ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1, Issue 2, 1995

ON EMBEDDING THE 1 : 1 : 2 RESONANCE SPACE IN A POISSON MANIFOLD

ÁGÚST SVERRIR EGILSSON

(Communicated by Frances Kirwan)

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}.

1. Introduction

In this paper we study certain singular Poisson spaces arising from Hamiltonian actions of Lie groups on symplectic manifolds. The singular Poisson spaces are embedded, using Hilbert maps, into Euclidean spaces and we prove that the singular Poisson structure cannot be extended to the Euclidean spaces. This disproves a conjecture raised by Cushman and Weinstein [12, 13].

For a Hamiltonian action of a Lie group G on a symplectic manifold (M, ω) with an equivariant momentum map J, Marsden and Weinstein define in [5] the reduced orbit space M_{μ}, for a value μ in the dual of the Lie algebra of G. The space M_{μ} is the quotient space J^{−1}(μ)/G_{μ} where G_{μ} is the isotropy group of μ with respect to the coadjoint action of G. For weakly regular values μ of J, if G_{μ} acts freely and properly on the manifold J^{−1}(μ), M_{μ} is a manifold and there is a unique symplectic structure on M_{μ} which lifts to i_{μ}^{∗}ω where i_{μ} is the inclusion map i_{μ} : J^{−1}(μ) → M.

The space M/G is assigned a smooth structure C^{∞}(M/G) by the G-invariant functions on M and M/G inherits a Poisson bracket from M making M/G a Poisson variety. Even when μ ∈ J(M) is not a weakly regular value the reduced orbit space M_{μ} has a smooth structure defined by restricting functions in C^{∞}(M/G) to J^{−1}(μ). It is proved in [1] that the space M_{μ} inherits, by restriction, the structure of a Poisson variety from M/G. For a compact Lie group G Sjamaar and Lerman show in [12] that the reduced orbit space M_{0} is a union of symplectic manifolds and moreover a stratified symplectic space.

If the Lie group G is compact and the action on M has only finitely many orbit types then by a theorem of Schwarz [11] there are functions f_{1}, …, f_{n} in

\begin{matrix}c \\ ◯ \end{matrix}1995 American Mathematical Society

Received by the editors May 8, 1995, and, in revised form, June 2, 1995.

1991 Mathematics Subject Classification. 53.

C^{∞}(M/G) such that F^{∗} C^{∞}(ℝ^{n}) = C^{∞}(M/G) for F = (f_{1}, …, f_{n}). This defines an embedding M/G → ℝ^{n} and in particular an embedding M_{μ} → ℝ^{n}. If a compact Lie group G acts orthogonally on a Euclidean space M then we can choose the invariants f_{1}, …, f_{n} to be generators of the algebra of invariant polynomials on M, in which case the embedding F is called a Hilbert map for the action and f_{1}, …, f_{n} are said to form a Hilbert basis for the algebra. It is known for many simple cases, see for example [12], that the Poisson structure on the image of M/G can be extended to all of ℝ^{n}. We will, however, outline a proof showing that for the 1 : 1 : −2 resonance action and a Hilbert map corresponding to a minimal homogeneous Hilbert basis, described below, defining the embedding M/G → ℝ^{11} there exists no smooth Poisson structure on ℝ^{11} extending the Poisson structure on the orbit space. Furthermore we show that the Poisson structure on the singular variety M_{0} cannot be extended to the hyperplane in ℝ^{11} induced by the momentum map for the action. The nonexistence of Poisson structures on ℝ^{11} extending the Poisson structure on the orbit space of the 1 : 1 : 2 resonance action is then established using results already obtained for the 1 : 1 : −2 resonance.

2. The 1 : 1 : −2 resonance

2.1. Preliminaries. Consider the space ℝ^{6} as ℂ^{3} and define an S^{1} action

⋆ : S^{1} × ℝ^{6} → ℝ^{6} by z ⋆ (u_{1}, u_{2}, u_{3}) = (zu_{1}, zu_{2}, z^{−2}u_{3})

where the coordinates on ℝ^{6} and ℂ^{3} are related by u_{i} = x_{i} + ıy_{i} for i = 1, 2, 3. Also define coordinates v_{i} = x_{i} − ıy_{i} on ℂ^{3} for i = 1, 2, 3, see [8].

The existence of a Hilbert basis for the invariant polynomials follows from a theorem of Hilbert [14], but for the action above it is easily established directly.

