Gel’fand, I. M.; Dorfman, I. Ya. Hamiltonian operators and the classical Yang-Baxter equation. (English. Russian original) Zbl 0527.58018 Funct. Anal. Appl. 16, 241-248 (1983); translation from Funkts. Anal. Prilozh. 16, No. 4, 1-9 (1982). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 15 Documents MSC: 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 53D05 Symplectic manifolds (general theory) 17B66 Lie algebras of vector fields and related (super) algebras 58A12 de Rham theory in global analysis 17B56 Cohomology of Lie (super)algebras 17B15 Representations of Lie algebras and Lie superalgebras, analytic theory Keywords:complex over a Lie algebra; generalized symplectic structure; Hamiltonian manifolds; Lie algebra of vector fields PDFBibTeX XMLCite \textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funct. Anal. Appl. 16, 241--248 (1983; Zbl 0527.58018); translation from Funkts. Anal. Prilozh. 16, No. 4, 1--9 (1982) Full Text: DOI References: [1] V. E. Zakharov and L. D. Faddeev, ”The Korteweg?de Vries equation is a completely integrable Hamiltonian system,” Funkts. Anal.,5, No. 4, 18-27 (1971). [2] C. S. Gardner, ”Korteweg?de Vries equation and generalizations, IV,” J. Math. Phys.,12, No. 8, 1548-1551 (1971). · Zbl 0283.35021 [3] I. M. Gel’fand and L. A. Dikii, ”Asymptotic resolvents of Sturm?Liouville equations and the algebra of Korteweg?de Vries equations,” Usp. Mat. Nauk,30, No. 5, 67-100 (1975). [4] I. M. Gel’fand and L. A. Dikii, ”Fractional powers of operators and Hamiltonian systems,” Funkts. Anal.,10, No. 4, 13-29 (1976). [5] I. M. Gel’fand and L. A. Dikii, ”Resolvents and Hamiltonian systems,” Funkts. Anal.,11, No. 2, 11-27 (1977). [6] O. I. Bogoyavlenskii and S. P. Novikov, ”Connection of the Hamiltonian formalisms of stationary and nonstationary problems,” Funkts. Anal.,10, No. 1, 9-13 (1976). [7] F. Magri, ”A simple model of the integrable Hamiltonian equation,” J. Math. Phys.,19, No. 5, 1156-1162 (1978). · Zbl 0383.35065 [8] I. M. Gel’fand and I. Ya. Dorfman, ”Hamiltonian operators and algebraic structures connected with them,” Funkts. Anal.,13, No. 4, 13-30 (1979). · Zbl 0428.58009 [9] I. M. Gel’fand and I. Ya. Dorfman, ”Schouten brackets and Hamiltonian operators,” Funkts. Anal.,14, No. 3, 71-74 (1980). · Zbl 0444.58010 [10] I. M. Gel’fand and I. Ya. Dorfman, ”Hamiltonian operators and infinite?dimensional Lie algebras,” Funkts. Anal.,15, No. 3, 23-40 (1981). · Zbl 0478.58013 [11] I. M. Gel’fand and D. B. Fuks, ”Cohomology of the Lie algebra of formal vector fields,” Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 2, 322-337 (1970). [12] I. M. Gel’fand and D. B. Fuks, ”Cohomology of the Lie algebra of vector fields on a circle,” Funkts. Anal.,2, No. 4, 92-93 (1968). · Zbl 0176.11501 [13] I. M. Gel’fand, B. L. Feigin, and D. B. Fuks, ”Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of bundles,” Funkts. Anal.,8, No. 2, 13-29 (1974). [14] P. P. Kulish and E. K. Sklyanin, ”Solutions of the Yang?Baxter equation,” in: Differential Geometry, Lie Groups and Mechanics, III (Zap. Nauchn. Semin. LOMI, Vol. 95) Nauka, Leningrad (1980), pp. 129-160. [15] A. A. Belavin and V. G. Drinfel’d, ”Solutions of the classical Yang?Baxter equation for simple Lie algebras,” Funkts. Anal.,16, No. 3, 1-29 (1982). [16] V. G. Drinfel’d, ”Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the classical Yang?Baxter equations,” Dokl. Akad. Nauk SSSR,266, (1982). [17] A. Lichnerowicz, ”Les vari?t?s de Poisson et leurs algebres de Lie associ?es,” J. Diff. Geom., No. 12, 253-300 (1977). · Zbl 0405.53024 [18] M. Flato and D. Sternheimer, ”Deformations of Poisson brackets, separate and joint analyticity in group representations, nonlinear group representations and physical applications,” in: Harmonic Analysis and Representations of Semisimple Lie Groups, Lect. NATO Adv. Study Inst., Liege 1977, Dordrecht (1980), pp. 385-448. [19] A. A. Kirillov, ”Local Lie algebras,” Usp. Mat. Nauk,31, No. 4, 57-76 (1976). · Zbl 0357.58003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.