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CLASSIFICATION OF NONRESONANT POISSON STRUCTURES

Published online by Cambridge University Press:  01 December 1999

J.-P. DUFOUR  and
M. ZHITOMIRSKII
J.-P. DUFOUR
Affiliation:
Getodim, URA 1407, GDR 144, Mathématiques, Université Montpellier II, Place E. Bataillon, 34095 Montpellier Cedex 05, France
M. ZHITOMIRSKII
Affiliation:
Department of Mathematics, Technion, 32000 Haifa, Israel
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Abstract

Unless otherwise explicitly stated all mappings and tensors in the paper are C∞. A Poisson structure on a (C∞) manifold M is a bracket operation (f, g) [map ] {f, g}, on the set of functions on M, which gives to this set a Lie algebra structure and which verifies the relation

formula here

An equivalent way to get such a structure is to give a 2-vector (that is, an antisymmetric two times contravariant tensor) P satisfying

formula here

where [,] is the Schouten bracket [7]. We then have

formula here

This paper is devoted to the local classification of these structures. The decomposition theorem of A. Weinstein [8, 9] reduces the problem to the case where P is a Poisson structure on Rn which vanishes at zero.

In this paper we will denote by PS(n) the set of germs at zero of Poisson structures on Rn vanishing at the origin.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 60 , Issue 3 , December 1999 , pp. 935 - 950
Copyright
The London Mathematical Society 1999

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