The orbit method for Poisson orders

A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of Spec$(Z)$. In this article we consider a Poisson order $A$ over a complex affine Poisson algebra $Z$. We begin by defining a stratification of the primitive spectrum Prim$(A)$ into symplectic cores, which should be thought of as families of coherent symplectic leaves on a non-commutative space. We define a category $A$-$\mathcal{P}$-Mod of $A$-modules adapted to the Poisson structure on $Z$, and we show that when the symplectic leaves of $Z$ are Zariski locally closed and $Z$ is regular, there is a natural homeomorphism from the spectrum of annihilators of simple objects in $A$-$\mathcal{P}$-Mod to the set of symplectic cores in Prim$(A)$ with its quotient topology. Applications of this result include a classification of annihilators of simple Poisson $Z$-modules when $Z =\mathbb{C}[\mathfrak{g}^*]$ where $\mathfrak{g}$ is the Lie algebra of a complex algebraic group, or when $Z$ is a classical finite $W$-algebra. The homeomorphism is constructed by defining and studying the Poisson enveloping algebra $A^e$ of a Poisson order $A$, an associative algebra which captures the Poisson representation theory of $A$. When $Z$ is a regular affine algebra we prove a PBW theorem for the enveloping algebra $A^e$ and use this to characterise the annihilators of simple Poisson modules in several different ways: we show that the annihilators of simple objects in $A$-$\mathcal{P}$-Mod, the Poisson weakly locally closed, Poisson primitive and Poisson rational ideals all coincide. This last statement can be seen as a semiclassical version of the Dixmier--M{\oe}glin equivalence.


The orbit method
Kirillov's orbit method appears in a wide variety of contexts in representation theory and Lie theory, and is occasionally referred to as a philosophy rather than a theory, on account of the fact that it serves as a guiding principle in many cases where it cannot be formulated as a precise statement. The original manifestation of the orbit method states that characters of simple modules for Lie groups can be expressed as normalisations of Fourier transforms of certain functions on coadjoint orbits [26], but perhaps the most concrete algebraic expression of the orbit method is a well-known theorem of Dixmier which asserts that when G is a complex solvable algebraic group and g = Lie(G), the primitive ideals of the enveloping algebra U (g) lie in natural one-to-one correspondence with the set-theoretic coadjoint orbit space g * /G; see [8,Theorem 6.5.12]. Dixmier's theorem fails for complex simple Lie algebras [16,Remark 9.2(c)]; however, progress has been made recently by Losev [28] using techniques from deformation theory to show that g * /G canonically maps to Prim U (g), and the map is an embedding in classical types. The image consists of a certain completely prime ideals and conjecturally it is always injective.
The Kirillov-Kostant-Souriau theorem asserts that the coadjoint orbits are actually the symplectic leaves of the Poisson variety g * , and so a broad interpretation of the orbit method philosophy is the following: suppose that Z is a Poisson algebra and A is a quantisation of Z, then the primitive spectrum Prim(A) should correspond closely to the set of symplectic leaves of Spec(Z); indeed, a slightly more general principle was suggested by Goodearl in [16, § 4.4]. There are several examples of quantum groups and quantum algebras where the correspondence we allude to here actually manifests itself as a bijection, or better yet a homeomorphism, once the set of leaves is endowed with a suitable topology; the reader should refer to [16] where numerous correspondences of this type are surveyed.

