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On Embedding Singular Poisson Spaces

This dissertation investigates the problem of locally embedding singular Poisson spaces. Specifically, it seeks to understand when a singular symplectic quotient V/G of a symplectic vector space V by a group G \subseteq Sp_2n(R) is realizable as a Poisson subspace of some Poisson manifold (R^n,{.,.}). The local embedding problem is recast in the language of schemes and reinterpreted as a problem of extending the Poisson bracket to infinitesimal neighborhoods of an embedded singular space. Such extensions of a Poisson bracket near a singular point p of V/G are then related to the cohomology and representation theory of the cotangent Lie algebra at p. Using this framework, it is shown that the real 4-dimensional quotient V/\ZZ_n (n odd) is not realizable as a Poisson subspace of any (R^{2n+6},{.,.}), even though the underlying variety algebraically embeds into R^{2n+6}. The proof of this nonembedding result hinges on a refinement of the Levi decomposition for Poisson manifolds to partially linearize any extension with respect to the Levi decomposition of the cotangent Lie algebra of V/G at the origin. Moreover, in the case n=3, this nonembedding result is complemented by a concrete realization of V/\ZZ_3 as a Poisson subspace of R^78.


Introduction
This dissertation studies the local Poisson geometry of singular symplectic quotients. It is particularly concerned with the problem of realizing such a quotient as an embedded Poisson subspace of some Poisson structure on R d . Such an embedding then realizes the quotient as a union of singular leaves of the symplectic foliation of the Poisson structure on R d .
The main results of this dissertation concern the singular symplectic quotient V /Z n of a 4dimensional symplectic real vector space V by the linear action of a cyclic group Z n ⊆ Sp 4 (R) of odd order. First, the Poisson structures on these quotients are shown to exhibit an obstruction to extension: Theorem 1.0.1. Let n be an odd integer greater than 2. The symplectic quotient V /Z n embeds into R 2n+6 , yet is not realizable as a Poisson subspace of R 2n+6 for any Poisson structure on R 2n+6 . Remark 1.0.2. The requirement that n be odd is likely unnecessary; the proof of the theorem requires this assumption at only one step.
This result provides further evidence for such rigidity of Poisson structures previously established by Egilsson [6] and Davis [2] (see Example 4.2.1).
Second, a procedure is given that constructs an explicit realization of V /Z 3 as a Poisson subspace of some R N . More specifically, Theorem 1.0.3. There is a Poisson structure on R 78 and an embedding V /Z 3 ֒→ R 78 that realizes V /Z 3 as a Poisson subspace of R 78 .
In conjunction, these two theorems paint a picture of a singular symplectic space that (1) smoothly embeds in R 12 , (2) whose Poisson structure cannot be extended to a Poisson structure on R 12 , yet (3) can be extended to a Poisson structure on R 78 . It is unclear why this dimensional obstruction disappears between 13 and 78. However, it seems to be a local property: in all known examples, the obstruction to embedding such singular quotients appears when attempting to infinitesimally extend the Poisson structure at a zero-dimensional symplectic leaf (a singular point with respect to both the smooth and Poisson structures).
While the motivating examples and corresponding results of this dissertation lie in the category of smooth manifolds, the techniques developed to handle them are algebraic in nature. Why would one answer questions about quotients of smooth symplectic manifolds by studying algebraic/formal Poisson structures? The more general category of Poisson structures is better equipped for studying such quotients for a number of reasons: 1. The symplectic quotients considered have isolated singularities. While a symplectic structure is defined in terms of a smooth structure on the space, Poisson structures have no such requirement for smoothness.
2. Every smooth Poisson structure has a (possibly singular) symplectic foliation. Thus, one may attempt to "desingularize" a symplectic quotient by realizing it as the union of (a finite number of) symplectic leaves of a smooth Poisson manifold. The "desingularization" is then a nearby, smooth symplectic leaf.
Remark 1.0.4. There is another notion of the desingularization of singular symplectic quotients, dual to the one above, called the "symplectic resolution of singularities." This approach to understanding symplectic singularities allows one to work exclusively with symplectic structures (though there is still value in considering their corresponding Poisson geometry -see [14]).
Our method for either finding, or proving the nonexistence of, a Poisson embedding is roughly as follows (see Chapter 4): Given a singular symplectic quotient X = V /G and an embedding X ֒→ K d for K = R or C, we attempt to extend the Poisson structure on X locally near the zero-dimensional symplectic leaf While the characterization above describes a (local) embedding of algebraic varieties V /G ֒→ K d , it is perhaps better to interpret it in the language of schemes. From this perspective, solving the degree k condition amounts to extending the Poisson bracket on X to a Poisson bracket on the k th infinitesimal neighborhood of X (which is not a variety, but is a scheme). The condition for each such extension is naturally expressed as the solution of a Maurer-Cartan type equation on the cotangent Lie algebra.
When working with examples (a large part of this dissertation) we will take advantage of the computational power of the first point of view, then reflect on the procedure and the result in the language of schemes.

Outline Chapter 2
Chapter 2 contains an introduction to the basic structures of Poisson geometry from an algebraic point of view -focusing on Poisson structures in the category of schemes. Following [14], we introduce singular symplectic structures and discuss the extent to which local properties of their Poisson structure carry over from the smooth case to the algebraic case. Lastly, using these concepts, we examine Poisson brackets on infinitesimal neighborhoods of a Poisson subscheme that will be used later as the framework in which one "infinitesimally extends" the Poisson bracket on a singular space.

Chapter 3
Chapter 3 describes the connection between the usual differential geometric description of a Poisson structure and the scheme theoretic development in Chapter 2. It begins by reviewing (following [5]) the correspondence, on smooth spaces, between Poisson brackets and bivector fields. Moreover, the definition of such bivector fields is extended to infinitesimal neighborhoods and related to the homogeneous decomposition of a bivector field.

Chapter 4
Chapter 4 serves an an introduction to the problem of embedding a singular symplectic space. The first section explains the problem and the extent to which the smooth embedding problem reduces to a algebraic/formal Poisson embedding problem. The second section consists of many examples -both of concrete embeddings and nonembedding results of singular Poisson spaces. The last section sketches a first approach to constructing an embedding: the Zariski cotangent space at the singular point has a natural Lie algebra structure, called the cotangent Lie algebra, that defines a Poisson-embedding up to first order. Any Poisson structure extending the given one is a deformation of the dual of this Lie algebra.

Chapter 5
Chapter 5 summarizes (following [5]) the linearization problem in Poisson geometry. In particular, the Levi decomposition for Poisson manifolds determines to what extent a potential extension of the Poisson structure on the quotient will be isomorphic to the dual of its cotangent Lie algebra. This semi-linearization is key to proving Theorem 1.0.1.

Chapter 6
Chapter 6 contains the proof of Theorem 1.0.1. Using the Levi decomposition for Poisson structures, the proof simplifies the "infinitesimal extension" conditions formulated in Chapter 4 to reveal an obstruction to extension in the (n − 1) th -infinitesimal neighborhood of the singular point.

Chapter 7
Chapter 7 reasons through a series of reductions in the "infinitesimal extension" conditions formulated in Chapter 4 to construct a concrete extension of the Poisson structure on V /Z 3 , for a 4-dimensional real vector space V (i.e. Theorem 1.0.3).

Appendix
The Appendix contains two parts: (1) a section reviewing the representation theory of sl 2 (C) and (2) a section explaining the Macaulay2 code used to construct the extension given in Chapter 7.
• A smooth manifold (M, C ∞ (M)) is a Poisson manifold if C ∞ (M) is a Poisson algebra.
• A complex manifold M is a complex-analytic Poisson manifold if the sheaf of complexanalytic functions is a sheaf of Poisson algebras.
• A formal scheme (X,Ô X ) is a formal Poisson scheme ifÔ X is a sheaf of Poisson algebras.
• An algebraic variety X is a Poisson variety if the sheaf of regular functions is a sheaf of Poisson algebras.
The only Poisson schemes considered in this thesis are affine Poisson schemes. Given a Poisson algebra A, Proposition 2.1.8 implies the affine scheme Spec(A) is in fact a Poisson scheme. Example 2.2.4. 2n-dimensional affine space A 2n k := Spec(K[x 1 , y 1 , . . . , x n , y n ]) is a Poisson scheme with the sheaf of Poisson algebras defined in Example 2.1.5. This is the standard symplectic structure on A 2n k . Example 2.2.5. Given a Lie algebra g over K, the Poisson bracket in Example 2.1.5 makes the dual space g * a Poisson scheme.
The remainder of the material from the previous algebraic section easily translates to their corresponding geometric properties: Definition 2.2.6. Let X be a Poisson scheme. A Poisson scheme Y is a Poisson subscheme of X if For smooth spaces, the above definition provides the link between Poisson structures and the symplectic structures defined below: Definition 2.2.14.
[Smooth symplectic structures] On a smooth scheme X, a closed, nondegenerate 2-form ω ∈ Ω 2 X is a smooth symplectic structure on X.
Remark 2.2.15. We append the adjective "smooth" to differentiate between the usual notion of a symplectic structure in manifold theory and the singular symplectic varieties considered in [14] and below.
There is a one-to-one correspondence between non-degenerate smooth Poisson structures and smooth symplectic structures given by the bundle isomorphism ♯ : Ω X → T X . Starting with a non-degenerate Poisson structure, define a 2-form ω x (u, v) =< ♯ −1 (u), v > where <, >: Ω X × T X → k is the usual non-degenerate pairing.

on A 2n
k corresponds to the constant 2-form ω = dx 1 ∧ dy 1 + . . . + dx n ∧ dy n Following [14], this correspondence between non-degenerate Poisson spaces and symplectic spaces may be used to extend the definition of a symplectic structure to a possibly singular algebraic variety. Let (X, {·, ·}) be an integral, normal, Poisson scheme of finite type over K (that is, X is a normal algebraic variety). Moreover, denote the maximal, open smooth subset of X by X sm . Remark 2.2. 18. The requirement that the variety be normal is motivated by the fact (see [11]) that normal varieties are the most general spaces for which an algebraic version of Hartog's theorem holds. This condition ensures that {·, ·} is determined by its restriction to the smooth locus X sm .
By [14], Definition 2.1, such a bracket is equivalent to a non-degenerate, closed 2-form on X sm that extends to a possibly degenerate 2-form on some resolution of singularities of X. If X is smooth, this clearly coincides with the definition of a smooth symplectic space.
Example 2.2. 19. Let X = A 2n k with the standard Poisson structure, and let G ⊆ Sp 2n (K) be a finite algebraic group acting linearly on X by Poisson isomorphisms. In the Chapter 4, it will be shown that the quotient X/G = Spec(O X (X) G ) is a singular symplectic variety.

Decompositions of Poisson structures
In the case of real manifolds, Poisson structures are built from non-degenerate, symplectic pieces. Weinstein proved [25] that any Poisson manifold is foliated by symplectic leaves: Theorem 2.2.20 (Weinstein's splitting theorem [25]). Let (M, {·, ·}) be a smooth Poisson manifold, and p ∈ M. Then there is a system of coordinates (x 1 , y 1 , . . . , x r , y r , z 1 , . . . , z d ) Then there exists a system of coordinates (x 1 , y 1 , . . . , x r , y r ) centered at p such that Thus, all symplectic manifolds are locally isomorphic to Example 2.1.4, and the study of local properties of Poisson manifolds reduces to the consideration of Poisson structures that vanish at the origin.
The analogous properties for algebraic Poisson structures do not hold as (1) the change in coordinates is rarely algebraic and (2) the symplectic leaves are rarely algebraic. However, in certain cases, Poisson schemes still have a stratification by symplectic leaves. For example, a smooth symplectic scheme trivially decomposes into a product of a single symplectic leaf and a trivial Poisson scheme. The next case -that of a singular symplectic variety -is the subject of Theorem 2.3 of [14].
Theorem 2.2.22 (Kaledin [14]). Let X be a singular symplectic variety. There exists a finite stratification X i ⊆ X where 1. each stratum X i is an irreducible Poisson subscheme, 2. the open smooth loci (X i ) sm are symplectic, 3. for any closed point x ∈ (X i ) sm , the formal completion of X decomposes as a product where Y x is a local formal Poisson scheme.
The completed strata (X i ) sm x are the symplectic leaves of X. The use of formal schemes in the theorem is unavoidable, as the symplectic leaves leaves are seldom algebraic. However, the theorem still gives a one-to-one correspondence between the algebraic strata X i and the symplectic leaves of X. Therefore, Corollary 2.2.23. There is a bijection between the Poisson prime ideals of a singular symplectic variety X and the symplectic leaves of X. Moreover, a singular symplectic variety has finitely many symplectic leaves.