A minimal homogeneous generating set for the invariant polynomials in

ℝ[x, y] = ℝ[x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}]

is given by the polynomials f_{1}, …, f_{11}, i.e.,

ℝ[x, y]^{S^{1}} = ℝ[f_{1}, …, f_{11}],

where

f_{1} = x_{1}^{2} + y_{1}^{2}, f_{2} = x_{2}^{2} + y_{2}^{2}, f_{3} = x_{3}^{2} + y_{3}^{2},
f_{4} + ıf_{5} = u_{1}v_{2}, f_{6} + ıf_{7} = u_{1}^{2}u_{3}, f_{8} + ıf_{9} = u_{2}^{2}u_{3} and
f_{10} + ıf_{11} = u_{1}u_{2}u_{3}.

A Hilbert map F : ℝ^{6} → ℝ^{11} corresponding to the Hilbert basis f_{1}, …, f_{11} is given by F = (f_{1}, …, f_{11}). We will refer to this particular choice of a Hilbert map for the 1 : 1 : −2 action as the Hilbert map for the 1 : 1 : −2 action in standard form.

2.2. Complex coefficients. On ℝ^{6} the standard Poisson bivector field ϱ is given by

ϱ = \frac{∂}{∂x_{1}} ∧ \frac{∂}{∂y_{1}} + \frac{∂}{∂x_{2}} ∧ \frac{∂}{∂y_{2}} + \frac{∂}{∂x_{3}} ∧ \frac{∂}{∂y_{3}} .

Denote by M = ℝ^{6} and by V = ℝ^{11} the spaces with linear coordinates

{x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}} and {l_{1}, l_{2}, l_{3}, R_{1}, I_{1}, …, R_{4}, I_{4}}

respectively and in which F : M → V is given by the formula F = (f_{1}, …, f_{11}). Now let M_{c} and V_{c} be real vector spaces with linear coordinates

{z_{1}, w_{1}, z_{2}, w_{2}, z_{3}, w_{3}} and {l_{1}, l_{2}, l_{3}, Z_{1}, W_{1}, …, Z_{4}, W_{4}}

respectively.

For the Euclidean space E = ℝ^{n} denote by X^{∗}(E) the C^{∞}-multivector fields on E, by X^{∗}[E] the multivector fields on E with polynomial coefficients and by X^{∗}[[E]] the multivector fields on E with formal coefficients.

Let T be the Taylor series operator

T : X^{∗}(V) → X^{∗}[[V]]

replacing each coefficient with its Taylor series at 0.

Define a ℂ-algebra isomorphism

κ : ℂ ⊗ X^{∗}[V] → ℂ ⊗ X^{∗}[V_{c}]

by the formulas l_{s} ↦ l_{s}, R_{t} ↦ \frac{1}{2}(Z_{t} + W_{t}) and I_{t} ↦ \frac{1}{2ı}(Z_{t} − W_{t}),

\frac{∂}{∂l_{s}} ↦ \frac{∂}{∂l_{s}} , \frac{∂}{∂R_{t}} ↦ \frac{∂}{∂Z_{t}} + \frac{∂}{∂W_{t}} and \frac{∂}{∂I_{t}} ↦ ı(\frac{∂}{∂Z_{t}} − \frac{∂}{∂W_{t}})

and similarly define a ℂ-algebra isomorphism

κ′ : ℂ ⊗ X^{∗}[M] → ℂ ⊗ X^{∗}[M_{c}]

by the formulas x_{t} ↦ \frac{1}{2}(z_{t} + w_{t}) and y_{t} ↦ \frac{1}{2ı}(z_{t} − w_{t}),

\frac{∂}{∂x_{t}} ↦ \frac{∂}{∂z_{t}} + \frac{∂}{∂w_{t}} and \frac{∂}{∂y_{t}} ↦ ı(\frac{∂}{∂z_{t}} − \frac{∂}{∂w_{t}}).

Let

τ : ℝ[V^{∗}] → ℝ[M^{∗}]

be the algebra morphism given by

τ(g) = g ∘ F

and let τ_{c} be the ℂ-algebra morphism induced by the diagram

ℝ[V^{∗}] ⟶^{τ} ℝ[M^{∗}]
↓κ  ↓κ′
ℂ[V_{c}^{∗}] ⟶^{τ_{c}} ℂ[M_{c}^{∗}]

and use τ_{c} to define the complex counterpart of the Hilbert map

F_{c} : ℂ ⊗ M_{c} → ℂ ⊗ V_{c}

with complex polynomial coordinate functions defined by

τ_{c}(p) = p ∘ F_{c},

furthermore let F_{c∗} be the ℂ-linear derivative of F_{c}.

