L’espace des orbites d’une action hamiltonienne linéaire du cercle et l’espace des orbites réduit en zéro, sont des exemples d’espaces de Poisson singuliers. L’espace des orbites hérite de l’algèbre de Poisson des fonctions qui sont invariantes pour l’action linéaire du cercle et l’espace des orbites réduit hérite de l’algèbre de Poisson obtenue par restriction à l’espace réduit des fonctions invariantes. Ces espaces vivent dans certaines variétés différentiables qui héritent aussi des quasi–structures de Poisson des variétés de Poisson. Dans cet article nous considérons la question de savoir si on peut trouver des algèbres satisfaisant l’identité de Jacobi parmi ces quasi-structures de Poisson. Nous prouvons que ce n’est pas le cas quand les poids de l’action satisfont à une relation simple. Une conséquence de cette relation est aussi que le nombre des générateurs qui sont nécessaires pour la génération de l’algèbre de fonctions invariantes est minimal.
The orbit space of a linear Hamiltonian circle action and the reduced orbit space, at zero, are examples of singular Poisson spaces. The orbit space inherits the Poisson algebra of functions invariant under the linear circle action and the reduced orbit space inherits the Poisson algebra obtained by restricting the invariant functions to the reduced space. Both spaces reside inside smooth manifolds, which in turn inherit almost Poisson structures from the Poisson varieties. In this paper we consider the question whether among these almost Poisson structures one can find algebras satisfying Jacobi identity. It is shown that this is not the case when the weights of the action satisfy a simple relation. A consequence of this relation is also that the number of generators needed to generate the algebra of invariant functions is minimal.
@article{AIF_2000__50_1_285_0,
author = {Egilsson, \'Ag\'ust Sverrir},
title = {Linear hamiltonian circle actions that generate minimal Hilbert bases},
journal = {Annales de l'Institut Fourier},
volume = {50},
year = {2000},
pages = {285-315},
doi = {10.5802/aif.1755},
mrnumber = {2001k:53156},
zbl = {0973.53067},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2000__50_1_285_0}
}
Egilsson, Ágúst Sverrir. Linear hamiltonian circle actions that generate minimal Hilbert bases. Annales de l'Institut Fourier, Tome 50 (2000) pp. 285-315. doi : 10.5802/aif.1755. http://gdmltest.u-ga.fr/item/AIF_2000__50_1_285_0/
[1] , and A universal reduction procedure for Hamiltonian group actions, in The Geometry of Hamiltonian Systems, Mathematical Sciences Research Institute Publications 22, Springer-Verlag 1991, 33-51. | MR 1123275 | MR 92h:58059 | Zbl 0742.58016
[2] , Proper group actions and symplectic stratified spaces, Pacific Journal of Mathematics, 181, n° 2 (1997), 201-229. | MR 1486529 | MR 98i:58085 | Zbl 0902.58008
[3] On embedding a stratified symplectic space in a smooth Poisson manifold, Ph. D. Thesis 1995, University of California at Berkeley.
[4] On embedding the 1:1:2 resonance space in a Poisson manifold, Electronic Research Announcements of the American Mathematical Society, 1, n° 2 (1995), 48-56. | MR 96j:58063 | Zbl 0849.58029
[5] Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry, 12 (1977), 253-300. | MR 58 #18565 | Zbl 0405.53024
[6] Le théorème de préparation en géométrie différentiable, Séminaire Henri Cartan, 15e année, 1962/1963, exposés 11-13, 22. | Numdam | Zbl 0119.28501
[7] and Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics, 5 (1974), 121-130. | MR 53 #6633 | Zbl 0327.58005
[8] Stability of C∞ mappings: I. The division theorem, Annals of Mathematics, 87 (1968), 89-104. | MR 38 #726 | Zbl 0159.24902
[9] Differentiable invariants, Topology, 16 (1977), 145-155. | MR 55 #9152 | Zbl 0376.58002
[10] Lectures on Hamiltonian systems, Memoirs of the American Mathematical Society, 81 (1968). | MR 37 #6060 | Zbl 0172.11401
[11] Jacobi-type identities for bilinear differential concomitants of certain tensor fields, Indagationes Math., 17 (1955), 390-403. | MR 17,661c | Zbl 0068.15001
[12] On the differential operators of first order in tensor calculus, Convengo di Geometria Differenziale, (1953), 1-7. | Zbl 0059.15301
[13] Smooth functions invariant under the action of a compact Lie group, Topology, 14 (1975), 63-68. | MR 51 #6870 | Zbl 0297.57015
[14] and Geometric Models for Noncommutative Algebras, Berkeley Mathematical Lecture Notes, Volume 10, 1999. | MR 2001m:58013 | Zbl 01515267
[15] and Stratified symplectic spaces and reduction, Annals of Mathematics, 134 (1991), 376-422. | MR 92g:58036 | Zbl 0759.58019
[16] The Classical Groups, Princeton University Press, 1946. | Zbl 1024.20502