Poisson Structures on k th -Infinitesimal Neighborhoods
The following proposition states that the k-infinitesimal neighborhood of a closed Poisson subscheme is again a Poisson subscheme. , is also a Poisson subscheme of X. Proof. It needs to be shown that I k is a Poisson ideal. In fact, given f ∈ A and g = g 1 . . . g k ∈ I k , the Leibniz identity implies that As each g i ∈ I and I is a Poisson ideal, it follows that each {f, g i } X ∈ I. Hence {f, g} X ∈ I k .
Thus, given a closed Poisson subscheme of an affine scheme, one automatically gets compatible Poisson structures on the k-infinitesimal neighborhoods of Y . That is, ∀k > 0 the map p k : A → A/I k+1 is a homomorphism of Poisson algebras. Furthermore, the Poisson structures (N k Y (X), {·, ·} k ) are compatible with each other in the following sense: Proposition 2.2.25. The natural projection φ k : A/I k+1 → A/I k is a homomorphism of Poisson algebras.
Proof. By the proposition above, the top arrows below are homomorphisms of Poisson algebras: v v n n n n n n n n n n n n Given f, g ∈ A/I k , letf ,g be lifts of f, g to A. By the diagram above, Using these lifts to A, and the fact that the p i are Poisson algebra homomorphisms, Thus, φ k is a homomorphism of Poisson algebras.
The propositions above are nicely summarized by the following commutative diagram of Poisson algebras: On the level of spaces, this diagram corresponds to a chain of Poisson embeddings: In subsequent chapters, the primary technique to (1) construct Poisson embeddings, or (2) show the impossibility of constructing a Poisson embedding will be as follows: Given a Poisson scheme (Y, {·, ·} Y ) and a closed embedding ι : Y ֒→ A n k into n-dimensional affine space, either (1) construct a Poisson structure on X (making ι a Poisson map) by extending {·, ·} Y to Poisson structures on each N k Y (A n k ), or (2) find an obstruction to such an extension, for some K.

The cotangent Lie algebra
Let X be a smooth Poisson scheme. Define a Lie bracket on the cotangent sheaf Ω X by defining it locally on exact 1-forms as follows: [df, dg] = d{f, g}. This Lie algebra structure turns the anchor map ♯ : Ω X → T X into a Lie algebra homomorphism. Moreover, this construction is functorial with respect to Poisson subschemes: Given Y ⊆ X with defining Poisson ideal I, the diagram below is a commutative diagram of Lie algebras: becomes an exact sequence of Lie algebras.
where φ 1 is the linear term of φ.

Poisson Bivectors and the Schouten Calculus
This section summarizes basic properties of smooth Poisson structures using multivector fields. As the majority of this material is applied to the case X = A n k , for the purpose of explicit computation, we'll adopt the coordinate intensive treatment of Dufour/Zung in [5].

The Poisson bivector and the Schouten bracket
Furthermore, when X is smooth, Hom O X ( 2 Ω X , O X ) may be identified with the sheaf of bivector fields 2 T X , and the defining relationship between the bivector and bracket becomes: where ι is the contraction operator, ♯ is the anchor (see Remark 2.2.2), and equality takes place in T X . It's clear that the properties of the bilinearity, skew-symmetry of {·, ·}, as well as the satisfaction of the Leibniz identity, are equivalent to π ∈ Γ(X, To see what property of π corresponds to {·, ·} satisfying the Jacobi identity, wee need to introduce the Schouten bracket.

Multivector fields and the Schouten bracket
This section introduces the theory of multivector fields and translates properties of Poisson brackets into the language of bivectors (see [5]). Throughout, (X, {·, ·}) will be a smooth Poisson scheme.
Denote the graded complex of multivector fields on X by * T X = k T X . The Lie bracket of vector fields on X uniquely extends to a graded Lie bracket on * T X satisfying the graded Leibniz rule. That is: There is a unique bilinear bracket [·, ·] of degree-(−1), called the Schouten bracket satisfying the following properties: for multivector fields α and β ∈ · T X , with respective degrees a, b ∈ N, γ ∈ c T X , and The Schouten bracket succinctly characterizes when a bivector corresponds to a Poisson bracket [5]: π corresponds to a Poisson bracket {·, ·} if and only if [π, π] = 0 Definition 3.1.2. Given a bivector field π ∈ 2 T X , the trivector field [π, π] ∈ 3 T X is called the Jacobiator of π.
The jacobiator of a bivector has the explicit characterization: [π, π](df, dg, dh) = 2 (π(df, dπ(dg, dh)) + c.p) An explicit formula for the Schouten bracket in local coordinates will prove useful for explicit calculations. We adopt the conventions in [5]. Given a local coordinate system (x 1 , . . . , x n ) in X, the O X -module * T X is generated by elements of the form ∂ To ease notation, we replace { ∂ ∂x 1 , . . . , ∂ ∂xn } with anti-commuting variables {ξ 1 , . . . , ξ n }. The Schouten bracket of α, β ∈ * T X is then locally written as a "super Poisson bracket." Remark 3.1.3. Defining ∂ ∂ξ i for anti-commuting variables takes a little care. We use the sign convention that where denotes a removed term.

Poisson Cohomology
Using the structures defined in the previous section, Lichnerowicz [16] defined a cohomology for Poisson manifolds. This construction transfers mostly unchanged to the category of smooth Poisson schemes. Definition 3.2.1. Let X be a smooth Poisson scheme with Poisson bivector π. We define the Poisson cohomology H * π (X, F ) of (X, π) with coefficients in a sheaf of Poisson-modules F to be the cohomology of the complex: 3. the second cohomology H 2 π (X) is the space of formal deformations modulo "trivial deformations". 4. the third cohomology H 3 π (X) consists of obstructions to deformations of π.
These Poisson cohomology groups tend to be difficult to compute, infinite dimensional groups -even for well-behaved spaces. In the category of smooth manifolds, the following theorem illustrates the range of possibilities for H * π (M).
Theorem 3.2.4 (Lichnerowicz). Given a Poisson manifold (M, π), the pullback of the anchor map descends to a homomorphism from the de Rham cohomology of M to the Poisson cohomology of M. Moreover: • ♯ * is, in general, neither injective nor surjective.
• If M is symplectic, ♯ * is an isomorphism.
This theorem reinforces the result of Darboux (Theorem 2.2.21): symplectic manifolds are locally rigid, with only their topology affecting their deformations.
Another example for which the Poisson cohomology groups are known is the case of the Lie-Poisson structure (g * , {·, ·}) for a Lie algebra g (see Example 2.1.5). In this case they are related to the familiar Chevalley-Eilenberg cohomology.
Theorem 3.2.5 ( [16]). Let (g * , π) be the Lie-Poisson structure of a Lie algebra g. Then there is an isomorphism of cohomology groups: where Sym(g) is the (infinite dimensional) representation given by the adjoint action of {·, ·}.
Remark 3.2.6. This theorem actually holds on the level of cochains.

Poisson cohomology and infinitesimal neighborhoods
Let Y = V (I) be a smooth Poisson subscheme of a Poisson scheme X. Recall from Section 2.2.3 that the ideal sheaves I k ⊆ O X are in fact Poisson ideal sheaves for all k ∈ N. Thus, each k gives rise to an exact sequence of sheaves of Poisson O X -modules: There is a one-to-one correspondence between Poisson structures {·, ·} (k) on Y k as defined in Section 2.2.3 and Poisson bivectors π k in T X ⊗ O Y k . Thus, although Y k is not smooth, it makes sense to talk about bivectors on Y k and to identify

Graded Poisson structures on A n k
Let π be a Poisson bivector on A n k vanishing at 0. The usual grading on A = K[x 1 , . . . , x n ] decomposes π into homogeneous components: One can now express the condition [π, π] = 0 for π to be a Poisson bivector in terms of the graded components: Listing these conditions by k ∈ N gives: Analyzing the condition in each degree gives: (1) The degree-1 condition, [π 1 , π 1 ] = 0, is equivalent to the linear component π 1 being a Poisson bivector. This linear Poisson structure is in fact the Lie-Poisson algebra associated to the cotangent lie algebra of the Poisson subscheme {0}.
(k) The k th -degree condition for k > 2: Thus a Poisson structure π as above may be viewed as an infinitesimal deformation of its linear part π 1 .

Extending Poisson structures to infinitesimal neighborhoods of 0
The graded decomposition above may be reinterpreted in terms of infinitesimal neighborhoods of the Poisson subscheme {0} ⊆ A k k . Using the notation from Section 2.2.3, denote the Poisson bivector on N k {0} (A n k ) corresponding to {·, ·} k by π (k) . A natural choice for a bivector on A n k that is p k -related to π (k) is given by π 1 + π 2 + . . . + π k ∈ 2 T A n k .
The second term (3.5) on the right-hand side vanishes, as each summand has coefficients in m k+1 0 . The first term (3.4) on the right-hand side vanishes exactly when the first k degree conditions for [π, π] = 0 are satisfied.
Moreover, using the fact that π 1 defines a Poisson structure, one can view each π (k) as a solution of an equation: . This isomorphism allows one to reduce the problem of solving the condition [π, π] = 0 to computing finite dimensional Chevalley-Eilenberg cohomology groups.

Chapter 4 The Poisson Embedding Problem
This chapter formulates and provides an initial approach to answering the motivating problem for this dissertation: This problem on smooth Poisson manifolds is then (locally) reduced to the following algebraic problem: Question 4.0.2. Given a Poisson scheme (X, {·, ·} X ) and an embedding (in the category of schemes) of X ֒→ A n R , when does there exist a Poisson structure on A n R extending {·, ·} X ? As our primary examples arise as quotients by subgroups of U(1), we will discuss a technique for taking advantage of the natural complex coordinate -further reducing the problem to finding complex algebraic Poisson embeddings of the complexification X C into C n . After discussing these reductions, we illustrate the problem with many examples framing the problem in the context of the previous chapters.

Embedding Poisson Structures: Preliminaries
In this chapter, we fix the following notation: 2. Let G ⊆ Sp(V ) be a compact group acting linearly on V by symplectic isomorphisms, and assume that 0 is an isolated fixed point.
The algebra of G-invariant polynomials on V is finitely generated, by the Hilbert basis theorem. Fixing a set of generators σ := {σ 1 , . . . , σ d } ⊆ Sym(V * ) G , we define the Hilbert map to be the map induced by the ring map: The corresponding map of algebraic varieties is given by: As this map separates the orbits of G, the Hilbert map induces an embedding F σ : V /G ֒→ R d of the quotient. This construction justifies the following definition: A useful characterization of the minimal embedding is given by the Zariski tangent space at the fixed point (see [11]):

Smooth structures
We give a definition, suitable for our purposes, of a smooth embedding of a (possibly singular) quotient: In the present context, the study of smooth embeddings reduces to the algebraic case described above: A theorem of Schwarz [22] states that C ∞ (V ) G is generated by a finite number of polynomial invariants. That is, there exist G-invariant polynomials {σ 1 , . . . , σ d } that generate smooth functions on V /G in the following sense: for all F ∈ C ∞ (V ) G , there exists anF ∈ C ∞ (R d ) such that F =F (σ 1 , . . . , σ d ). Thus, any smooth embedding factors through a minimal algebraic embedding.
In more general contexts, the model for the smooth functions on a singular space is more subtle and understood using stratifications (see [19]).

