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On embedding the $1:1:2$ resonance space in a Poisson manifold


Author: Ágúst Sverrir Egilsson
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 48-56
MSC (1991): Primary 53
DOI: https://doi.org/10.1090/S1079-6762-95-02001-4
MathSciNet review: 1350680
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Abstract: The Hamiltonian actions of $\S^{1}$ on the symplectic manifold ${\Bbb R}^{6}$ in the $1:1:-2$ and $1:1:2$ resonances are studied. Associated to each action is a Hilbert basis of polynomials defining an embedding of the orbit space into a Euclidean space $V$ and of the reduced orbit space $J^{-1}(0)/\S^{1}$ into a hyperplane $V_{J}$ of $V$, where $J$ is the quadratic momentum map for the action. The orbit space and the reduced orbit space are singular Poisson spaces with smooth structures determined by the invariant functions. It is shown that the Poisson structure on the orbit space, for both the $1:1:2$ and the $1:1:-2$ resonance, cannot be extended to $V$, and that the Poisson structure on the reduced orbit space $J^{-1}(0)/\S^{1}$ for the $1:1:-2$ resonance cannot be extended to the hyperplane $V_{J}$.


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Ágúst Sverrir Egilsson
Email: egilsson@math.berkeley.edu

DOI: https://doi.org/10.1090/S1079-6762-95-02001-4
Received by editor(s): May 8, 1995
Received by editor(s) in revised form: June 2, 1995
Article copyright: © Copyright 1995 American Mathematical Society