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On Embedding the 1:1:2 Resonance Space in a Poisson Manifold
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On embedding the 1:1:2 resonance space in a Poisson manifold
Ágúst Sverrir Egilsson
Abstract.
The Hamiltonian actions of $\S^{1}$ on the symplectic manifold
$\R^{6}$ 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 $V_{J}$.
Copyright American Mathematical Society 1995
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Article Info
- ERA Amer. Math. Soc. 01 (1995), pp. 48-56
- Publisher Identifier: S 1079-6762(95)02001-4
- 1991 Mathematics Subject Classification. 53.
- Received by the editors May 8, 1995, and, in revised form, June 2, 1995
- Communicated by Frances Kirwan
- Comments (When Available)
Ágúst Sverrir Egilsson
University of Iceland, Department of Mathematics,
101 Reykjavik, Iceland.
E-mail address: egilsson@math.berkeley.edu
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