Poisson embedding
Poisson structures behave very well with respect to quotients by group actions. As G acts on V by symplectomorphisms, the Poisson bracket on V descends to a Poisson bracket on V /G: If φ, ψ are G-invariant functions on V , then {φ, ψ} V is also a G-invariant function on V , as  If no embedding exists, then we say the C-Poisson embedding dimension is infinity.
In [23], R. Cushman conjectured that given an embedding φ : V /G ֒→ R n , there is a Poisson structure on R n making φ a Poisson embedding. A. Egilsson provided a negative answer to this conjecture in [6], using an example that will be described in the following Section 4.1.2.
The answer to the following related question is still not known: In fact, the answer is unknown in every category considered in this dissertation.

Reduction to the Algebraic Case
This reduction follows the same technique developed in [2]. In the context of this dissertation, the singular quotients V /G behave well enough that their smooth functions C ∞ (V /G) as defined in Section 4.1 are locally well-approximated by power series. By Theorem 4.2.15 in [2], such an isomorphism is obtained any time the algebra A is 1. a subalgebra of a polynomial algebra 2. finitely generated as an algebra by homogeneous polynomials.
These conditions are guaranteed by Schwarz [22] as described above.
Within the context of embeddings, this isomorphism implies that any embedding φ : V /G ֒→ R n , after applying the completion functor, gives rise to a local embedding: Moreover, as the horizontal isomorphisms respect the bracket structures (see Section 2.1, or [2]), if φ is a Poisson embedding, the above diagram induces a local Poisson embedding. Thus we obtain a relationship between smooth and formal embedding dimensions. This theorem allows us to restrict our attention to algebraic and formal Poisson structures, as either: 1. we find an algebraic or formal Poisson embedding of V /G ֒→ R n , or 2. we find an obstruction to formally extending the Poisson structure on V /G, which by the above theorem is also an obstruction to smoothly extending the Poisson structure on V /G.
Remark 4.1.11. It is possible that such a procedure could result in a third possibility: a formal Poisson structure on a neighborhood V /G that is not extendable to a Poisson structure on the whole space. Such an example has yet to be found, but would be quite interesting.

The Complexification Trick
Many of the motivating examples of Poisson quotients arise from actions of subgroups G ⊆ U(n). When working with such actions, the natural coordinates to consider are the complex coordinates. However, making use of this extra structure requires a development of how a Poisson embedding behaves under complexification (see also [6]).
Recall from Proposition 2.1.12, that the tensor product of Poisson algebras over a field K is a Poisson algebra. Applying this proposition to a Poisson algebra (A, {·, ·}) over R and the trivial Poisson algebra C over R, we obtain a Poisson algebra A ⊗ R C over R. One may also consider A ⊗ R C as a Poisson algebra over C, and the resulting bracket {·, ·} C is precisely the C-linear extension of {·, ·}. Thus, if (X, {·, ·}) is a real affine Poisson variety, its complexification (X C , {·, ·} C ) is naturally a complex affine Poisson variety. Moreover, this construction is functorial: given a Poisson map of real Poisson varieties F : X → Y , the induced map between their complexifications F C : X C → Y C is a Poisson map.
Example 4.1.12 (Kähler coordinates). Let V 1 and V 2 be real 2n-dimensional vector spaces (considered as affine varieties), with coordinate rings We give V 1 the standard symplectic structure and define the map between the complexifications: As complex coordinates are often more natural, we will usually work with the induced embedding of complex varieties (X C , {·, ·} C ) and use this one-to-one correspondence between Poisson embeddings of the real variety X ֒→ R d and certain Poisson embeddings of the complexification X C ֒→ C d .

Examples of Poisson Embeddings
The following example, considered in [23], describes a quotient that Poisson embeds into a linear Poisson structure: Example 4.2.1 (Quadratic invariants). Assume that the G-invariant functions on V are generated by quadratic polynomials σ 1 , . . . , σ n . As the Poisson bracket {·, ·} V has degree-(−2), the Hamiltonian vector fields X σ i := {·, σ i } are linear; they span a Lie subalgebra h ⊆ sp(V ). The moment map µ : V → h * is precisely the Hilbert map F σ . Moreover, as moment maps are homomorphisms of Poisson algebras, the induced map on the quotient µ : V /G ֒→ h * ≃ R n is a Poisson embedding, identifying V /G as a nilpotent cone in h * .

Embedding 2-Dimensional Symplectic Orbifolds
The Hilbert map associated to a low dimensional dimensional Poisson quotient often defines a Poisson embedding. The following treats 2-dimensional (real and complex) orbifolds.

Real Symplectic Orbifolds
Every local model R 2 /Γ for a real 2-dimensional symplectic orbifold Poisson embeds into R 3 . The 2-dimensional local models for general orbifolds are quotients R 2 /Z n and R 2 /D 2n with the standard real representations. However, only the action of Z n is compatible with the symplectic structure on R 2 .
A minimal generating set (i.e. a Hilbert basis) for the algebra of invariant functions consists of polynomials: These define an embedding into R 3 via the Hilbert map: F σ (z, z) = (σ 0 (z, z), σ 1 (z, z), σ 2 (z, z)) and the quotient is realized as the vanishing locus V (y 2 + z 2 − 4x n ) ⊆ R 3 . The Poisson bracket on the algebra of invariants is given by: which extends to the Poisson structure on R 3 given by: That this bracket satisfies the Jacobi identity is easily verified on the generators {x, y, z}: 3. If n = 2, then the resulting quotient V /Z 2 falls within the assumptions of Example 4.2.1. The resulting Poisson structure on R 3 is isomorphic to the Lie-Poisson structure on sl 2 (R) * and the image of the embedding is the nilpotent cone.

Holomorphic Symplectic Orbifolds
Other low dimensional examples arise as holomorphic Poisson structures on 2-dimensional complex orbifolds. Let V be a 2 dimensional complex symplectic vector space and Γ ⊆ Sp(2, C) a finite group. The resulting quotient V /Γ is a local model for a holomorphic symplectic orbifold.
Proposition 4.2.4. Every such "holomorphic symplectic orbifold" of dimension 2 has a Poisson embedding into C 3 .
As dim V = 2, the groups Sp(V ) and SL(V ) are equal. Quotients of finite subgroups Γ ⊆ SL(V ) in dimension two are the "Kleinian singularities" and are classified by their singularity type (i.e. ADE-singularities). For all such quotients, the algebraic embedding dimension is equal to the Poisson embedding dimension. The proof of this equality follows from two specific properties of these quotients: 1. The vanishing ideals of the quotients, by Krull's Hauptidealsatz, are principal (i.e. generated by a single polynomial), forcing both the resulting singularity and the moduli space of extensions to be well-behaved.
2. The low dimension of the embedding space allows one to easily verify the Jacobi identity of any extended bracket.
The computations of the invariants are classical (e.g. see Chapter 2 of [15]).
Example 4.2.5 (A n -singularities). Let Γ = Z n act on V by multiplication: A Hilbert basis for the action consists of polynomials: The induced Hilbert map F σ , whose dual is given by Checking the Jacobi identity by hand verifies that this defines a Poisson bracket on C 3 . In Example 4.3.3 all Poisson structures on C 3 extending the one above are classified.
Example 4.2.6 (D (n+2) -singularities). Let the binary dihedral group BD 4n ⊆ Sp(V ) be the finite group generated by the transformations: The BD 4n -invariant polynomials must be invariant under both generators R n and T . A minimal generating set for the polynomials invariant under the action of R n is given by {x n , xy, y n }, while the analogous set for T is given by {x 2 − y 2 , xy(x 2 + y 2 ), (xy) 2 }. Thus a Hilbert basis of the BD 4n -invariant polynomials is given by: The Hilbert map F σ realizes V /BD 4n as the vanishing locus V (4u n+1 + w 2 − uv 2 ) ⊆ C 3 . The Poisson bracket on V /BD 4n extends to the Poisson bracket on C 3 : One can easily check directly that this extended bracket satisfies the Jacobi identity.
Example 4.2.7 (E 6 -singularities). The 2-dimensional E 6 -singularities are realized by the action of the Binary Tetrahedral group BT 24 , generated by the following matrices: A Hilbert basis of the BT 24 -invariant polynomials is given by: The Hilbert map F σ realizes V /BT 24 as the vanishing locus V (108u The Poisson bracket on V /BT 24 extends to the Poisson bracket on C 3 : Example 4.2.8 (E 7 -singularities). The 2-dimensional E 7 -singularities are realized by the action of the Binary Octahedral group BO 48 , generated by BT 24 (i.e. the matrices S, T, and . Noting that R 2 8 = S, we find that the Hilbert basis is given by a subset of the BT 24 -invariant polynomials: The Hilbert map F σ realizes V /BO 48 as the vanishing locus V (uv 3 − 108u 3 − w 2 ) ⊆ C 3 . The Poisson bracket on V /BO 48 extends to the Poisson bracket on C 3 : Example 4.2.9 (E 8 -singularities). The 2-dimensional E 8 -singularities are realized by the action of the Binary Icosahedral group BI 120 , generated by: A Hilbert basis of the BI 120 -invariant polynomials is given by: The Hilbert map F σ realizes V /BI 120 as the vanishing locus V (u 2 + σ 3 1 − 1728w 5 ) ⊆ C 3 . The Poisson bracket on V /BI 120 extends to the Poisson bracket on C 3 : Remark 4.2.10. Higher dimensional orbifolds do not behave so nicely. For example, in dimension 4, Theorem 1.0.1 proves that any minimal embedding R 4 /Z n ֒→ R 2n+6 cannot be a Poisson embedding.

S 1 -resonance Spaces
Denote the coordinate functions on R 2n by {x 1 , y 1 , . . . , x n , y n } and fix (d 1 , . . . , d n ) ∈ Z n . Define the (d 1 , . . . , d n )-resonance action on R 2n to be the linear, symplectic S 1 -action given by a block matrix built from 2-dimensional rotation matrices Using the change of coordinates in example 2.3 and passing to complex coefficients rewrites the action as The quotient from a general S 1 -resonance action is a cone over weighted projective n-space CP(d 1 , . . . , d n ); Hilbert bases are determined in [7].
This basis embeds the quotient Additionally, there is a Poisson structure on R 4 making this a Poisson embedding: The Jacobi identity can be checked directly.
Example 4.2.13 ((1,1,2)-resonance space). Unlike in the examples above, the Poisson embedding dimension of the (1, 1, 2)-resonance space does not equal the smooth embedding dimension. Any Hilbert basis for the (1, 1, 2)-action has 11 elements, giving rise to an embedding F : C 3 /S 1 ֒→ R 11 (the minimal smooth embedding dimension). In [6], A. Egilsson proves that there cannot exist a Poisson structure on R 11 extending the Poisson structure on the embedded variety. Later, in [3], B. Davis strengthens this result: Given an embedding of the (1, 1, 2)-resonance space into R 12 , there does not exist a Poisson structure on R 12 extending the Poisson structure on the quotient.