Poisson orders and their modules
In deformation theory, Poisson algebras arise as the semi-classical limits of quantisations, as we briefly recall. If A is a torsion-free C[q]-algebra where q is a parameter, such that A 0 := A/qA is commutative, then A 0 is equipped with a Poisson bracket by setting {π(a 1 ), π(a 2 )} := π(q −1 [a 1 , a 2 ]), (1.1) where π : A → A 0 is the natural projection and a 1 , a 2 ∈ A. Of course, we do not need to assume that A 0 is commutative to obtain a Poisson algebra since formula (1.1) endows the centre Z(A 0 ) with the structure of a Poisson algebra regardless. In fact, something stronger is true: by choosing π(a 1 ) ∈ Z(A 0 ) and π(a 2 ) ∈ A 0 formula (1.1) endows A 0 with a biderivation which restricts to a Poisson bracket {·, ·} : Z(A 0 ) × Z(A 0 ) → Z(A 0 ). In [6] Brown and Gordon axiomatised this structure in cases where A 0 is a Z(A 0 )-module of finite type by saying that A 0 is a Poisson order over Z(A 0 ). The precise definition will be recalled in § 2.2, and a slightly more general approach to constructing Poisson orders in deformation theory will be explained in § 2.3. The bracket (1.2) induces a map H : Z(A 0 ) → Der C (A 0 ) and the image is referred to as the set of Hamiltonian derivations of A 0 . In op. cit., they proved some very attractive general results with the ultimate goal of better understanding the representation theory of symplectic reflection algebras. In this paper, we pursue the themes of the orbit method in the abstract setting of Poisson orders. When Z is a Poisson algebra and A is a Poisson order over Z, we define a Poisson A-module to be an A-module with a compatible action for the Hamiltonian derivations H(Z); see § 2.2. In the case where A = Z, these modules are closely related to D-modules over the affine variety Spec(Z) (see Remark 2.4), and they have appeared in the literature many times (see [1,10,19,27,32], for example). In the setting of Poisson orders, a similar category of modules was studied in [35]. (1. 3) The set of annihilators of simple Poisson A-modules will always be equipped with its Jacobson topology.

Symplectic cores versus annihilators of simple Poisson modules
A primitive ideal of A is the annihilator of a simple A-module and the set of such ideals equipped with their Jacobson topology is called the primitive spectrum, denoted as Prim(A). It is often the case that simple A-modules cannot be classified but Prim(A) can be described completely, which offers good motivation for studying primitive spectra. The Poisson core of an ideal I ⊆ A is the largest ideal P(I) of A contained in I which is stable under the Hamiltonian derivations, and we define an equivalence relation on the set Prim(A) by saying I ∼ J if P(I) = P(J). The equivalence classes are called the symplectic cores of Prim(A), the set of symplectic cores is denoted by Prim C (A) and the symplectic core of Prim(A) containing I is denoted by C(I). We view Prim C (A) as topological space endowed with the quotient topology. In case Z is an affine Poisson algebra such that Spec(Z) has Zariski locally closed symplectic leaves, Proposition 3.6 in [6] shows that the symplectic leaves coincide with the symplectic cores of Spec(Z); see also Proposition 2.7. Thus, the cores of Prim(A) can occasionally be regarded as non-commutative analogues of symplectic leaves. The hypothesis that the symplectic leaves of Spec(Z) are algebraic can be replaced with something strictly weaker; see Remark 6.2. Our theorem has obvious parallels with Joseph's irreducibility theorem [23], which states that for g a complex semisimple Lie algebra and M a simple g-module, the variety {χ ∈ g * | χ(gr Ann U (g) (M )) = 0} contains a unique dense nilpotent orbit. Other closely related results can be found in [13,29], although all of the papers we cite here apply to Poisson structures which have finitely many symplectic leaves -this is a hypothesis we do not require. It is natural to wonder whether our first theorem might serve as a starting point for a new proof of the irreducibility theorem.

First Theorem. Suppose that Spec(Z) is a smooth complex affine Poisson variety with Zariski locally closed symplectic leaves, and A is a Poisson order over Z. For every simple Poisson
In order to illustrate in what sense our theorem is an expression of the orbit method philosophy, we record the following special case where A = Z = C[g * ] is the natural Poisson structure arising on the dual of an algebraic Lie algebra. that there exists a bijection P-Prim(A) ↔ Prim C (A). The first and second steps are really consequences of our second main theorem, which gives a detailed comparison of different types of H(Z)-stable ideals in Poisson orders, as we now explain.

