Linear hamiltonian circle actions that generate minimal Hilbert bases
Annales de l'Institut Fourier, Volume 50 (2000) no. 1, p. 285-315

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.

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.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {1},
     year = {2000},
     pages = {285-315},
     doi = {10.5802/aif.1755},
     zbl = {0973.53067},
     mrnumber = {2001k:53156},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_1_285_0}
}
Egilsson, Ágúst Sverrir. Linear hamiltonian circle actions that generate minimal Hilbert bases. Annales de l'Institut Fourier, Volume 50 (2000) no. 1, pp. 285-315. doi : 10.5802/aif.1755. http://www.numdam.org/item/AIF_2000__50_1_285_0/

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