Infinitesimal Extensions of Embedded Poisson Varieties
As discussed in the previous section, it is generally quite difficult to determine when an embedding can be turned into a Poisson embedding. The remainder of this thesis is concerned with this question: Given an embedding of a singular symplectic quotient F : V /G ֒→ R n , when is there a Poisson structure on R n such that φ becomes a Poisson embedding?
As a first approach to finding such a Poisson structure (R n , {·, ·}), one can attempt to extend the Poisson structure on V /G ≃ Im(F ) ⊆ R n to the infinitesimal neighborhoods N k V /G (R n ) of V /G in R n . As mentioned in Chapter 3, a Poisson embedding F : V /G ֒→ R n gives rise to a chain of Poisson maps: , where I := ker(F * ). A procedure for building successive extensions goes as follows: We begin by extending the Poisson bracket {·, ·} V /G on V /G arbitrarily to a bilinear bracket {·, ·} on R n (that almost surely will not satisfy the Jacobi identity) and denote the bivector corresponding to {·, ·} by π. All extensions of the Poisson structure on V /G are of the form π + π I , where π I has coefficients in I. Therefore we may try to solve the equation [π + π I , π + π I ] = 0 for π I . Moreover, we may also attempt to solve for π I step-by-step: 0. The bivector π may be considered the zeroth extension π (0) := π, as the Schouten bracket vanishes [π, π] = 0 mod I. Thus, π (0) defines a Poisson bracket on N 0 1. The first extension would be a bivector π (1) := π (0) + π 1 I such that (a) π (1) defines a Poisson structure on N 1 V /G (R n ). In other words, [π (1) , π (1) ] = 0 mod I 2 ).
(b) π 1 I has coefficients in I, so that π (1) is an extension of π (0) . k. Similarly, the k th extension is a bivector π (k) := π (k) +π k I such that π (k) defines a Poisson structure on N k V /G (R n ) -that is, [π (k) , π (k) ] = 0 mod I k ) -and π k I has coefficients in I.
The limit π (∞) := lim ← − π (k) then defines a Poisson structure on a formal neighborhood of V /G ⊆ R n . Remark 4.3.2. This procedure is in general very difficult, as the ideals I k are usually not easy to describe. However, various refinements of this procedure are used later.

Classification of Minimal Poisson Embeddings of
Poisson structure on C 3 is defined so that the embedding F σ : C 2 /Z n ֒→ C 3 is a Poisson embedding. One can further ask if it is possible to determine all Poisson structures extending the quotient Poisson structure on C 2 /Z n .

Multivector fields on C 3
To investigate this problem, we take advantage of a few peculiarities of the deRham sequence of 3-dimensional affine space (see [20] for details): Let A = C[u, v, w] be the algebra of polynomials on C 3 and denote the k-multivectors on C 3 by X k (C 3 ) := k T C 3 . Recall that X k (C 3 ) ≃ A for k = 0, 3 and X j (C 3 ) ≃ A 3 for j = 1, 2. Using these identifications, define the following operations (using the analogous, familiar, vector operations on C 3 ):

Given a vector field
Using the identification of multi-derivations and differential forms X 3−k (C 3 ) ≃ Ω k (C 3 ) given by the star operator * : , there is an isomorphism of exact sequences: 0

Characterization of extensions
We recall the set up for the Poisson embedding F σ : C 2 /Z n ֒→ C 3 . The Hilbert basis is given by {σ 0 = xy, σ 1 = x n , σ 2 = y n }, and the map F σ is defined by This embedding realizes the quotient as the vanishing locus V (φ) ⊆ C 3 , where φ = u n − vw. Recall that the extension of {·, ·} C 2 /Zn given by the bivector below defines a Poisson structure: This is a very special type of Poisson bivector: The Poisson cohomology and deformation theory of such Poisson structures are computed in [20]. Now, consider another bivector π ′ corresponding to an extension of {·, ·} C 2 /Zn . As π and π ′ agree on V /G, it follows that π ′ = π + π φ , where π φ vanishes on V /G. Such a bivector π φ has the form φA for an arbitrary bivector A. That is The following proposition gives criteria for when π ′ is also a Poisson bivector: The set of Poisson structures on C 3 extending the Poisson structure (C 2 /Z n , {·, ·} C 2 /Zn ) is given by Proof. The proof relies heavily on the following formula for the Schouten bracket of bivectors: Given bivectors A, B ∈ X 2 (C 3 ), their Schouten bracket is given by: The bivector π + π φ , by construction, extends the bracket given by Poisson structure on the quotient. It defines a Poisson structure if and only if where [π, π] = 0 because π itself is a Poisson structure. Recall, that As * is an isomorphism, this expression is zero if and only if dφ ∧ dα + φα ∧ dα = 0 Note that if dα = 0, then π + φ( * α) is a Poisson structure. As the De Rham complex for C 3 is exact, any closed form α = df is exact. Thus, functions give a wealth of Poisson structures lifting {·, ·} V /G : In this case, as the proof demonstrates, d π π φ = −[π, π φ ] = 0, and so π+π φ is a deformation of π. Using the cohomological computations in chapter 3 of [20], conditions for when such deformations π + φ( * df ) are trivial can be explicitly determined. Using the notation from above, denote The deformation π + π φ is a trivial deformation of π if and only if The vector field which generates the deformation is then given by: The nontrivial deformations of π are generated by: Remark 4.3.5. This sort of analysis also applies to the other 2-dimensional orbifolds as well.

Infinitesimal Extensions around 0
This section applies the analysis of infinitesimal extensions of Poisson structures in Section 2.2.3 to the Poisson embedding problem (Section 4.1). Guided by the fact that a set of invariant polynomials {σ 1 , . . . , σ n } defines an embedding F σ : V /G → R n if and only if their images span m 0 /m 2 0 , it is natural to consider local problem of extending a Poisson bracket to infinitesimal neighborhoods of 0.
Any embedding F : V /G ֒→ R n such that φ(0) = 0 gives rise to embeddings into the infinitesimal k-neighborhoods of the origin N k {0} (R n ): Moreover, by Section 3.3.1, any Poisson structure on R n making F a Poisson embedding in turn makes the diagram above a commutative diagram of Poisson maps. Thus, one may attempt to construct a Poisson structure on R n by iteratively extending Poisson structures on each N k {0} (R n ). This procedure, of course, begins with the bottom square: In the following proposition (1) describes the Poisson structures on N 1 {0} (R n ) that extend the given one on N 1 {0} (V /G), and (2) describes the conditions for each subsequent extension: where π l has homogeneous coefficients of degree l, and let g ≃ R n be the Lie algebra corresponding to the Lie-Poisson structure (R n , π 1 ). Then 1. The Lie algebra g is a Lie algebra extension of m 0 /m 2 0 2. Each extension π (k) satisfies the following g-cohomological equations: Remark 4.4.2. The proposition above may also be thought of as interpreting the Lie algebra extension 0 → K → g → m 0 /m 2 0 → 0 as a first approximation to the Poisson embedding F .
That is, F is only a Poisson map up to first order. The subsequent extensions are then nonlinear deformations of the Lie-Poisson structure π 1 to a Poisson structure π that makes F a Poisson embedding.
Proof. The pullback map F * : R[x 1 , . . . , x n ] ։ Sym(V * ) is a surjective homomorphism of Poisson algebras. Moreover, the pullback restricts to a surjective map F * (x 1 , . . . , x n ) ։ m 0 , and thus descends to a surjective linear map of vector spaces F * : Span{x 1 , . . . , x n } ։ m 0 /m 2 0 . As F * is a homomorphism of Poisson algebras, the induced map F * is a Lie-algebra homomorphism. Denoting the kernel of this linear map K := ker(F * ), the Lie bracket on the Zariski cotangent space of R n is the Lie algebra extension of m 0 /m 2 0 given by the exact sequence of Lie algebras: The proof of (2) then follows from Section 3.3 and the fact that the linear Poisson structure (R n , {·, ·} 1 ) is isomorphic to the Lie-Poisson structure on g.
Remark 4.4.3. This procedure describes, in different terminology, the procedure used by Egilsson [6] and Davis [3] to prove nonembedding results for the (1, 1, 2)-resonance space , the first order approximation of any embedding F : . Then, using explicit computation, Egilsson proved the impossibility of solving the condition d g π (2) ii) Similarly, an embedding F : C 3 /S 1 ֒→ R 12 is completely determined by the data attached to a 1-dimensional central extensiong of g ≃ m 0 /m 2 0 . For suchg, Davis showed the impossibility of solving dgπ (2) iii) This procedure quickly becomes intractable as the dimension of the target space increases. To use this technique to prove that no Poisson embedding F : C 3 /S 1 ֒→ R 11+n exists, one first has to classify n-dimensional Lie algebra extensions of m 0 /m 2 0 (which is no easy task), and then prove the insolvability of the higher degree equations for each extension of m 0 /m 2 0 .
The two approaches to infinitesimally extending a Poisson structure on an embedded variety V (I) ⊆ R n considered in this chapter have opposing limitations.
• Extending a Poisson structure to formal neighborhoods of the variety itself N k V (I) (R n ) is a very tractable problem, in the sense that there are not many possible extensions at each step. On the other hand, finding explicit descriptions of the extensions involves determining membership in the ideal I, which is often a difficult problem to solve.
• Extending the germ of a Poisson structure to formal neighborhoods of the origin N k {0} (R n ) involves performing calculations degree-by-degree (that is, working modulo m k 0 ), which is computationally easy. However, there are many extensions at each step, and the task of determining which of these arise from the Poisson structure results in having to perform a very large number of (easier) computations.
In the subsequent chapters, we will take the latter view, using Lie theoretic techniques to make the large number of computations manageable.

Chapter 5 Linearization of Poisson Structures
One of the difficulties of studying extensions of Poisson algebras is that one must extend both the Poisson algebra as a Lie algebra and as an associative algebra, and satisfy compatibility between the two structures. In the previous chapter, we laid out a procedure that (1) first extends the associative algebra, then (2) attempts to extend the Poisson bracket. In other words, one finds a (perhaps infinitesimal) embedding of the Poisson space then tries to extend the Poisson structure.
The final criteria for a Poisson embedding given in Proposition 4.4.1 first establishes an embedding of a Poisson quotient space Q up to first order into (g * , π 1 ), for an extension g of its cotangent Lie algebra m 0 /m 2 0 . Then, it states criteria for deforming the Lie-Poisson structure π 1 into a Poisson structure that extends the full Poisson structure on Q. The nonlinear deformations of (g * , π 1 ) are greatly influenced by the Lie algebra cohomology of g. One extreme is the result of Weinstein [25]: (Weinstein). Let π = π 1 +π 2 +. . . be formal Poisson structure that vanishes at the origin with corresponding linear Poisson structure (g * , π). If g is semi-simple, then the Poisson structures (g * , π) and (g * , π 1 ) are formally isomorphic.
This theorem may be applied to the embedding problem. Suppose that the cotangent Lie algebra g := m 0 /m 2 0 of (Q, {·, ·} Q ) is semi-simple. Then there is a Poisson embedding into the minimal embedding space g * if and only if there is a Poisson embedding into (g * , π 1 ), where the linearization π 1 of π is the linear Poisson structure on g * . In fact, it seems likely (see Section 6.2.1) that, if g is semi-simple, then the natural map Q ֒→ N 1 In general, when g is not semi-simple, the Poisson structure (g * , π 1 ) tends to have nonlinear deformations. However, one still has some control of the form of such deformations via the Levi decomposition for Poisson structures (see [5], [24]).
Definition 5.0.2 (Levi decomposition). Let g be a finite dimensional Lie algebra, r be its radical and s := g/r its semi-simple quotient. By the Levi-Malcev Theorem, the exact sequence 0 → r → g → s → 0 splits. The decomposition g ≃ s ⋉ r given by this splitting is called the Levi decomposition of g.
One may partially linearize a Poisson bracket with respect to a Levi decomposition: Theorem 5.0.3 (Wade). Let π = π 1 + π 2 + . . . be a formal Poisson structure that vanishes at the origin with corresponding linear Poisson structure (g * , π), and let g = s ⋉ r be the Levi decomposition of g, with coordinates (s 1 , . . . ,s n ,r 1 , . . . ,r m ) and Lie brackets Then there exists a formal change of coordinates (s 1 , . . . , s n , r 1 , . . . , r m ) such that: 1. The change of coordinates is the identity up to first order.
2. The resulting Poisson structure has the form: This theorem's proof relies on either Hermann's linearization of vector fields [12] or on the Hochschild-Serre spectral sequence [13], depending on the point of view one wants to take (See [5] and the references therein). Additionally, this semi-linearization improves the embedding criteria given by Proposition 4.4.1 as follows: {0} (R n ) by π (k) . Let g ≃ R n be the Lie algebra corresponding to the Lie-Poisson structure (R n , π (1) ). Then 1. The Lie algebra g is a Lie algebra extension of m 0 /m 2 0 .
2. If the Levi decomposition of g ≃ s ⋉ r, then the extended Poisson structure π (k) on N k {0} (R n ) is a solution to the following r-cohomological equations: where we may consider π (i) ∈ C 2 (r, Sym ≤i (g)) s via the identification 2 T N k {0} (R n ) ≃ C 2 (g, Sym i g) given in Theorem 3.2.5.
To prove the nonembedding theorem in Chapter 6, we need to make a slight improvement to the semi-linearization given in the Levi decomposition. For a relevant class of Lie algebras, this improvement will give a normal form for some of the nonlinear P ij in Theorem 5.0.3. This improvement is based on a technique developed by J.P. Dufour and N.T. Zung in [4], and employs the Poincaré-Dulac normal form of a vector field.