The Poisson-Dixmier-Moeglin equivalence for Poisson orders
Let k be any field, Z be an affine Poisson k-algebra and A a Poisson order over Z. As usual Spec(A) denotes the set of prime two-sided ideals, and we have Prim(A) ⊆ Spec(A). The Poisson ideals of A are the two-sided ideals which are stable under the Hamiltonian derivations H(Z). We write P-Spec(A) for the set of all Poisson ideals of A which are also prime. The set P-Spec(A) endowed with its Jacobson topology is referred to as the Poisson spectrum of A, and the elements are known as Poisson prime ideals of A. If A is prime and Z is an integral domain, then we can form the field of fractions Q(Z), and since A is a Z-module of finite type, the tensor product A ⊗ Z Q(Z) is isomorphic to Q(A) the division ring of fractions of A; in particular Q(A) exists. When I ∈ Spec(A), we have I ∩ Z ∈ Spec(Z) (see [30,Theorem 10.2.4], for example) and so we can form the division ring Q(A/I). If I is a Poisson ideal, then the set of derivations H(Z) acts naturally on A/I, and the action extends to an action on Q(A/I) by the Leibniz rule where δ ∈ H(Z) and a, b ∈ A/I with b = 0. The centre of Q(A/I) will be written CQ(A/I) and we define the Poisson centre of Q(A/I) to be the subalgebra Let I ∈ P-Spec(A) be a Poisson prime ideal. We say that:   When the symplectic leaves of Z are locally closed in the Zariski topology conditions (I) and (II) hold for Z as a Poisson order over itself. If these equivalent conditions hold for Z, then they also hold for A.
The equivalence of (i), (ii) and (iii) is known as the weak Poisson Dixmier-Moeglin equivalence (PDME). It has recently been proven for Poisson algebras by Bell et al. [3], and our approach is to lift the theorem to the setting of Poisson orders using the close relationships between prime and primitive ideals in finite centralising extensions. Part (a) is similarly a well-known fact in the setting of Poisson algebras, and the same proof works here. When the Poisson primitive ideals of a Poisson algebra Z are Poisson locally closed, we say that the PDME holds for Z. Generalising this rubric, we shall say that the PDME holds for A when conditions (I) and (II) hold for A. It was an open question from [6] as to whether every affine Poisson algebra satisfies the PDME; however, recently counterexamples have been discovered [3]. We may now rephrase the last sentence of the second theorem: we have shown that if the PDME holds for a complex affine Poisson algebra Z, then it holds for every Poisson order A over Z.

The universal enveloping algebra of a Poisson order
To conclude our statement of results, it remains to offer some commentary on (b) and (c) of the second main theorem. Statement (c) was proposed in the case A = Z in [32] although the proof contains an error † . The converse was conjectured at the same time and proven in case Z is a polynomial algebra in [33]. Since our results are stated in the setting of Poisson modules over Poisson orders, we require new tools and new methods. Our main technique is to define and study the (universal) enveloping algebra of a Poisson order. This is an associative algebra A e generated by symbols {m(a), δ(z) | a ∈ A, z ∈ Z} subject to certain relations (3.2)-(3.5) such that category A e -Mod of left modules is equivalent to the category of Poisson A-modules. Using this construction, we are able to define localisation of Poisson modules over Poisson orders, which is our main tool in proving part (b) of the second main theorem. In order to prove part (c), we show that when Z is regular, A e is a free (hence faithfully flat) A-module (Corollary 3.14), which implies that the ideals of A e are closely related to the ideals of A (cf. Lemma 4.1). The fact that A e is A-free follows quickly from our last main theorem of this paper, which we view as a Poincarée-Birkhoff-Witt (PBW) theorem for the enveloping algebras of Poisson orders. There is a natural filtration A e = i 0 F i A e , defined by placing generators {m(a) | a ∈ A} in degree 0 and {δ(z) | z ∈ Z} in degree 1, which we call the PBW filtration of A e . The associated graded algebra is denoted by gr A e . The statement and proof of our third and final main theorem are quite similar to Rinehart's PBW theorem for Lie algebroids [34].
Third Theorem. Suppose that Z is affine and regular over a field. Then the natural surjection induced by multiplication in gr A e is an isomorphism.