There is a one-to-one relationship between bivector fields π ∈ X^{2}[V] which are F -related to ϱ

F_{∗} ∘ ϱ = π ∘ F

and bivector fields π_{c} ∈ ℂ ⊗ X^{2}[V_{c}] which are F_{c}-related to ϱ_{c}

F_{c∗} ∘ ϱ_{c} = π_{c} ∘ F_{c}

and satisfy κ^{−1}(π_{c}) ∈ X^{2}[V]. The relationship is given by π = κ^{−1}(π_{c}).

Extend the Schouten-Nijenhuis [9, 10] bracket [[[,]]] by ℂ-bilinearity to the spaces ℂ ⊗ X^{∗}[V] and ℂ ⊗ X^{∗}[V_{c}], then the identity

[[[κ(X), κ(Y)]]] = κ([[[X, Y]]]) for X, Y ∈ X^{∗}[V]

follows easily.

Let

ϱ_{c} = κ′(ϱ) = −2ı ∑_{n} \frac{∂}{∂z_{n}} ∧ \frac{∂}{∂w_{n}}

then the commutative diagram below

\begin{matrix}(ℝ[V^{∗}], π) & ⟶^{τ} & (ℝ[M^{∗}], ϱ) \\ ↓κ & & ↓κ′ \\ (ℂ[V_{c}^{∗}], π_{c}) & ⟶^{τ_{c}} & (ℂ[M_{c}^{∗}], ϱ_{c}) \end{matrix}  (1)

demonstrates the setting for finding a bivector field π ∈ X^{2}[V] which is F -related to ϱ.

For the above diagram we have

[[[π, π]]] = 0 if and only if [[[π_{c}, π_{c}]]] = 0

or equivalently [3, 9] that π ∈ X^{2}[V] is a Poisson bivector field if and only if π_{c} = κ(π) is a Poisson bivector field.

2.3. Embeddings. The 1 : 1 : −2 resonance action above has momentum map J given by the quadratic polynomial

J = \frac{1}{2}(x_{1}^{2} + y_{1}^{2} + x_{2}^{2} + y_{2}^{2} − 2x_{3}^{2} − 2y_{3}^{2}).

There exists a unique functional J in V^{∗} satisfying

J = J ∘ F

and F maps the space J^{−1}(0) into the hyperplane

V_{J} = J^{−1}(0) ⊂ V.

The smooth structure on the reduced orbit space J^{−1}(0)/S^{1}, obtained, see [1], by restricting the invariant functions on M to J^{−1}(0), is described by

C^{∞}(J^{−1}(0)/S^{1}) = (η ∘ F)^{∗} C^{∞}(V_{J})|_{J^{−1}(0)}

where η is the projection

η : V → V_{J}

onto V_{J} along the vector J^{∗} = \frac{1}{2}(l_{1}^{∗} + l_{2}^{∗} − 2l_{3}^{∗}).

Extending the Poisson structure on M/S^{1} to all of V is equivalent to finding a bivector field ζ in X^{2}(V) satisfying the conditions 1a) [[[ζ, ζ]]] = 0 and

1b) ζ ∘ F = F_{∗} ∘ ϱ.

Extending the Poisson structure on J^{−1}(0)/S^{1} to all of V_{J} is equivalent to finding a bivector field ξ′ in X^{2}(V_{J}) satisfying 2a) [[[ξ′, ξ′]]] = 0 and

2b) ξ′ ∘ (η ∘ F)|_{J^{−1}(0)} = (η ∘ F)_{∗} ∘ ϱ|_{J^{−1}(0)}.

Further a Poisson bivector ξ′ in X^{2}(V_{J}), i.e., [[[ξ′, ξ′]]] = 0, extending the structure on J^{−1}(0)/S^{1} can be extended to a Poisson bivector ξ on all of V satisfying 3a) ξ ∘ ι = ι_{∗} ∘ ξ′,

3b) ξ ∘ F |_{J^{−1}(0)} = F_{∗} ∘ ϱ|_{J^{−1}(0)},

3c) [[[ξ, J]]] = 0, i.e., J is Casimir, and where ι is the inclusion map V_{J} → V.

