Abstract
In this paper there is given a geometric scheme for constructing integrable Hamiltonian systems based on Lie groups, generalizing the construction of M. Adler. The operation of this scheme is considered for parabolic decompositions of semisimple Lie groups. Fundamental examples of integrable systems are connected with graded Lie algebras. Among them are the generalized periodic chains of Toda, multidimensional tops, and the motion of a point on various homogeneous spaces in a quadratic potential.
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Literature cited
A. A. Kirillov, “Unitary representations of nilpotent Lie groups,” Usp. Mat. Nauk,17, No. 4, 57–101 (1962).
V. I. Arnol'd, Ergodic Problems of Classical Mechanics, W. A. Benjamin (1968).
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Matematika,13, No. 5, 128–150 (1969).
V. E. Zakharov and L. D. Faddeev, “Korteweg-de Vries equation of a completely integrable Hamiltonian system,” Funkts. Anal.,5, No. 4, 18–27 (1971).
M. Adler, “On a trace functional for formal pseudodifferential operators and the symplectic structure for Korteweg-de Vries type equations,” Invent. Math.,50, 219–248 (1979).
S. V. Manakov, “Complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, No. 2, 543–555 (1974).
H. Flashchka, “Toda lattice II,” Progr. Theor. Phys.,51, 703–716 (1974).
J. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,” Adv. Math.,16, 197–220 (1975).
P. van Moerbeke, “The spectrum of Jacobi matrices,” Invent. Math.,37, 45–81 (1976).
O. I. Bogoyavlensky, “On perturbations of the periodic Toda lattice,” Commun. Math. Phys.,51, 201–209 (1976).
S. V. Manakov, “Integration of Euler equations of the dynamics of an n-dimensional rigid body,” Funkts. Anal.,10, No. 4, 93–94 (1976).
M. A. Ol'shanetskii and A. M. Perelomov, “Explicit solutions of some completely integrable Hamiltonian systems,” Funkts. Anal.,11, No. 1, 75–76 (1977).
S. P. Novikov, “Periodic Korteweg-de Vries problem I,” Funkts. Anal.,8, No. 3, 54–66 (1974).
B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, “Linear equations of Korteweg-de Vries type, finite-zonal linear operators and Abelian manifolds,” Usp. Mat. Nauk,31, No. 1, 55–136 (1976).
D. Kazhdan, B. Kostant, and S. Sternberg, “Hamiltonian group actions and dynamical systems of Calogero type,” Commun. Pure Appl. Math.,31, No. 4, 481–508 (1978).
M. A. Olshanetsky and A. M. Perelomov, “Explicit solutions of the classical generalized Toda models,” Invent. Math.,54 (1979).
A. G. Reyman and M. A. Semenov-Tian-Shansky, “Reduction of Hamiltonian systems, affine Lie algebras and Lax equations,” Invent. Math.,54, 81–100 (1979).
I. M. Krichever, “Algebraic curves and nonlinear difference equations,” Usp. Mat. Nauk,33, No. 4, 215–216 (1978).
P. van Moerbeke and D. Mumford, “The spectrum of difference operators and algebraic curves,” Acta Math.,143, 93–154 (1979).
B. Kostant, “The solution to a generalized Toda lattice and representation theory,” Adv. Math.,34, 195–338 (1979).
A. A. Kirillov, Elements of the Theory of Representations [in Russian], Nauka, Moscow (1972).
D. R. Lebedev and Yu. I. Manin, “Equations of lengths of Benni waves. II. Lax representation and conservation laws,” Preprint ITÉF-89 (1979).
I. M. Gel'fand and L. A. Dikii, “Fractional powers of operators and Hamiltonian systems,” Funkts. Anal.,10, No. 4, 13–29 (1976).
B. Kostant, “On Whittaker vectors and representation theory,” Invent. Math.,48, No. 2, 101–184 (1978).
B. Kostant, “The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,” Am. J. Math.,81, 973–1032 (1959).
I. C. Gohberg and I. A. Fel'dman, Convolution Equations and Projection Methods for Their Solution, Amer. Math. Soc. (1975).
A. S. Mishchenko and A. T. Fomenko, “Integration of Euler equations on semisimple Lie algebras,” Dokl. Akad. Nauk SSSR,231, No. 3, 536–538 (1976).
A. S. Mishchenko and A. T. Fomenko, “Euler equations on finite-dimensional Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,42, 396–415 (1978).
F. Magri, “A simple model of the integrable Hamiltonian equation,” J. Math. Phys.,19, 1156–1162 (1978).
P. P. Kulish and A. G. Reiman, “Hierarchy of sympletic forms for Schrödinger and Dirac equations on the line,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,77, 134–147 (1978).
I. M. Gel'fand and I. Ya. Dorfman, “Hamiltonian operators and algebraic structures connected with them,” Funkts. Anal.,13, No. 4, 13–30 (1979).
M. Adler and P. van Moerbeke, “Completely integrable systems, Kas-Moody Lie algebras, and curves,” Preprint (1979).
J. Moser, “Geometry of quadrics and spectral theory,” Preprint (1979).
A. G. Reiman, M. A. Semenov-Tyan-Shanskii, and I. B. Frenkel', “Graded Lie algebras and completely integrable dynamical systems,” Dokl. Akad. Nauk SSSR,247, 802–805 (1979).
I. M. Krichever, “Periodic non-Abelian Toda chain and its two-dimensional generalization,” Submitted to Dokl. Akad. Nauk SSSR.
M. Duflo, “Operateurs differentiels bi-invariants sur un groupe de Lie,” Ann. Sci. École Norm. Sup., 4-e Ser.,10, 265–288 (1977).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 95, pp. 3–54, 1980.
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Reiman, A.G. Integrable Hamiltonian systems connected with graded Lie algebras. J Math Sci 19, 1507–1545 (1982). https://doi.org/10.1007/BF01091461
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DOI: https://doi.org/10.1007/BF01091461