Structure of the paper
We now describe the structure of the current paper. In § 2 we state the definition of a Poisson order: our definition is very slightly different to the one originally given in [6], although a careful comparison is provided in Remark 2. , and prove the equivalence of (I) and (II), as well as the subsequent two assertions of second main theorem. In § 3 we introduce the enveloping algebra of a Poisson order. We state the universal property in § 3.1 and prove a criterion for A e to be noetherian. In § 3.2 we use A e to define and study localisations of Poisson A-modules, whilst in § 3.3 we prove the PBW theorem and state some useful consequences. In § 4 we prove (b) and (c) of the second theorem using the tools developed in of § 3. In § 5 we prove (a) and the equivalence of (i), (ii) and (iii) in the second main theorem. Following [15], we observe that results, such as the PDME, can be studied in the slightly more general context of finitely generated algebras equipped with a set of distinguished derivations, and it is in this setting that we prove the results of § 5. Finally, in § 6 we show that the second theorem implies the first. In § 6.3 we make a careful comparison between Dixmier's bijection g * /G → Prim U (g) and our bijection {annihilators of simple Poisson C[g * ]-modules} → g * /G in the case where g is a solvable, and finally in § 6.4 we discuss some famous examples of Poisson orders arising in deformation theory to which our first main theorem can be applied. We conclude the article by posing some questions about their Poisson representation theory.

A discussion of related results and new directions
It is worth mentioning that our first main theorem is very close in spirit to a conjecture of Hodges and Levasseur [21, § 2.8, Conjecture 1] which seeks to relate the primitive spectrum of the quantised coordinate ring of a complex simple algebra group O q (G) in the case where q is a generic parameter, to the Poisson spectrum of the classical limit O(G); see [16, § 4.4] for a survey of results. Although the spectra are always known to lie in natural bijection, this bijection is only known to be a homeomorphism in case G = SL 2 (C) and SL 3 (C) [12]. By contrast, our bijection is always a homeomorphism, however our results only apply to these families of algebras when the parameter is a root of unity. It would be natural to attempt to strengthen this comparison. Although our results are fairly comprehensive we expect that part (c) of the second main theorem should hold without the hypothesis that Z is regular, and so the first main theorem should hold true without assuming Spec(Z) is smooth. Note that the symplectic leaves of a singular Poisson variety can be defined, thanks to [6, § 3.5]. This would constitute an extremely worthwhile development, as there are important examples of Poisson orders over singular Poisson varieties, for example, rational Cherednik algebras. At least this should be achievable for Poisson orders over isolated surface singularities using the methods of [27, § 3.4] along with our proof of Theorem 4.2, which only depends upon the PBW theorem for A e .
Another motivation for this work is the following: there appear to be deep connections between the dimensions of simple modules of a Poisson order A over Z and the dimensions of its symplectic leaves of Z. We expect that the Poisson representation theory of A will be closely related to the representation theory of A, and so the current paper will lay the groundwork for such relationships to be understood in a broader context.

Notations and conventions
For the first and second sections, we let k be any field whilst in subsequent sections we shall work over C. When the ground field is fixed, all vector spaces, algebras and unadorned tensor products will be defined over this choice of field.
When we say that A is an algebra, we mean a not necessarily commutative unital k-algebra. When we say that A is affine, we mean that it is semiprime and finitely generated. By an A-module we mean a left module, unless otherwise stated. By a primitive ideal we always mean the annihilator of a simple left A-module. The category of all A-modules is denoted by A -Mod and the subcategory of finitely generated A-modules is denoted by A -mod.
When we say that A is filtered, we mean that there is a non-negative Z-filtration As usual, the associated graded algebra of a filtered algebra is gr A := Furthermore, A is said to be almost commutative if gr A is commutative.

Poisson orders and their modules
A Poisson algebra is a commutative algebra Z endowed with a skew-symmetric k-bilinear biderivation {·, ·} : Z × Z → Z which makes Z into a Lie algebra. Let A be a Z-algebra which is a module of finite type over Z. We say that A is a Poisson order (over Z) if the Poisson bracket on Z extends to a map  [6] is slightly weaker than the one given here, as they only assume property (ii) of H in the case where x, y, a ∈ Z. Our justification for choosing this definition is twofold: firstly, the most interesting examples which arise in deformation theory satisfy these slightly stronger properties; see § 2.3. Secondly the stronger definition suggests a stronger definition for a Poisson A-module, and the enveloping algebra for this category of modules satisfies the PBW theorem of § 3, which is fundamental to all of our results.
When Z is a fixed Poisson algebra and A is a Poisson order over Z, we define a Poisson A-module to be an A-module M together with a linear map such that for all x, y ∈ Z, all a ∈ A and all m ∈ M , we have The morphisms of Poisson A-modules are defined in the obvious manner, and the category of all Poisson A-modules will be denoted by A-P -Mod. Since Poisson A-modules are Poisson Z-modules by restriction, we are considering a special class of flat Poisson connections.
Remark 2.2. It is not true that simple Poisson A-modules are necessarily finitely generated over A. For example, when A = Z = C[g * ] and g is a simple Lie algebra, it is not hard to see that simple Poisson A-modules annihilated by the augmentation ideal (g) A are the same as simple g-modules, and these are often infinite-dimensional. We thank Ben Webster for this useful observation.

