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Hamiltonian operators and associative algebras with a derivation. (English) Zbl 0837.16034
Summary: We prove that an algebraic structure proposed by I. M. Gel’fand and I. Ya. Dorfman [Funct. Anal. Appl. 13, 248–262 (1980); translation from Funkts. Anal. Prilozh. 13, No. 4, 13–30 (1979; Zbl 0428.58009)] in studying Hamiltonian operators is equivalent to an associative algebra with a derivation under a unitary condition.
MSC:
16W25Derivations, actions of Lie algebras (associative rings and algebras)
37K30Relations of infinite-dimensional systems with algebraic structures
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
References:
[1]Balinskii, A. A. and Novikov, S. P.: English transl.Soviet Math. Dokl. 32, 228-231 (1985).
[2]Dolan, L., Goddard, P., and Montague, P.:Nuclear Phys. B338, 529-601 (1990).
[3]Gel’fand, I. M. and Dikii, L. A.:Russian Math. Surveys 30(5), 77-113 (1975). · Zbl 0334.58007 · doi:10.1070/RM1975v030n05ABEH001522
[4]Gel’fand, I. M. and Dikii, L. A.:Funct. Anal. Appl. 10, 16-22 (1976). · Zbl 0347.49023 · doi:10.1007/BF01075767
[5]Gel’fand, I. M. and Dorfman, I. Ya.:Funct. Anal. Appl. 13, 248-262 (1979). · Zbl 0437.58009 · doi:10.1007/BF01078363