Poincaré -Dulac normal form
A thorough exposition of this subject matter may be found in Appendix A of [5]. Roughly stated, the Poincaré-Dulac normal form is a decomposition of a vector field into a linear part and a resonant part. This section will make this statement precise.
Let K n have coordinates {v 1 , . . . , v n } and let X be formal a vector field on K n that vanishes at 0. In a neighborhood of 0, we may decompose X into a linear and nonlinear part X = X 1 + X nl . Furthermore, the linear vector field decomposes into semi-simple and nilpotent parts X 1 = X 1 ss + X 1 n . Then X is in Poincaré-Dulac form if [X, X 1 ss ] = 0. In particular, when K = C, the semi-simple part diagonalizes as: We write X in coordinates as . . , k n , k) is multi-index notation. The condition that [X, X 1 ss ] = 0 then becomes Therefore, a vector field is in Poincaré-Dulac normal form if the nonlinear coefficients (x k 1 1 x k 2 2 . . . x kn n ) are resonant with respect to the eigenvalues (λ 1 , . . . , λ n ) of X 1 : Theorem 5.0.5 (Poincaré -Dulac). Let X be a formal vector field vanishing at the origin. Then there is a formal change of coordinates that puts X into Poincaré-Dulac normal form.
Remark 5.0.6. Under an additional diophantine condition, a theorem of Bruno actually gives an analytic version of the Poincaré-Dulac theorem (see [5]).
Example 5.0.7. If the nonlinear part of X is nonresonant, then the Poincaré-Dulac theorem states that X is linearizable.
The example of g = (K ⊕ s) ⋉ r We now prove, for a certain Lie algebra g, the existence of a slight improvement of the semi-linearization given by Theorem 5.0.3. Let g = (K ⊕ s) ⋉ r be the semi-direct product of (1) a 1-dimensional central extension of a semi-simple Lie algebra s and (2) a nilpotent Lie algebra r. Denote the basis of g = (K ⊕ s) ⋉ r by {t,s 1 , . . . ,s k ,r 1 , . . . ,r m }.
Example 5.0.8. Let the semi-simple part be s = sl n (C) and the Lie ideal be r = C n . If the action of s = sl n (C) on C n is the usual one and C acts via scalar multiplication, then g is the affine Lie-algebra aff(C n ). This example, considered in [4], originally necessitated the use of the Poincaré-Dulac normal form to improve the Levi decomposition.
Proposition 5.0.9. Let π = π k be a formal Poisson structure on K n that vanishes at the origin, with corresponding linear Poisson structure (g * , π 1 ), where g = (K ⊕ s) ⋉ r. There exists a formal change of coordinates (t, s 1 , . . . , s k , r 1 , . . . , r m ) such that 1. The change of coordinates is the identity up to first order.
2. The resulting Poisson structure has the form: where a k ij , b k ij , λ k ∈ K, the P ij are formal nonlinear functions and the R k are nonlinear functions that are resonant with respect to the action of K on r.
Proof. By the Levi decomposition, we may assume that π is already in the form of Theorem 5.0.3 -that is, π has the form in the proposition, where the R k are not necessarily resonant. The Hamiltonian vector fields {X s 1 , . . . , X s k } form a Lie subalgebra of Ham(g) isomorphic to s. Moreover, the Hamiltonian vector field is invariant under the action of s (as {t, s i } 1 = 0), and so we may s-equivariantly apply the Poincaré-Dulac theorem to obtain coordinates such that Furthermore, this coordinate change does not affect the Levi decomposition -that is, Example 5.0.10 ( The cotangent Lie algebra g = gl 2 (C) ⋉ (V n ⊕ V n ) ). This example will apply Proposition 5.0.9 to the cotangent Lie algebra of the 4-dimensional symplectic quotient V /Z n considered in Chapter 6. Let g = (C ⊕ sl 2 (C)) ⋉ (V n ⊕ V n ) be the Lie algebra with basis {t, H, E, F, x 0 , . . . , x n , y 0 , . . . , y n } and Lie brackets given by: 1. The standard gl 2 (C) = (C ⊕ sl 2 (C)) structure on the first factor.
2. The action of the semi-simple factor sl 2 (C) on V n ⊕ V n given by ρ n ⊕ −ρ n , where ρ n : sl 2 (C) → V n is the standard (n + 1)-dimensional irreducible representation of sl 2 (C).
3. The action of C on V n ⊕ V n is given by Id ⊕ −Id.

Chapter 6
A Nonembedding Result for V /Z n This chapter consists of the proof of Theorem 1.0.1, which asserts that the Poisson embedding dimension of the singular symplectic quotient V /Z n of 4-dimensional real affine space V by the cyclic group Z n of odd order is strictly greater than its smooth embedding dimension. To prove such a result, we need a description of the quotient and its Poisson structure.

Basic Properties of V /Z n
Throughout this chapter, let G = Z n be the cyclic group of order n. Let V = R 4 be equipped with the standard symplectic structure (see Example 2.1.4), and let Z n ⊆ S 1 ⊆ Sp(V ) act diagonally on V = R 2 ×R 2 , with the action of Z n on R 2 given in Example 4.2.2. As in Example 4.1.12, denote the complexified coordinates of V C by {z, z, w, w}. The resulting linear action of Z n on the 4-dimensional complex vector space V C is given by: ζ · (z, z, w, w) = (ζz, ζ −1 z, ζw, ζw −1 ), for ζ ∈ Z n ⊆ C.
(6.1) By Section 4.1.2, if there is no Poisson structure on C N extending the Poisson structure on V C /Z n , then there cannot be a Poisson embedding V /Z n ֒→ R N . We will take this approach and therefore only be concerned with V C . Thus, to reduce notational clutter, we will drop the C from V C .
V is now a complex vector space equipped with the action in Equation 6.1. However, to prevent a proliferation of i = √ −1 throughout the chapter, we will scale this Poisson bracket by −1 2i so that {z, z} = {w, w} = 1.
As this action is a Poisson action, the quotient V /Z n inherits a natural Poisson structure. Denote this Poisson variety (V /Z n , {·, ·} V /Zn ) by (Q, {·, ·} Q ).
The following lemma will be used repeatedly to prove properties of the quotient Q. It arises from the fact that the Z n -action is a restriction of the standard action of U(2) on V and therefore defines a homomorphism of (complexified) Lie algebras: where µ is the moment map for the U(2)-action, and Ham(f ) = {·, f } is the map that sends a function to its Hamiltonian vector field. Explicitly, the homomorphism is given by: We may identify gl 2 (C) with a subalgebra of T V via the map Such an identification of gl 2 (C) ⊆ T V induces an action of gl 2 (C) on Sym(V * ) and Sym(V * ) Zn . Therefore Lemma 6.1.2. The action above makes Sym(V * ) and Sym(V * ) Zn representations of the Lie algebra gl 2 (C). Moreover, 1. The action decomposes Sym(V * ) and Sym(V * ) Zn into t-eigenspaces and H-eigenspaces.
2. As the Poisson bracket on V has degree-(−2) and gl 2 (C) is generated by (the Hamiltonian vector fields of ) quadratic functions, this action preserves the natural grading on Sym(V * ) = k Sym k (V * ).

A Hilbert Basis for V /Z n
In this section, we will determine a Hilbert basis for the quotient Q = V /Z n -that is, a minimal generating set for the algebra of invariant polynomials Sym(V * ) Zn . The previous section provides three different gradings on the algebras Sym(V * ) and Sym(V * ) Zn . The polynomial algebra decomposes as This decomposition gives a useful characterization of the invariants Sym(V * ) Zn ⊆ Sym(V * ): Thus p is Z n -invariant if and only if the t-eigenvalue of p is divisible by n.
Now we present a Hilbert basis for Q: Proposition 6.1.4. A Hilbert basis for the action of Z n on V given above is given by: {zz, ww, zw, zw, w n , zw n−1 , . . . z n , w n , zw n−1 , . . . , z n } Proof. One can easily verify (using the proposition above) that each of these are Z n -invariant and that no member of this list can be written in terms of the other. Thus, it only remains to show that any Z n -invariant monomial can be written in terms of those on the list. Let p be a Z n -invariant monomial, so that we may write There are four cases to consider: (1) Suppose a > b and c < d. We factor p = (zz) b (ww) c z a ′ w d ′ . As t · p ≡ 0 mod n, and t · (zz) = t · (ww) = 0, we have that t · (z a ′ w d ′ ) = 0. Thus, a ′ − d ′ ≡ 0 mod n and there exists a k ∈ Z such that (a ′ − d ′ ) = kn. If k ≥ 0, then a ′ = kn + d ′ ≥ 0 and so z a ′ w d ′ = z kn+d ′ w d ′ .
Thus p may be written as the product of elements in the proposed Hilbert basis: If k < 0, then d ′ = (−k)n + a ′ > 0 and so z a ′ w d ′ = z n|k|+a ′ w d ′ . Thus, p may be written as (2) Suppose a > b and c > d. We can factor p in a similar manner to the previous case: . Then, writing a ′ = nk a + r a and b ′ = nk b + r a with 0 ≤ r a , r b < n, one can further factor p: Moreover, as a ′ + b ′ ≡ 0 mod n, it follows that r a + r b ≡ 0 mod n. As 0 ≤ r a , r b < n, it follows that either Cases (3) and (4) are proved analogously. Thus, an arbitrary p ∈ Sym(V * ) Zn may be written as a product of elements in the list and therefore the list forms a Hilbert basis for the Z naction on V .

The defining ideal
It will prove useful to have explicit description of the defining ideal of the image of Q ⊆ C 4+2(n+1) . Table 6.1 lists the generators of the ideal I := ker(F * ) ⊆ Sym(gl 2 (C) × W ).
Remark 6.1.5. These generators are given in this form because of their representation theoretic properties. It is tedious, but straightforward to check that these generate ker(F * ).