Examples of Poisson orders and their modules
Every Poisson algebra is a Poisson order over itself. Furthermore, for Z fixed there are several constructions which allow us to construct new Poisson orders over Z from old ones. Let A be a Poisson order over Z. Then: (1) Mat n (A) is a Poisson order for any n > 0, with (2) the opposite algebra A opp is a Poisson order; (3) the tensor product A ⊗ Z B of two Poisson orders is again a Poisson order with The above constructions are very suggestive of a theory of a Brauer group over Z adapted to the theory of Poisson orders. This is a theme we hope to pursue in future work.
All other examples of Poisson orders which we will be interested in arise in the context of deformation theory. We follow [24] closely. Let R be a commutative associative algebra and let ( ) ⊆ R be a principal prime ideal, and write k = R/( ). Consider the k-algebra A := A/ A with centre Z and write N for the preimage of Z in A under the natural projection π : A A . For a ∈ N and b ∈ A, we have [a, b] ∈ A and so we may define generalising (1.1). When A is finite over Z , the bracket (2.2) makes Z into a Poisson algebra. If Z ⊆ Z is any Poisson subalgebra such that A is a Z-module of finite type, then A becomes a Poisson order over Z. Notable examples include.
(1) When R = Z, = p ∈ Z is any prime number and g Z is the Lie algebra of a Z-group scheme, then U (g Z )/pU (g Z ) ∼ = U (g p ) where g p := g Z ⊗ Z F p and F p := Z/(p). It is well known that g p is a restricted Lie algebra and the calculations of [24] show that the p-centre Z p (g) is a Poisson subalgebra of the centre of U (g p ), naturally isomorphic to F p [g * p ] with its Lie-Poisson structure. Since U (g p ) is finite over the p-centre, we see that U (g p ) is a Poisson order over Z p (g).
(2) Let R = C[t ±1 ], = (t − q 0 ) for some primitive th root of unity q 0 ∈ C, and A is any of the following: a quantised enveloping algebra of a complex semisimple Lie algebra, a quantised coordinate ring of a complex algebraic group, any quantum affine space. It is well known that the th powers of the standard generators of A generate a central subalgebra Z 0 over which A is a finite module, and (2.2) equips Z 0 with the structure of a complex affine Poisson algebra, so A is a Poisson order over Z 0 .
We continue with A a Poisson order over Z and list some elementary examples of Poisson modules. The first example of a Poisson A-module is A, with map ∇ defined by ∇(z)a := {z, a}.
If I A is any left ideal which is also Poisson, then both I and the quotient A/I admit the structure of a Poisson A-module. A natural way to construct such ideals is to consider those of the form AI where I Z is any Poisson ideal. A method for constructing Poisson A-modules from Poisson Z-modules occurs as a special case of the following crucial lemma.
Proof. To see that ∇ A is well defined, we must check that the kernel of the natural For the rest of the proof, tensor products a ⊗ m will be taken over B. The first axiom of a Poisson A-module follows from the calculation where x, y ∈ Z, a ∈ A and m ∈ M . The second axiom of a Poisson module is a consequence of the next calculation, in which a, b ∈ A and x, m are as before The third axiom of a Poisson module only regards the Lie algebra structure and so follows from the Hopf algebra structure on the universal enveloping algebra of the Lie algebra Z, since A and M are Poisson Z-modules.
Remark 2.4. It was observed in [10, Proposition 1.1] that when Z is a symplectic affine, Poisson algebra over C every Poisson Z-module arises from a unique D-module on Spec(Z).