The above Poisson bivector field ξ is obtained from ξ′ by defining ξ to be constant along the fibres of the projection η.

Theorem 1. Let F : M → V be the Hilbert map in standard form for the 1 : 1 : −2 resonance action ⋆ : S^{1} ×M → M. Then the Poisson structure on the singular orbit space M/S^{1}, embedded into V by F, cannot be extended to V. Furthermore the Poisson structure on the singular reduced orbit space J^{−1}(0)/S^{1} cannot be extended to V_{J}.

Before proving the above theorem we establish the lemmas below.

Choose a bivector field π in X^{2}[V] such that

π ∘ F = F_{∗} ∘ ϱ,  (2)

e.g. use diagram (1); the existence of a bivector field satisfying (2) follows from the definition of a Hilbert basis.

We can further assume that

[[[π, J]]] = 0

and using the natural grading in X^{2}[V] write

π = π^{1} + π^{2+}

where π^{1} has coefficients of degree 1 and π^{2+} has coefficients of degrees 2 and higher.

Then π^{1} is uniquely determined by (2) and is already a Poisson bivector,

[[[π^{1}, π^{1}]]] = 0.

Using the Poisson bivector π^{1} we define the π^{1}-cohomology coboundary operator, see [3], on X^{∗}(V) by

δ = −[[[π^{1}, ·]]].

Also define π_{c}, π_{c}^{1} and π_{c}^{2+} in ℂ ⊗ X^{2}[V_{c}] as the images of π, π^{1} and π^{2+} under κ respectively, and define δ_{c} to be the coboundary operator on ℂ ⊗ X^{∗}(V_{c}) given by

δ_{c} = −[[[π_{c}^{1}, ·]]].

Furthermore define Γ to be the ideal in ℂ[V_{c}^{∗}] generated by the terms

J , l_{3}, Z_{1}, W_{1}, Z_{2}, W_{2}, Z_{3}, W_{3}, Z_{4}, W_{4}^{2}

and all the monomials of degree 3 and higher.

For X ∈ ℂ ⊗ X^{3}[V_{c}] denote the coefficient of \frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{2}} ∧ \frac{∂}{∂W_{4}} in X by X_{ϕ}, i.e.,

X_{ϕ} = X(dZ_{2} ∧ dW_{2} ∧ dW_{4}),

and for Y ∈ ℂ ⊗ X^{2}[V_{c}] define Ỹ by

Ỹ = Y_{Z_{2}W_{2}} \frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{2}} + Y_{Z_{2}W_{4}} \frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{4}} + Y_{W_{2}W_{4}} \frac{∂}{∂W_{2}} ∧ \frac{∂}{∂W_{4}}

where Y_{Z_{2}W_{2}} is the coefficient of \frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{2}} in Y.

Lemma 1. Let ϵ_{c} be a bivector in ℂ ⊗ X^{2}[V_{c}]. Then δ_{c}(ϵ_{c})_{ϕ} ≡ −[[[φ_{c}, ϵ̃_{c}]]]_{ϕ} (mod Γ) where φ_{c} is the bivector field given by the formula

\begin{aligned}φ_{c} & = −2ı(W_{4} \frac{∂}{∂l_{1}} ∧ \frac{∂}{∂W_{4}} + W_{4} \frac{∂}{∂l_{2}} ∧ \frac{∂}{∂W_{4}} + W_{4} \frac{∂}{∂l_{3}} ∧ \frac{∂}{∂W_{4}} \\ & +(l_{2} − l_{1})\frac{∂}{∂Z_{1}} ∧ \frac{∂}{∂W_{1}} + 2W_{4} \frac{∂}{∂Z_{1}} ∧ \frac{∂}{∂W_{2}} + 2W_{4} \frac{∂}{∂W_{1}} ∧ \frac{∂}{∂W_{3}} ). \end{aligned}

Proof: All the coefficients of the basis vectors \frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{2}} , \frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{4}} and \frac{∂}{∂W_{2}} ∧ \frac{∂}{∂W_{4}} in π_{c}^{1} and φ_{c} are zero and the bivector field π_{c}^{1} − φ_{c} is in the ideal generated by Γ·X^{2}[V_{c}]. From this and the definition of the Schouten-Nijenhuis bracket it follows that δ_{c}(ϵ_{c})_{ϕ} ≡ −[[[φ_{c}, ϵ̃_{c}]]]_{ϕ} (mod Γ). □

One can choose π_{c}^{2+} to be homogeneous of degree 2 and an example of π̃_{c}^{2+} is given by

π̃_{c}^{2+} = −2ı((l_{1}^{2} + 4l_{1}l_{3})\frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{2}} + (l_{1}Z_{1} + 2l_{3}Z_{1})\frac{∂}{∂Z_{2}} ∧ \frac{∂}{∂W_{4}} ).