The Lie-Poisson Structure of the Minimal Embedding
Recall that the Hilbert basis in Section 6.1.1 has a Lie algebra structure (the cotangent Lie algebra) inherited from the Poisson bracket on Sym(V * ) Zn . Moreover, as was proved in Chapter 3, this structure determines the linear part of any extension of the Poisson structure on Q to C 4+2(n+1) (one may also see this directly; there are no linear terms in I). Direct computation gives that the cotangent Lie algebra is isomorphic to

Name
Index Range where V n is the (n + 1)-dimensional irreducible representation of sl 2 (C). The action is given by ρ n ⊕ n · (Id) on the first V n -factor and − [ρ n ⊕ n · (Id)] on the second V n -factor (see Appendix A for an explanation of the notation). The corresponding Lie-Poisson structure determined by the cotangent Lie algebra is given by the follow skew-symmetric coefficient matrix: Remark 6.1.6. The gl 2 (C) subalgebra of the Lie-Poisson structure above is related to the usual basis by: In the following, we will identify the elements of gl 2 (C) with this subalgebra.
The sl 2 (C)-action and gl 2 (C)-action The Lie algebra sl 2 (C) ≃ Span{a 0 − a 1 , a 2 , a 3 } ⊆ Sym(gl 2 (C) ⋉ W ) acts on Sym(gl 2 (C) ⋉ W ) via their Hamiltonian actions with respect to the Lie-Poisson structure: As the Lie-Poisson bracket has degree-(−1), this action preserves polynomial degree and thus restricts to actions on each Sym k (gl 2 (C) ⋉ W ). In particular, it acts on the generators of I, decomposing them into direct sums of irreducible representations: Proposition 6.1.7. The generators given in Table 6.1 span the following representations of sl 2 (C): Similarly, the central element t = (a 0 + a 1 ) ∈ gl 2 (C) acts on Sym(gl 2 (C) ⋉ W ) via t · f := {f, t} 1 for f ∈ Sym(gl 2 (C) ⋉ W ) which, as before, gives a notion of t-degree on Sym(gl 2 (C) ⋉ W ).

A non-Poisson extension of {·, ·} Q
We will now fix an extension of {·, ·} Q to a bilinear bracket {·, ·} β with corresponding bivector β = β 1 + β n−1 . The linear part β 1 (with corresponding bracket {·, ·} 1 ) must be the Lie-Poisson bivector on g * in the previous section. We define the nonlinear summand explicitly by the formula where β n−1 is defined to be zero on all other basis elements of g * ∧ g * . This extension of {·, ·} Q does not define a Poisson structure on g * ≃ C 4+2(n+1) , as the vanishing condition on the Schouten bracket is not satisfied: The discussion in Section 6.1.2 implies that [β 1 , β 1 ] = 0. The last term [β n−1 , β n−1 ] = 0 also vanishes, which follows from the fact that β n−1 is nonzero only on elements of the form dx i ∧dy j ∈ g * ∧g * , while β n−1 has coefficients in Sym(gl 2 (C)) -i.e. the image of the bracket corresponding to β n−1 is a subset of its kernel. On the other hand, one can directly note that [β 1 , β n−1 ](da 2 , dx 1 , dy 3 ) = 0.
Proposition 6.1.8. A partial description of the jacobiator [β, β] is given by: [β, β](du, dv, dw) =    r(a)C, with r ∈ Sym n−1 (gl 2 (C)), (da i , dx j , dy k ), i = 2, 3 p(a)A l + q(a)B l with p, q ∈ Sym n−1 (gl 2 (C)), (dx i , dy j , dy k ) p(a)A l + q(a)B l with p, q ∈ Sym n−1 (gl 2 (C)), (dy i , dx j , dx k ) For those (du, dv, dw) not included above, or not achievable from the above using permutations, the jacobiator is zero. In particular, we have the contractions with a 0 , a 1 are zero: Proof. A direct computation using the definition of β gives that the jacobiator is zero on those triples not on the above list. That the jacobiator takes the form above follows from two observations. For the case (da i , dx j , dy k ), direct computation using the definition of β shows [β, β](da i , dx j , dy k ) ∈ Sym(gl 2 (C)): The remaining nonzero terms are all in Sym(gl 2 (C)) by definition of β. However, [β, β] has coefficients in I, as β corresponds to the extension of a Poisson bracket on V (I) ≃ Q. The only way [β, β](da i , dx j , dy k ) ∈ Sym(gl 2 (C)) ∩ I is for [β, β](da i , dx j , dy k ) = r(a)C as given in the proposition. The same reasoning applies to the other triples.
To prove the last statement, it is enough to show the contractions are zero for t = a 0 + a 1 and H = a 0 − a 1 , as contraction is linear. However, from the equality above, [β, β](dt, dx j , dy k ) = 0 if and only if {t, β n−1 (dx j , dy k )} 1 = β n−1 (dx j , d{t, y k } 1 ) + β n−1 (d{t,

Proof of the Nonembedding Result
In this section, we begin the proof of Theorem 1.0.1: Theorem 1.0.1. Given n > 2 with n odd, there does not exist a Poisson bracket on R 4+2(n+1) extending the Poisson structure on Q = V /Z n . Remark 6.2.1. The proof given only requires n to be odd at one step, which is indicated in bold. It is likely that the result also holds for even n. 1. For the sake of contradiction, we assume the existence of a Poisson structure π extending the Poisson structure on Q.
2. Then, we apply the semi-linearization techniques of Chapter 5 to obtain a change of coordinates that transforms π into an easy-to-work-with form.
3. We obtain an explicit description of the resulting embedding. In particular, we show it coincides with the Hilbert map F in the section above on the gl 2 (C)-coordinates. Such a description in turn yields a tangible description of π in terms of the "non-Poisson extension" β and a bivector α that vanishes on Q.

4.
We analyze the jacobiator [π, π] in terms of the description obtained in the step above. This analysis results in [π, π] = 0, contradicting the assumption that π is a Poisson structure. This final step consists of three parts: (a) We define a grading on Sym(g) that corresponds to infinitesimal neighborhoods N k gl 2 (C) (C 4+2(n+1) ) and examine the restriction of [π, π] to We obtain a refinement of our description of π on the neighborhood in the previous step. By examining the restriction of [π, π] applied to (dt, dw i , dw j ), we conclude the term [π, π](dt, dw i , dw j ) vanishes on this neighborhood if and only if the restriction of α vanishes on V (C) = V (a 0 a 1 − a 2 a 3 ).
(c) Using this refinement of α, we determine that the restriction of [π, π](dw i , dw j , dw k ) on N 1 gl 2 (C) (C 4+2(n+1) ) ∩ N n−1 {0} (C 4+2(n+1) ) cannot vanish, contrary to our first assumption. Essentially, this contradiction follows from the fact that an α vanishing on V (C) cannot cancel the nonzero terms of the jacobiator [β, β] computed in the section above. in which the Poisson bivector takes the form π = β 1 + π nl , where π nl is the nonlinear part: where the P ij are nonlinear, and w i ∈ {x i , y i } Denote by F φ the embedding with pullback φ. By the semi-linearization procedure in Proposition 5.0.9, F coincides with F φ up to first order. In the next section, we will determine F φ in higher orders. However, in the case at hand, the first change of coordinates is unnecessary:
Proof. The Levi factor sl 2 (C) is unique up to conjugation by an element of Sym(g). Thus the change of coordinates that linearizes {·, ·} on sl 2 (C) is given by u → u + {p ss , u} 1 where p ss ∈ Sym(g) and u ∈ Sym(g).
For this change of coordinates to define a Poisson bracket extending {·, ·} Q , the function {p ss , u} ∈ I must vanish on V /Z n for all u ∈ Sym(g). We claim that the p ss has no linear terms. If this were the case, then {p ss , u} 1 would have linear terms (by considering a linear u). As {p ss , u} 1 ∈ I and I contains no linear terms, this is impossible. Thus, p ss has no linear terms. We now use the fact that the two Levi factors are conjugate to obtain conditions on p ss . Let s i ∈ sl 2 (C), and writes i = s i + {p ss , s i } 1 . As the span ofs i is isomorphic to sl 2 (C), we arrive at the following equalities: where on the left hand side we used that Now suppose that the first nonzero term in p ss has degree d ss . Then the minimal degree of the left-hand term 2{p ss , {s i , s j } 1 } 1 is also d ss , whereas the minimal degree of the right-hand term {{p ss , s i } 1 , {p ss , s j } 1 } 1 is 2d ss − 1. This only occurs if d ss = 1 or both sides are zero. As, p ss = 0, it follows that 2{p ss , {s i , s j } 1 } 1 = 0.
As sl 2 (C) is semi-simple, it follows that p ss commutes with every element of sl 2 (C). Thus the coordinate transformation u → u + {p ss , u} 1 = u + 0 = u is the identity transformation.
We can in fact improve the above lemma: Lemma 6.2.4. The surjection φ sends the subalgebra Sym(gl 2 (C)) to the subalgebra Sym(gl 2 (C)) ⊆ Sym(V * ) Zn generated by quadratic terms. That is: Proof. As sl 2 (C) acts on both Sym(g) and Sym(V * ) Zn , the lemma above implies that φ is a homomorphism of sl 2 (C)-representations. This imposes conditions on the image of the coordinates {t, x i , y j } under φ: where V k is the (k + 1)-dimensional irreducible representation of sl 2 (C). As the 1-dimensional irreducible representations of sl 2 (C) in Sym(V * ) Zn are spanned by powers of (zz + ww), the map φ(t) = p t (zz + ww) is a polynomial in (zz + ww) with zero constant term and linear term equal to (zz + ww).
We need to show that the polynomial p t is linear. As φ is a Poisson map, we have: where K = (k, k 1 , k 2 ) is a multi-index with k > 0. Writing φ = F * + ψ as the sum of the first order terms and higher order terms, we examine the term φ( K λ k (x k 1 y k 2 ) k ): Expanding the terms inside each summand, the term (z k 1 w n−k 1 )(z k 2 w n−k 2 ) has degree strictly less than the terms involving ψ. Furthermore, (z k 1 w n−k 1 )(z k 2 w n−k 2 ) has t-degree 0 (i.e. an equal number of {z, w}-terms and {z, w}-terms). The right hand side of Equation 6.2 has no t-degree 0 component as {t, ·} Q vanishes on t-degree zero functions. It follows that each λ k (z k 1 w n−k 1 )(z k 2 w n−k 2 ) term must be cancelled by the φ(x i ) term. However, any nontrivial cancellation is impossible as: 1. Span{φ(x l ); l = 0 . . . n} ≃ V n and n is odd,

while each term
is contained in a direct sum of even-dimensional irreducible representations.
Thus φ( K λ k (x k 1 y k 2 ) k ) = 0 and Equation 6.2 becomes The only way for this equality to hold is if p t (zz + ww) = zz + ww.
Finally, we give a more explicit description of φ(x i ) and φ(y i ). Using the lemma above, we see that −nφ( This implies: ww, zw, zw] has t-degree 0. 2. As φ is a homomorphism of sl 2 (C)-representations, the image {φ(x i ); i = 0 . . . n} spans a subspace isomorphic to the irreducible representation V n .

Span{z
It follows that Thus p i = p(zz + ww) is a polynomial in zz + ww with constant term 1. Making the same argument with the y i implies: where q is a polynomial in zz + ww with constant term 1. The following proposition summarizes the above: Proposition 6.2.5. The map φ defining the embedding F φ has the form: where p, q are polynomials of (zz + ww) with constant term 1.
From the above description of the embedding F Φ , we easily see: Corollary 6.2.6. The ideal ker(φ) agrees with the naïve ideal of definition I = ker(F * ) given in Table 6.1, with the exception of the M ij which are deformed to Corollary 6.2.7. The Poisson bivector π has the decomposition π = (β + α) +π where 1. β = β 1 + β n−1

α vanishes on ker(φ)
3.π = k≥n π k is the sum of homogeneous components of π greater than or equal to n.
Proof. We will show To prove this, simply calculate the term: However, ψ(x i ), ψ(y j ) ∈ Sym(V * ) Zn both have degree greater than or equal n + 2 as 1. both have degree greater than n, and 2. the algebra of invariants Sym(V * ) Zn ⊆ Sym(V * ) is generated by elements of degrees 2 and n.
Therefore, as the Poisson bracket {·, ·} Q has degree −2, the right-hand side has degree greater than or equal to n. The other cases are analogous.