Symplectic cores in primitive spectra
We continue with an affine Poisson k-algebra Z and a Poisson order A over Z. If S is any collection of ideals of A, then we can endow S with the Jacobson topology by declaring the sets {I ∈ S | J∈S J ⊆ I} to be closed, where S ⊆ S is any subset. We will refer to such a set S as a space of ideals to suggest that we are equipping it with the Jacobson topology. The spaces of prime ideals and primitive ideals of A are denoted as Spec(A) and Prim(A), respectively. A ring is Jacobson if every prime ideal is an intersection of primitive ideals; clearly this property is equivalent to the statement that Prim(A) is a topological subspace of Spec(A), not just a subset. It is well known that Z is Jacobson, since it is affine and commutative, and so it follows from [30, 9.1.3] that A is a Jacobson ring.
The H(Z)-stable ideals of Z and A are called Poisson ideals and the space of prime Poisson ideals is called the Poisson spectrum, denoted as P-Spec(Z) and P-Spec(A), respectively. Recall that for any ideal I ⊆ A, the Poisson core P(I) is the largest Poisson ideal contained in I; by [8, 3.3.2], we have P(I) prime whenever I is prime, and the same holds for Z.
Lemma 2.5. Let A be a Poisson order over Z and I ⊆ J ⊆ A are any ideals with I Poisson. Denote the quotient map π : A → A/I. We have: Proof. To prove (i), it suffices to observe that P(J ∩ Z) ⊆ P(J) ∩ Z ⊆ J ∩ Z, by the definition of P, whilst (ii) follows from the fact that π defines an inclusion preserving bijection between the set of Poisson ideals of A/I and the set of Poisson ideals of A which contain I.
Remark 2.6. It is not hard to see that the topology on P-Spec(A) is the subspace topology from the embedding P- Our purpose now is to define the symplectic stratification of the primitive spectrum Prim(A) used in the statement of the first main theorem. Consider the following diagram, which is commutative by part (i) of Lemma 2.5: The vertical arrows denote contraction of ideals I → I ∩ Z. The fibres of the map Prim(Z) → P-Spec(Z) are called the symplectic cores of Prim(Z), and they were first studied by Brown and Gordon in [6]. We define the symplectic cores of Prim(A) to be the fibres of the map Prim(A) → P-Spec(A). For m ∈ Prim(Z) we write C(m) for the symplectic core of m, and for I ∈ Prim(A) we write C(I) for the symplectic core of I. The following result shows that the symplectic cores of Prim(Z) are closely related to the symplectic leaves; the first part was proven in [6, Proposition 3.6], and the second statement in [16,Theorem 7.4(c)].
where the union is taken over all n ∈ Prim(Z) such that L(n) C(m).
Thus, we think of the symplectic cores of Prim(A) as being something similar to the symplectic leaves of the primitive spectrum. If the Poisson primitive ideals of A are Poisson locally closed, then we say that the PDME holds for A. Later on in the paper (Lemma 5. Furthermore, if the PDME holds for Z as a Poisson order over itself, then it also holds for A. Proof. If I ∈ Prim(A) then, using Lemma 5.2, {P(I)} is a locally closed subset of P-Prim(A) if and only if the intersection N I properly contains P(I), so (i) ⇔ (ii). We point out that the lemma just cited does not depend on any of the results of this article which precede it, and follows straight from the definitions. It is not hard to see that for I ∈ Prim(A), we have if and only if N I = P(I), from which the equivalence of (ii) and (iii) follows. Now suppose that P(I ∩ Z) P(m) where the intersection is taken over all ideals m ∈ Prim(Z) such that P(I ∩ Z) P(m). Using Lemma 2.5, we deduce that P( where the intersection is taken over all m ∈ Prim(Z) such that I ∩ Z P(m). By the incomparability property over essential extensions [11,Theorem 6.3.8], we see that P(I) P(J) implies P(I ∩ Z) P(J ∩ Z) and so from P(m) ⊆ N I ∩ Z, we deduce that P(I) N I . We conclude from (iii) ⇒ (i) that the PDME holds for A.

The universal enveloping algebra of a Poisson order
Throughout this entire section, we work over an arbitrary field k. Let Z be a Poisson k-algebra and let A be a Poisson order over Z.

