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Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations

Published online by Cambridge University Press:  20 November 2018

Yucai Su*
Affiliation:
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, P. R. China email: [email protected]
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Abstract

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Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2003

References

[DZ] Dokovic, D. and Zhao, K., Derivations, isomorphisms and second cohomology of generalized Block algebras. Algebra Colloq. 3(1996), 245272.Google Scholar
[F] Farnsteiner, R., Derivations and central extensions of finitely generated graded Lie algebras. J. Algebra 118(1988), 3345.Google Scholar
[J] Jia, Y., Second cohomology of generalized Cartan type H Lie algebras in characteristic 0. J. Algebra 204(1998), 312323.Google Scholar
[K1] Kac, V. G., A description of filtered Lie algebras whose associated graded Lie algebras are of Cartan types. Math. USSR-Izv. 8(1974), 801835.Google Scholar
[K2] Kac, V. G., Classification of infinite-dimensional simple linearly compact Lie superalgebras. Adv. Math. 139(1998), 155.Google Scholar
[K3] Kac, V. G., Infinite Dimensional Lie Algebras. 3rd edition, Cambridge University Press, 1990.Google Scholar
[LW] Li, W. and Wilson, R. L., Central extensions of some Lie algebras. Proc. Amer.Math. Soc. 126(1998), 25692577.Google Scholar
[O] Osborn, J. M., New simple infinite-dimensional Lie algebras of characteristic 0. J. Algebra 185(1996), 820835.Google Scholar
[OZ] Osborn, J. M. and Zhao, K., Generalized Poisson brackets and Lie algebras for type H in characteristic 0. Math. Z. 230(1999), 107143.Google Scholar
[S1] Su, Y., 2-Cocycles on the Lie algebras of generalized differential operators. Comm. Algebra 30(2002), 763782.Google Scholar
[S2] Su, Y., On the low dimensional cohomology of Kac-Moody algebras with coefficients in the complex field. Adv. in Math. (Beijing) 18(1989), 346351.Google Scholar
[S3] Su, Y., 2-Cocycles on the Lie algebras of all differential operators of several indeterminates. (Chinese) Northeast. Math. J. 6(1990), 365368.Google Scholar
[SX] Su, Y. and Xu, X., Central simple Poisson algebras. To appear.Google Scholar
[SXZ] Su, Y., Xu, X. and Zhang, H., Derivation-simple algebras and the structures of Lie algebras of Witt type. J. Algebra 233(2000), 642662.Google Scholar
[SZ] Su, Y. and Zhao, K., Second cohomology group of generalized Witt type Lie algebras and certain representations. Comm. Algebra 30(2002), 32853309.Google Scholar
[X1] Xu, X., Generalizations of Block algebras. Manuscripta Math. 100(1999), 489518.Google Scholar
[X2] Xu, X., New generalized simple Lie algebras of Cartan type over a field with characteristic 0. J. Algebra 224(2000), 2358.Google Scholar
[Z] Zhang, H., The representations of the coordinate ring of the quantum symplectic space. J. Pure Appl. Algebra 150(2000), 95106.Google Scholar
[Zh] Zhao, K., A class of infinite dimensional simple Lie algebras. J. London Math. Soc. (2) 62(2000), 7184.Google Scholar