Call an element ρ in X^{∗}[V] a relation if ρ ∘ F = 0 and define

Φ^{∗}[V] = {ρ′ ∈ X^{∗}[V] : ρ′ ∘ F = 0}.

Lemma 2. Let ζ ∈ X^{2}[V] be a bivector of the form

ζ = π + ρ + J γ

for a relation ρ in Φ^{2}[V], a bivector γ ∈ X^{2}[V] and such that at least one of the two conditions below are satisfied

i) [[[ζ, J]]] contains no terms of degree 2, ii) γ contains no constant terms.

Then [[[ζ, ζ]]] has nonzero terms of degree 2.

Proof: Let ζ_{c} = κ(ζ). Using Lemma 1 we calculate that an ϵ in Φ^{2}[V] + J X^{2}[V] satisfies the formula δ_{c}(κ(ϵ))_{ϕ} ≡ 0 (mod Γ). Furthermore the formula [[[π_{c}^{1}, π_{c}^{2+}]]]_{ϕ} ≢ 0 (mod Γ) holds. Now we conclude that if either of conditions i) or ii) holds then

[[[ζ_{c}, ζ_{c}]]]_{ϕ} ≡ 2[[[π_{c}^{1}, π_{c}^{2+}]]]_{ϕ} (mod Γ)

and the lemma follows. □

Lemma 3. Let Δ ∈ X^{∗}(V) satisfy Δ ∘ F |_{J^{−1}(0)} = 0. Then there exists a formal multivector field γ ∈ X^{∗}[[V]] such that T (Δ) − J γ is a formal relation, i.e.,

T (Δ) − J · γ ∈ Φ^{∗}[[V]] = {ρ ∈ X^{∗}[[V]] : ρ ∘ F = 0}.

This lemma is proved by using the Malgrange-Mather division theorem [4, 6].□

Outline of the proof of Theorem 1: Let ζ ∈ X^{2}(V) be a bivector field such that ζ and ϱ are F -related, i.e., ζ ∘ F = F_{∗} ∘ ϱ.

Now write T (ζ) − π = ρ′ + ρ′^{3+} where ρ′ ∈ Φ^{2}[V] has polynomial coefficients and such that ρ′^{3+} ∈ Φ^{2}[[V]] contains only terms of degrees greater than two.

Now since π + ρ′ satisfies condition ii) of Lemma 2, deduce that the Taylor series of [[[ζ, ζ]]] has nonzero terms of degree 2. Hence ζ is not Poisson.

Assume now that there exists a Poisson bivector field ξ′ in X^{2}(V_{J}) extending the Poisson structure of J^{−1}(0)/S^{1} to all of V_{J}.

By 3a, 3b and 3c we extend ξ′ to a Poisson bivector field ξ on V in such a way that ξ extends the Poisson structure of J^{−1}(0)/S^{1} in V and has J for a Casimir function.

Use Lemma 3 to write

T (ξ) = π + ρ + J γ + ρ^{3+} + J γ^{2+}

where as before ρ is in Φ^{2}[V] and ρ^{3+} is in Φ^{2}[[V]] and has only coefficients of degrees greater than two, γ is in X^{2}[V] and γ^{2+} is a formal bivector field with coefficients of degrees greater than one. Conclude, since π+ρ+J γ satisfies condition i) of Lemma 2, that the Taylor series of [[[ξ, ξ]]] has nonzero terms of degree 2. Thus contradicting the existence of the Poisson bivector field ξ′.□

3. The 1 : 1 : 2 resonance

3.1. Preliminaries. To analyze the relationship between the n_{1} : ⋯: n_{k} resonance and the σ_{1}n_{1} : ⋯: σ_{k}n_{k} resonance, σ_{i} ∈ {1, −1} for i = 1, …, k, we define an automorphism, see [2], ψ^{σ} on the space ℂ ⊗ X^{∗}[V]. Here we will define ψ^{σ} for the 1 : 1 : −2 resonance and the σ = (1, 1, −1) case.