The W -grading
While our approach often uses the usual grading on the polynomial algebra Sym(g), we will now introduce a grading on the coordinate algebra of (gl 2 (C) ⋉ W ) * ≃ C 4+2(n+1) that takes advantage of the semi-linearization of π. Definition 6.2.8. Define the W -grading on Sym(g) by giving the generators the following degrees: deg W (a i ) = 0 and deg W (w i ) = 1 for all a i ∈ gl 2 (C) and w i ∈ W . This grading results in the decomposition where each Sym k (W ) is considered as a Sym(gl 2 (C))-algebra via the associative multiplication. To distinguish between the W -grading and the usual polynomial grading, we will denote the k th W -graded homogeneous component of Sym(g) by Sym(g) k . Thus Sym C (g) k = Sym k gl 2 (C) (W ).
Recall from Definition 2.1.15, that a grading on an algebra of functions induces a grading on the space of brackets and the space of multivectors. We will use subscripts to denote the decomposition of a multivector X with respect to the W -grading: Moreover, with respect to a grading on the space of multivectors, the Schouten bracket [·, ·] preserves the grading.
While the usual graded decomposition of π corresponds to extensions of Poisson structures on infinitesimal neighborhoods of the origin N k {0} (g * ), the W -grading corresponds to extending Poisson structures on infinitesimal neighborhoods of gl 2 (C) * -i.e. on N k gl 2 (C) * (g * ). We will need two properties with respect to this grading: Proposition 6.2.9. The linear Poisson bivector β 1 has W -deg 0, and therefore the β 1differential d β 1 preserves the W -grading.
Proof. This follows from the fact that gl 2 (C) ⋉ W is a semi-direct product and {W, W } 1 is identically zero.
Moreover, for any k, the decomposition of the space of k-vectors with respect to W -degree is bounded below: Proposition 6.2.10. The minimal W -degree of a W -homogeneous p-vector is −p.
Proof. This proposition is a consequence of the Leibniz identity.

Degree Conditions
When expressing π as a sum of its homogeneous components (with respect to the usual polynomial degree), the vanishing of the Schouten bracket becomes Note that for each k, the sum inside the parentheses on the right hand side must be zero. In this section, we will examine the W -degree (−2) and (−1) components of [π, π] for each k < n.
The component π k −2 preserves the t-degree for k < n Proposition 6.2.12 (Facts about the bivector π). Let 1 < k < n. The following easily verifiable facts of the bivector π k = β k + α k will be used repeatedly: Proof. Fact 1 follows from the normal form given by the Levi decomposition: the subalgebra gl 2 (C) is a 1-dimensional central extension of the Levi factor sl 2 (C) (on which π is linear). Fact 2 follows directly from the definition of the W -grading: as the W -degree of t is 0, the bivector α k d (dt, dw i ) ∈ Sym k (g) ≤1 . However, the Levi-decomposition implies that α k (dt, dw i ) = (x i y j ) l ∈ Sym(g) ≥2 . Finally, Fact 3 follows from the description of ker(φ) given in Section 6.1.1: ker(φ) ∩ Sym(gl 2 (C)) = C .
We will prove that π k −2 preserves the t-degree, which will determine the form of π k −2 for k < n. Proposition 6.2.13. The k th component π k −2 preserves the t-degree for all k < n. That is: For all k < (n − 1), we have: Applying these three terms to an element of the form (dt, dw i , dw j ) yields the following lemmas: Lemma 6.2.14 (Term I). The first term in Equation 6.3 measures the extent to which α k −2 preserves the t-degree: Calculating the Schouten bracket gives Proof. We write out the second term: The fourth term is zero by Fact 1 and the other terms vanish by Fact 2 of Proposition 6.2.12. Similar reasoning implies that term III (i.e. [α l −1 , α k−l −1 ](dt, dw i , dw j )) is zero. In other words,[π, π] k −2 = 0 if and only if α k −2 : W ∧ W → Sym k (gl 2 (C)) ∩ I preserves the t-degree for k < (n − 1). The k = (n − 1) case is similar, as there is only one new term in the vanishing condition: However, as was shown in Section 6.1.3, β n−1 preserves the t-degree and proving Proposition 6.2.13.
That α k −2 preserves the t-degree forces α k −2 (dw i , dw j ) to have the following form: Proposition 6.2.16. For all k < n, the W -degree (−2) summand takes the values Proof. To prove the first equality, we use that the t-degree of x i is (−n) and α k −2 (dw i , dw j ) ∈ Sym(gl 2 (C)): The second equality is proved similarly, while the third equality follows from Fact 3 of Proposition 6.2.12.
The component π k −1 preserves the t-degree mod C for k < n As in the W -degree (−2) case, the following (slightly weaker) condition on the t-degree will determine the form of π k −1 for k < n.
Proposition 6.2.17. In the case of W -degree (−1), the component π k −1 only preserves the t-degree up to a factor in the ideal C : Proof. One proves this proposition in the same way as Proposition 6.2.13. Consider the W -degree (−1) component of the Schouten bracket [π, π] k : Applying these three terms to an element of the form (dt, dw i , dw j ) yields three more lemmas: preserves the t-degree: Proof. This is identical to the proof of Lemma 6.2.14 above.
Lemma 6.2.19 (Term II). The second term vanishes identically: Proof. This follows from Fact 1 of Proposition 6.2.12.
Proof. We examine each summand [α l −2 , α k−l 1 ] on (dt, dw i , dw j ). Using Fact 1 from Proposition 6.2.12, the nonzero terms are: The result then follows from applying Fact 3 of Proposition 6.2.12.
As in the W -degree −2 case, the above lemmas easily imply the conclusion of Proposition 6.2.17.
That π k −1 almost preserves the t-degree still greatly determines the form of π k −1 : Proof. Proposition 6.2.17 implies Therefore, any component As any nonzero element of Span{A l , B l , A l , B l } ⊗ Sym(gl 2 (C)) has t-degree ±n, the component α k −1 must be zero -proving the proposition.
The degree (n − 1) condition Propositions 6.2.21 and 6.2.16 imply that π has the following form: Proposition 6.2.22. Recall that π = β + α + π, where α vanishes on the quotient Q and π vanishes up to degree n. The previous sections imply that α has the following form: where p k ij (g) ∈ Sym k−2 (g).
Proof. We use the semi-explicit description of α n−1 −2 obtained earlier: The first two terms in the above sum are: The last term in the sum {y 0 , α n−1 −2 (dx i , dx j )} 1 vanishes as α n−1 −2 (dx i , dx j ) = 0.
Lemma 6.2.25 (3rd and 4th terms). The third and fourth terms of the "degree (n-1) condition" send (dx i , dx j , dy 0 ) into the ideal generated by C. That is Proof. The third term is identically zero. For all k < (n − 1), Since α k −1 (dx i , a l ) = 0 for all a l ∈ gl 2 (C) (Proposition 6.2.22), a simple application of the Leibniz rule implies α k −1 (dx i , dα n−k −1 (dx j , dy 0 )) = 0. Moreover, this holds for all the cyclic permutations in the third term. Since α l −2 (dw i , dw j ) ∈ Sym l (gl 2 (C)) and α l 0 (da, dw) = 0 for all l < (n − 1), a ∈ gl 2 (C), w ∈ W , the composition α k 0 (dw i , dα n−k −2 (dw j , dw k )) = 0. Thus As α k −2 (dw i , dw j ) ∈ Span{C} ⊗ Sym(g), the result follows. One can now verify that the (W -degree −2 part of the) "degree (n − 1) condition" cannot possibly be satisfied, as Remark 6.2.26. This last conclusion is most easily seen by noticing the following: define the ideal J := C, A j , . . . A i , B j , . . . , B i . Then Thus, we arrive at a contradiction to the assumption that π is a Poisson structure extending the Poisson structure on Q, as the jacabiator [π, π] of any such Poisson structure must be identically zero.

Chapter 7 Constructing Explicit Poisson Embeddings
This section will explore the problem of constructing an explicit Poisson embedding of one of the singular symplectic quotients considered in the previous chapter. This discussion will result in an explicit Poisson embedding of V /Z 3 into R 78 . Existence of such an embedding, along with the non-embedding theorem of the previous chapter, implies the following theorem: Theorem 7.0.1. The four dimensional symplectic orbifold V /Z 3 has the following embedding properties: • The minimal smooth embedding dimension of V /Z 3 is 12.
• The minimal Poisson embedding dimension of V /Z 3 is 12 < d ≤ 78.
Thus, the Poisson structure (and not the geometric structure) on the singular quotient C 3 /Z 3 has an obstruction to extension in R 12 that disappears in R 78 . It's not known whether there exists a Poisson embedding of V /Z 3 into a Euclidean space of smaller dimension.

Generalities on Constructing Poisson Embeddings
This section will continue to use the notation of Chapter 5 and later specialize to the setup in Chapter 6. Fix a symplectic linear action of a compact group on the standard 2n-dimensional symplectic vector space V (denoting the resulting quotient (Q, {·, ·} Q )), and a minimal embedding Q ֒→ g * that is a Poisson embedding up to first order. Recall the following theorem proved in Chapter 4: 2. The extended Poisson structure π is constructed from the following g-cohomological equations: In the following section we will consider the obvious "extension" of the fixed minimal embedding: for some finite dimensional vector space Z, with coordinates {z i }, and fix the following extension of {·, ·} Q : • Let {·, ·} be an extension of {·, ·} Q to (g × Z) * , and denote by π the corresponding Poisson bivector. This Poisson bracket is the extension that we are attempting to determine.
• Let α := (π−β) be the difference of the above bivectors. We note that α has coefficients in the defining ideal I of Q ≃ V (I) ⊆ (g × Z) * . From this we may deduce that α 1 has coefficients in Z.

