## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.math.psu.edu/era/.
**

On Embedding the 1:1:2 Resonance Space in a Poisson Manifold
**This journal is archived by the American Mathematical
Society. The master copy is available at
http://www.ams.org/era/
**

## 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*

**Retrieve entire article **
#### 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`

*Electronic Research Announcements of the AMS *Home page