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A note on the algebra of Poisson brackets

Published online by Cambridge University Press:  24 October 2008

C. J. Atkin
C. J. Atkin
Affiliation:
Department of Mathematics, Victoria University of Wellington, New Zealand
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Extract

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 1 , July 1984 , pp. 45 - 60
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

[1] Avez, A., Diaz-Miranda, A. and Lichhnerowicz, A.. Sur l'algèbre des automorphismes infinitésimaux d'une variété symplectique. J. Differential Geom. 9 (1974), 1–40.CrossRefGoogle Scholar
[2] Bayen, F., Flato, M., Fronsdahl, C., Lichnerowicz, A. and Sternheimer, D.. Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111 (1978), 61–110.CrossRefGoogle Scholar
[3] Bayen, F., Flato, M., Fronsdahl, C., A. Lichnerowicz and D. Sternheimer. Deformation theory and quantization. II. Physical applications. Ann. Physics 111 (1978), 111–151.CrossRefGoogle Scholar
[4] Grabowski, J.. Isomorphisms and ideals of the Lie algebras of vector fields. Invent. Math. 50 (1978), 13–33.CrossRefGoogle Scholar
[5] Jacobson, N.. The theory of rings. Amer. Math. Soc. Mathematical Surveys, vol. ii. (Amer.Math. Soc, 1943).CrossRefGoogle Scholar
[6] Lichnerowicz, A.. Sur l'algèbre de Lie des champs de vecteurs. Comment. Math. Helv. 51 (1976), 343–368.CrossRefGoogle Scholar
[7] Lichnerowicz, A.. Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geom. 12 (1977), 253–300.CrossRefGoogle Scholar
[8] Omori, H.. Infinite-dimensional Lie transformation groups. Lecture Notes in Mathematics 427. (Springer 1974).CrossRefGoogle Scholar
[9] Pursell, L. and Shanks, M.. The Lie algebra of a smooth manifold. Proc. Amer. Math. Soc. 5 (1954), 468–472.Google Scholar
[10] Urwin, R.. Lie algebras which determine a symplectic manifold (preprint).Google Scholar
[11] Wojtyński, W.. Automorphisms of the Lie algebra of all real analytic vector fields on a circle are inner. Bull. Acad. Polon. Sci. Sir. Sci. Math., Phys., Astronom. 23 (1975), 1101–1105.Google Scholar
[12] Zariski, O. and Samuel, P.. Commutative Algebra. vol. 1. (Van Nostrand, 1958).Google Scholar