The General Degree 1 Condition
We will attempt to satisfy condition (1) in the proposition above by finding a (possibly nonabelian) Lie algebra extension: The degree 1 vanishing condition has the following characterization: 2. The bivector α 1 defines a linear map ρ Z : g → End(Z) whose image lies within the derivations on Z, denoted by Der(Z) ⊆ End(Z).
3. Furthermore, ρ Z : g → Der(Z) → Out(Z) defines a homomorphism of Lie algebras (and is therefore "a g-representation modulo inner derivations") 4. α 1 defines a cocycle c : g ∧ g → Z with respect to the above data. Moreover, this cocycle measures the failure of ρ Z to be a true g-representation: Proof. As the vanishing ideal of Q in g * contains no linear terms, the linear components have the form: Define the Lie bracket on Z by [z i , z j ] Z := k a k ij z k . This Lie bracket satisfies the Jacobi identity precisely when [π 1 , π 1 ](dz i , dz j , dz k ) = 0, as β 1 vanishes on Z.
Remark 7.1.2. These data appear in different guises in the literature. The first appearance of this cohomological perspective is due to Hochschild [13]. One may also package these data into a morphism of Lie 2-algebras g → DER(Z) (see HDA6 [1]).
Example 7.1.3. Suppose that Z is 1-dimensional. Then π 1 on (g × Z) * is just a one dimensional extension of β 1 on g * . The conditions in the proposition above become: 1. (Z, [·, ·] Z ) is necessarily abelian.
3. ρ Z is the trivial representation.
4. c is a genuine cocycle, with respect to the trivial representation.
Example 7.1.4. Consider the example of the (1, 1, 2)-resonance space C 3 /S 1 . In [6], Egilsson proved that there is no Poisson structure on R 11 extending the Poisson structure on C 3 /S 1 . In this case R 11 is a minimal embedding and R 11 ≃ g * = (R ⊕ gl 2 (R)) * ⋉ (R 3 ⊕ R 3 ) * ). This result was later strengthened by Davis [2], who proved there does not exist a Poisson embedding of the quotient into R 12 . Davis implicitly considered all possible cocycles (as in the above example), in order to prove the degree 2 condition could not be satisfied. When Z is of dimension greater than one, the problem of determining when the degree 1 condition is satisfied quickly becomes intractable. After all, the first criteria alone demands that one keeps track of all, possibly nonabelian, extensions of g. Thus, in an effort to construct explicit embeddings of (Q, {·, ·} Q ), we will assume that (Z, [·, ·] Z ) is an abelian Lie algebra. Such an assumption imposes quite a bit of structure on any resulting extension and so we will expect to have the dimension of Z get quite large before finding any embeddings.
The case of an abelian Z When we impose the condition that Z be an abelian Lie algebra, the criteria in Proposition 7.1.1 become:

Introduction
The contruction of the Poisson embedding of Q = V /Z 3 follows a simple idea. As above, fix an algebraic embedding Q ⊆ g × Z into an n-dimensional affine space and an extension {·, ·} β of the Poisson structure (Q, {·, ·} Q ) on the quotient to a skew-symmetric bilinear bracket on Sym(g × Z). Finding a Poisson structure that extends the one on Q amounts to solving [β + α, β + α] = 0 for a bivector α that vanishes on Q. This requires tackling three issues: 1. choosing the dimension of the embedding space g × Z, 2. fixing the extension of {·, ·} Q up to first order, and 3. determining the higher-order bivector (α 2 + α 3 + α 4 + . . .) The first two issues are closely related and are largely determined by the simplifying assumption that the Lie algebra g × Z defined by the choice of π 1 = (β 1 + α 1 ) is an abelian extension of the Lie algebra g. The approach to the third issue is very crude: we write α 2 as a bivector in general form and use a computer to calculate [β + α, β + α]. We then set the resulting expression equal to 0 and attempt to find constraints on the coefficients of α 2 . At the start, this method is extraordinarily unfeasible: the potential embedding space is 78-dimensional, making the vector space of quadratic bivectors (18, 270, 252)-dimensional. However, choosing such a large candidate for the embedding space has two advantages: 1. The aim is to merely find one Poisson structure. As g × Z is so large, one may hope there is probably more Poisson structure extending {·, ·} Q . This allows us to merely try to cancel the non-zero terms of [β, β] when choosing α.
2. The large size of g × Z seems to allow the Poisson structure on Q to impose a clear structure on α without being overly restrictive. By choosing g × Z to be an abelian extension of g, all of the relevant structures (g × Z, α, I) inherit a grading from g, and all such structures must be g-compatible as described below.
Thus, by choosing a large embedding dimension, we allow ourselves the freedom to both apply reductions and take advantage of structure imposed by g, in order to restrict the form of α to something computationally manageable. Using the techniques mentioned above, the (18, 270, 252)-dimensional vector space of quadratic bivectors becomes 199-dimensionalsmall enough for a computer to handle.
Remark 7.2.1. The computations in this chapter are done with Macaulay2 [10], Python Scripts, and Maple TM [18]. The majority of the work -including all the invariant theoretic calculations and the Schouten bracket computations -are done using Macaulay2. The Python scripts are used both to construct α 2 according to a set of rules (defined by a grading) and to scan very long expressions for certain coefficients (to aid in eliminating extraneous coefficients). Lastly, Maple TM is used only a handful of times, to symbolically solve systems of linear equations. A detailed explanation of the process and code used is included in Appendix B.

The setup and minimal embedding
As in Chapter 6, for the case of Q = V /Z 3 we will work with the complexification of the vector spaces involved -that is, while the minimal embedding space of Q is R 12 ≃ g * , we will consider the complexification g C ≃ gl 2 (C) ⋉ (C 4 ⊕ C 4 ) when doing calculations -which we will simply denote by g to ease notation. As before, denote the basis of g by g = gl 2 (C) ⋉ (C 4 ⊕ C 4 ) ≃ Span{a 0 , a 1 , a 2 , a 3 , x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 } and the embedding φ * : V /Z 3 ֒→ g * is the one induced by the algebra map φ : Sym(g * ) → Sym(V * ) Z 3 defined on generators by: φ : The previously considered extension β = β 1 + β 2 is given explicitly by Recalling the isomorphism of Lie algebras gl 2 (C) → Span{a 0 , a 1 , a 2 , a 3 } given by {H → (a 0 − a 1 ), E → a 2 , F → a 3 , t → (a 0 + a 1 )} one notes that β 1 has the structure: The quadratic term β 2 has the structure: where the p(x i , y j ) are quadratic in the a i .

The Degree 1 Condition
An explicit embedding of Q = C 3 /Z 3 into g * ≃ C 12 was fixed in the above section. However, this embedding is not a Poisson embedding -in fact, by Theorem 1.1, there cannot exist a Poisson embedding into g * .
Reduction 7.2.2. Following the above section, consider an embedding Q ֒→ (g × Z) * and an abelian Lie algebra extension of g. That is, 1. a g-representation ρ Z : g → End(Z), and 2. a cocycle c : g ∧ g → Z.
These conditions become quite strong when one uses the Levi decomposition g ≃ sl 2 (C)⋉ (C ⊕ C 4 ⊕ C 4 ). We can thus restrict ρ Z to the Levi factor sl 2 (C), and use the natural action of sl 2 (C) on g to deduce: 1. the representation (Z, ρ Z | sl 2 ) is an sl 2 (C)-representation, 2. c : g ∧ g → Z is sl 2 (C)-equivariant, (i.e. a homomorphism of sl 2 (C)-representations). Therefore, a natural starting candidate for an embedding space would be to take and to assume that c : g ∧ g → g ∧ g restricts to scalar multiples of the identity on the irreducible subrepresentations of g ∧ g (see Proposition A.0.2). So as not to confuse the vector spaces Z and g ∧ g, fix the following basis for Z: Z = Span{. . . (aa) ij . . . , (ax) ij . . . , (ay) ij . . . , (xx) ij . . . , (xy) ij . . . , (yy) ij . . .} The choice of notation, while perhaps unfortunate for exposition, was chosen to ease computations in Macaulay2. Such a first order extension yields a bivectors α 1 = α 1 ρ + α 1 c . While the explicit formulae for this bivector is included in Appendix B, the following computations are intended to explain the structure of the formulae and how they are computed: 1. The first term α 1 ρ is defined via the natural representation of g on g ∧ g. For example α 1 ρ (dxy 02 , da 3 ) is defined as follows: 2. The second term α 1 c : g ∧ g → g ∧ g is a multiple of the identity on the irreducible sl 2 (C)-subrepresentations of g ∧ g. For example:
Reduction 7.2.4. If there exists a Poisson structure π = β + α on (g × Z) * with linear Poisson structure π 1 = β 1 + α 1 , then there exists a semi-linearized Poisson structure with respect to the Levi decomposition (sl 2 (C) ⋉ (C ⊕ W ⊕ Z)) * . Therefore, we need only consider π 2 = β 2 + α 2 that are supported on the radical of g × Z. That is, The statement in this reduction only holds for our embedding up to order (n − 1) = 2, by the proof in Lemma 6.2.1. However, our α will conveniently be a second order extension α = α 1 + α 2 .

Reductions of α 2 via H-grading
The most effective reduction in reducing the possible choices of α 2 comes from the Hgrading defined by the natural action of sl 2 (C). The following properties were either proved in Chapter 6 (or may be verified by inspection or Macaulay2): π 1 = β 1 + α 1 preserves the H-grading. β 2 preserves the H-grading.
Reduction 7.2.5. As we are attempting to solve equation (7.1) above, only the summands in α 2 that preserve the H-grading need to be considered.
This observation, along with an analysis of the H-grading on the vanishing ideal I of Q ⊆ g * , will provide the first attempt at determining α 2 . Recall that in degree 2 part of the vanishing ideal I is spanned by: Remark 7.2.6. The sub-sections of the above table actually form irreducible representations of sl 2 (C).
Reduction 7.2.7. Additionally there are degree 2 elements in I of the form: However, these elements do not appear in the coefficients of the jacobiator [β 1 , β 2 ], and therefore will not appear in the coefficients of α 2 .
With the above descriptions of the H-gradings of g×Z and I, we are almost ready to make a first attempt at defining α 2 . However, we will first make a few more preliminary reductions guided by the idea that we are merely trying to cancel the nonzero part of [β 1 , β n−1 ]: Reduction 7.2.8. The coefficients of α 2 are in the vanishing ideal I (and not in Sym(Z)) and α 2 preserves the H-grading.
• The following terms are of the form: as the H-degree of the terms dx i ∧ dy j , dx i ∧ dx j , dy i ∧ dy j are even and C 0 is the only even quadratic element of I in the table above.
• Analogous reasoning implies that the following have odd H-degree: Reduction 7.2.9. Through an analysis of the coefficients of the jacobiator [β 1 , β 2 ], we can improve upon the reduction above: • Elements of the form α 2 (dxx ij , dy k ) and α 2 (dxy ij , dx k ) lie in Span{A l , B l }.

Representations of sl 2 (C)
The purpose of this section is to fix notation concerning the representation theory of the complex semi-simple Lie algebra sl 2 (C). We follow the treatment in [9]. Denote the standard basis of The standard representation of sl 2 (C) is given by ρ 2 : In each dimension, there is exactly one irreducible representation. We denote the (n + 1)dimensional representation by V n with corresponding basis {v 0 , . . . , v n }. The representation ρ n : sl 2 (C) → End(V n ) is given as follows: The Cartan element H acts diagonally, decomposing V n into eigenspaces V n = n i=0 E i , and the action of sl 2 (C) acts on each one dimensional eigenspace: As sl 2 (C) is semi-simple, a finite dimensional representation V is the direct sum of the irreducible representations described above: V = V d i , for d i ∈ N. We will often have sl 2 (C)-equivariant linear maps of two representations and use Schur's Lemma to deconstruct the map: Lemma A.0.1 (Schur). Suppose that V and W are irreducible representations of a complex Lie algebra g and φ : V → W is a g-module homomorphism. Then This implies the following corollary, which we use liberally throughout chapters 6 and 7: Corollary A.0.2. Let V and V ′ be finite dimensional sl 2 (C) representations, and φ : V → V ′ be a g-module homomorphism. Denote the decomposition of V = i V d i and V ′ = j V c j into their irreducible representations. Then φ decomposes as This corollary is often applied to the tensor product, symmetric product, and wedge product of representations: Proposition A.0.3. Let V n be an irreducible representation of sl 2 (C). The decompositions of the following representations into irreducible sub-representations are: Example A.0.4. A simple, yet typical application of this is the case n = 1, then: V 1 ⊗ V 1 = V 2 ⊕ V 0 and V 1 ∧ V 1 = V 0 , and so any sl 2 (C)-equivariant map φ : and sends the complementary subspace to zero. That is, φ : Remark A.0.5. Using Corollary A.0.2 as in the example above, one can actually explicitly determine the extensions up to order (n − 1) in Chapters 6 and 7.

Macaulay2 Computations
This appendix contains an explicit description, using Macaulay2 [10], of the extension of the Poisson structure on V /Z 3 given in Chapter 7 to R 78 . A sketch of the procedure used to construct this Poisson structure is also given (using the syntax/notation of Macaulay2), as it is likely that such a process is applicable to other examples (e.g. V /Z n for other n, or for the (2, 1, 1)-resonance action considered in [6,3]). Throughout, we use Q for the coefficient ring to avoid decimals and issues associated to floating point.