichenherr, H., Forger, M.: On the dual symmetry of the nonlinear sigma models. Nucl. Phys. B155, 381–393 (1979) More about nonlinear sigma models on symmetric spaces. Nucl. Phys. B164, 528–535 (1980); B282, 745–746 (1987) (Erratum) Higher local conservation laws for nonlinear sigma models on symmetric spaces. Commun. Math. Phys.82, 227–255 (1981)
Faddeev, L.D., Takhtajan, L.A.: Hamiltonian methods in the theory of solitons., Berlin, Heidelberg New York: Springer 1987
Forger, M.: Nonlinear sigma models on symmetric spaces. In: Nonlinear partial differential operators and quantization procedures. Proceedings, Clausthal, Germany 1981, Andersson, S.I., Doebner, H.D. (eds.); Lect. Notes in Math.1037. Berlin: Springer 1983
Forger, M., Laartz, J., Schäper, U.: Current algebra of classical non-linear sigma models. Commun. Math. Phys.146, 397–402 (1992)
Freidel, L., Maillet, J.M.: Quadratic algebras and integrable systems. Preprint LPTHE-24/91; On classical and quantum integrable field theories associated to Kac-Moody current algebras. Preprint LPTHE-25/91
Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York: Academic Press 1978
Laartz, J.: The extension structure of 2D massive current algebras. Freiburg University preprint THEP 91/21, to be published in Mod. Phys. Lett. A
Maillet, J.-M.: Kac-Moody algebra and extended Yang-Baxter, relations in theO(N) non-linear sigma model. Phys. Lett. B162, 137–142 (1985)
Maillet, J.-M.: Hamiltonian structures for integrable classical theories from graded Kac-Moody algebras. Phys. Lett. B167, 401–405 (1986)
Maillet, J.-M.: New integrable canonical structures in two-dimensional models. Nucl. Phys. B269, 54–76 (1986)
Avan, J., Talon, M.: GradedR-matrices for integrable systems. Nucl. Phys. B352, 215 (1991)
Avan, J.: Current algebra realization ofR-matrices associated toZ 2-graded Lie algebras. Phys. Lett. B252, 230 (1990)
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Communicated by N. Yu. Reshetikhin
Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1
Supported by the Studienstiftung des Deutschen Volkes
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Bordemann, M., Forger, M., Laartz, J. et al. The Lie-Poisson structure of integrable classical non-linear sigma models. Commun.Math. Phys. 152, 167–190 (1993). https://doi.org/10.1007/BF02097062
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DOI: https://doi.org/10.1007/BF02097062