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Hamiltonian operators and infinite-dimensional Lie algebras. (English) Zbl 0487.58008

Translation from Funkts. Anal. Prilozh. 15, No. 3, 23–40 (1981; Zbl 0478.58013).

MSC:

37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
35Q53 KdV equations (Korteweg-de Vries equations)
17B80 Applications of Lie algebras and superalgebras to integrable systems
17B66 Lie algebras of vector fields and related (super) algebras

Citations:

Zbl 0478.58013
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References:

[1] I. M. Gel’fand and I. Ya. Dorfman, ”Hamiltonian operators and algebraic structures connected with them,” Funkts. Anal. Prilozhen.,13, No. 4, 13-30 (1979). · Zbl 0428.58009
[2] I. M. Gel’fand and I. Ya. Dorfman, ”The Schouten bracket and Hamiltonian operators,” Funkts. Anal. Prilozhen.,14, No. 3, 71-74 (1980). · Zbl 0444.58010
[3] I. M. Gel’fand and L. A. Dikii, ”Asymptotic of the resolvent of the Sturm?Liouville equations and the algebra of the Korteweg?de Vries equations,” Usp. Mat. Nauk,30, No. 5, 67-100 (1975). · Zbl 0334.58007
[4] I. M. Gel’fand and D. B. Fuks, ”Cohomologies of the Lie algebra of vector fields on the circle,” Funkts. Anal. Prilozhen.,2, No. 4, 92-93 (1968). · Zbl 0176.11501
[5] M. Adler, ”On a trace functional for formal pseudodifferential operators and the sympletic structure of the Korteweg?de Vries equation,” Invent. Math.,50, No. 3, 219-248 (1979). · Zbl 0393.35058
[6] D. P. Lebedev and Yu. I. Manin, ”The Gel’fand?Dikii Hamiltonian operator and the coadjoint representation of the Volterra group,” Funkts. Anal. Prilozhen.,13, No. 4, 40-46 (1979). · Zbl 0441.58007
[7] I. M. Gel’fand and L. A. Dikii, ”Fractional powers of operators and Hamiltonian systems,” Funkts. Anal. Prilozhen.,10, No. 4, 13-29 (1976).
[8] B. A. Kupershmidt and Yu. I. Manin, ”Equations of long waves with a free surface. I. Conservation laws and solutions,” Funkts. Anal. Prilozhen.,11, No. 3, 31-42 (1977), ”II. Hamiltonian structure and higher equations,” Funkts. Anal. Prilozhen.,12, No. 1, 25-32 (1978). · Zbl 0364.35043
[9] D. R. Lebedev, ”Benney’s long waves equations. Hamiltonian formalism,” Lett. Math. Phys.,3, No. 6, 481-488 (1979). · Zbl 0415.35068
[10] L. Martinez Alonso, ”A new infinite-dimensional Kirillov structure related with nonlinear evolution equations” (submitted to Lett. Math. Phys.). · Zbl 0488.58012
[11] F. A. Berezin, ”Some remarks about the associative envelope of a Lie algebra,” Funkts. Anal. Prilozhen.,1, No. 2, 1-14 (1967). · Zbl 0227.22020
[12] A. A. Kirillov, ”Local Lie algebras,” Usp. Mat. Nauk,31, No. 4, 57-76 (1976). · Zbl 0357.58003
[13] M. Flato, A. Lichnerowicz, and D. Sternheimer, ”Algebras de Lie attachees a une variete canonique,” J. Math. Pures Appl. (9),54, No. 4, 455-480 (1975).
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