For the 1 : 1 : 2 resonance the Hilbert map F^{σ} : M → V in standard form, by definition here, is obtained by interchanging u_{3} and v_{3} in the definitions of the invariants f_{1}, …, f_{11}. Let σ = (σ_{1}, σ_{2}, σ_{3}) = (1, 1, −1) and define

ω^{σ} : {l_{1}, l_{2}, l_{3}, Z_{1}, W_{1}, …, Z_{4}, W_{4}} → {1, −1, ı, −ı}

by the formula

ω_{γ}^{σ} = ı^{deg_{z_{3}} τ_{c}(γ)} · ı^{deg_{w_{3}} τ_{c}(γ)}.

Now, using ω^{σ}, define the automorphism

ψ_{c}^{σ} : ℂ ⊗ X^{∗}[V_{c}] → ℂ ⊗ X^{∗}[V_{c}]

by the formulas l_{s} ↦ ω_{l_{s}}^{σ}l_{s}, Z_{t} ↦ ω_{Z_{t}}^{σ}Z_{t} and W_{t} ↦ ω_{W_{t}}^{σ}W_{t},

\frac{∂}{∂l_{s}} ↦ _{\frac{1}{ω_{ls}^{σ}}} \frac{∂}{∂l_{s}} , \frac{∂}{∂Z_{t}} ↦ _{\frac{1}{ω_{Z_{t}}^{σ}}} \frac{∂}{∂Z_{t}} and \frac{∂}{∂W_{t}} ↦ _{\frac{1}{ω_{W_{t}}^{σ}}} \frac{∂}{∂W_{t}}

and also define

ψ^{σ} : ℂ ⊗ X^{∗}[V] → ℂ ⊗ X^{∗}[V] by ψ^{σ} = κ^{−1} ∘ ψ_{c}^{σ} ∘ κ.

The map ψ^{σ} is a Schouten-Nijenhuis morphism defining a cochain map between the complexes

(ℂ ⊗ X^{∗}[V], δ) ⟶^{ψ^{σ}} (ℂ ⊗ X^{∗}[V], δ^{σ})

where δ = −[[[π^{1}, ·]]] and δ^{σ} = −[[[ψ^{σ}(π^{1}), ·]]].

Lemma 4. Let π′ be a bivector field in X^{2}[V]. The bivectors π′ and ϱ are F -related if and only if the bivectors ψ^{σ}(π′) and ϱ are F^{σ} related.

Proof: [2].□

Thus ψ^{σ} allows us to apply results for the 1 : 1 : −2 resonance to the 1 : 1 : 2 resonance case.

3.2. Embeddings.

Theorem 2. Let F^{σ} : M → V be the Hilbert map in standard form for the 1 : 1 : 2 resonance action ⋆ : S^{1} ×M → M. Then the Poisson structure on the singular orbit space M/S^{1}, embedded into V by F^{σ}, cannot be extended to V.

Proof: Follows from applying the morphism ψ^{σ} to results for the 1 : 1 : −2 resonance. □

4. Generalizations

In order to analyze which circle actions induce Poisson embeddings of the orbit spaces one should study the semigroup structures obtained from the invariant polynomials. The semigroup structure arises from complexifying the phase space and the ambient Euclidean space. A proper framework for Poisson structures and maps for these semigroups would be helpful in resolving the general case. The author is currently working in this direction.

5. Acknowledgements

I would like to thank my advisor, Alan Weinstein, for sharing his ideas on the subject at hand. I would also like to thank Alexander Givental for many discussions related to the content of this paper and Vinay Kathotia and the referee for many helpful comments. Much of the work presented here was formulated during my visit to the CENTRE International de Recherche EMILE BOREL in Paris for their symplectic geometry semester in 1994. I would like to take this opportunity to thank them for their hospitality.

Moreover I would like to thank IPC in Iceland for their generous financial support. The research described here was partially supported by a grant from the Thor Thors fund and by DOE grant DE-FG03-93ER25177.

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E-mail address: egilsson@math.berkeley.edu

University of Iceland, Department of Mathematics, 101 Reykjavik, Iceland.