Daletskij, Yu. L. Hamiltonian operators in graded formal calculus of variations. (English. Russian original) Zbl 0609.58010 Funct. Anal. Appl. 20, 136-138 (1986); translation from Funkts. Anal. Prilozh. 20, No. 2, 62-64 (1986). In this paper, continuing his previous investigations, the author considers the graded variant of the Gel’fand-Dikij formal variational calculus and generalizes some results of Gel’fand and Dorfman regarding the description of the Lie algebra of the Poisson brackets. The obtained description turns out to be connected with the polynomial Poisson algebras, considered by Sklyanin. On the other hand, from it one can derive the superanalogue theorem of Dubrovin-Novikov on the hydrodynamic Hamiltonians. Reviewer: Pan Yanglian Cited in 1 Document MSC: 58E30 Variational principles in infinite-dimensional spaces 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Keywords:Hamiltonian operator; Lie superalgebra structure; graded variant of the Gel’fand-Dikij formal variational calculus; Lie algebra of the Poisson brackets; hydrodynamic Hamiltonians PDFBibTeX XMLCite \textit{Yu. L. Daletskij}, Funct. Anal. Appl. 20, 136--138 (1986; Zbl 0609.58010); translation from Funkts. Anal. Prilozh. 20, No. 2, 62--64 (1986) Full Text: DOI References: [1] I. M. Gel’fand and Yu. L. Daletskii, ”Some formal differential structures related to Lie superalgebras,” Preprint Institute of Applied Mechanics (IPM), No. 85, Moscow (1984). [2] Yu. L. Daletskii (Yu. L. Daletsky), Lie Superalgebras in a Hamilton Operator Theory of Nonlinear and Turbulent Processes, Gordon and Breach, New York (1984). [3] Yu. L. Daletskii and B. L. Tsygan, Funkts. Anal. Prilozhen.,19, No. 4, 82-83 (1985). [4] I. M. Gel’fand and L. A. Dikii, Usp. Mat. Nauk,30, No. 5, 67-100 (1975). [5] I. M. Gel’fand and L. A. Dikii, Funkts. Anal. Prilozhen.,10, No. 1, 18-25 (1975). [6] I. M. Gel’fand and I. Ya. Dorfman, Funkts. Anal. Prilozhen.,13, No. 4, 13-30 (1979). [7] I. M. Gel’fand and I. Ya. Dorfman, Funkts. Anal. Prilozhen.,14, No. 3, 71-74 (1980). [8] I. M. Gel’fand and I. Ya. Dorfman, Funkts. Anal. Prilozhen.,15, No. 3, 23-40 (1981). [9] E. K. Sklyanin, Funkts. Anal. Prilozhen.,16, No. 4, 27-34 (1982). [10] B. A. Dubrovin and S. P. Novikov, Dokl. Akad. Nauk SSSR,270, No. 4, 781-785 (1983). [11] D. A. Leites, The Theory of Supermanifolds [in Russian], Izd. Karel’sk. Filiala Akad. Nauk SSSR, Petrozavodsk (1983). 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.