Definition and first properties of the enveloping algebra
Poisson A-modules can be thought of as modules over a non-associative algebra due to the action of the derivations ∇(Z), and one encounters elementary technical problems with dealing with such modules. For example, if M is a simple Poisson A-module, then it is not necessarily finitely generated over A (cf. Remark 2.2); this contrasts with the situation for simple A-modules where any such module is generated by any non-zero element. To remedy this problem we take a viewpoint which is common in universal algebra: we write down an associative algebra whose module category is equivalent to A-P -Mod and we use this new algebra to study simple Poisson A-modules and their annihilators.
The Poisson enveloping algebra A e of the Poisson order A over Z is the k-algebra with generators and relations α : A → A e is a unital algebra homomorphism; for all x, y ∈ Z and all a ∈ A. Recall that the Poisson algebra Z is a Poisson order over itself and we write Z e for the enveloping algebra of Z. The algebra Z e has been extensively studied in the mathematical literature, although the first results appeared in [34], since Poisson algebras are examples of Lie-Rinehart algebras. Our next observation follows straight from the relations.
There is a natural homomorphism Z e → A e which sends the elements {α(z), δ(z) | z ∈ Z} of Z e to the elements of A e with the same names.
Next, we record some criteria for A e to satisfy the ascending chain condition on ideals.
Proof. The Artin-Tate lemma shows that when A is finitely generated so too is Z, and so Z is noetherian. It suffices to prove that when Z is noetherian so too are A and A e . The extension Z ⊆ A is centralising in the sense of [30, 10.1.3] and so Corollary 10.1.11 of that book shows that A is noetherian. Now by the relations the map of rings α : A → A e is almost normalising in the sense of [30, 1.6.10] and so the lemma follows from Theorem 1.6.14 of the same book.
When a ∈ A e we write ad(a) for the derivation of A e given by b → ab − ba. The following two statements can be proven by induction on (3.4) and (3.5), respectively. Lemma 3.3. For x 1 , . . . , x n ∈ Z and a ∈ A, we have: We define a filtration on A e by placing A in degree 0 and δ(z) in degree 1 for all z ∈ Z. We call the resulting filtration the PBW filtration on A e and as usual we denote the associated graded algebra by gr A e . One of our main tools in this paper is a precise description of gr A e , which we give in Theorem 3.12. For now we record a precursor to that result which will be needed when describing localisations of Poisson modules.
where i 1 , . . . , i n ∈ I and j 1 · · · j m lie in J. The same statement holds with the elements α(a) occurring after the elements δ(x).
Proof. It follows from relations (3.2) and part (ii) of Lemma 3.3 that the algebra A e is generated by the set {α(a i ), δ(x j ) | i ∈ I, j ∈ J}. Therefore, the lemma will follow from the claim that gr Z e is central in gr A e . This is clear upon examining the top graded components of relations (3.3) and (3.4).
We now record the universal property of A e which allows us to view Poisson A-modules as A e -modules. Consider the category U whose objects are triples (B, α , δ ) where B is an associative algebra with unital algebra homomorphism α : A → B and Lie algebra homomorphism δ : Z → B satisfying (3.4) and (3.5), and where the morphisms (B, α , δ ) → (C, α , δ ) between two objects in U are the algebra homomorphisms β : B → C making the diagrams below commute. (1) (A e , α, δ) is an initial object in the category U ; (2) There is a category equivalence Remark 3.6. We may now define the category A-P -Mod, of finitely generated Poisson A-modules, to be the essential image of A e -mod in A-P -Mod under the above equivalence.

Localisation of Poisson A-modules
It is well known that if S ⊆ Z is any multiplicative subset containing no zero divisors, then the localisation Z S := Z[S −1 ] carries a unique Poisson algebra structure such that the natural map Z → Z S is a Poisson algebra homomorphism. Briefly, this structure is defined by extending the Hamiltonian derivations to Z S via (1.4). The reader may refer to the proof of [25,Lemma 1.3] for the precise formula. In the same manner, when a multiplicative set S ⊆ Z consists of nonzero divisors of A, the algebra A S := A ⊗ Z Z S carries a unique structure of a Z S -Poisson order and A S is a faithful Z S -module. Let A e S denote the Poisson enveloping algebra of A S .
Now we use the following characterisation of simple A e -modules: they are precisely the modules which are generated by any non-zero element. Pick 0 = s −1 m ∈ M S and let t −1 n be any other element. Since M is a simple A e -module, there is a ∈ A e such that am = n, and it follows that t −1 as(s −1 m) = t −1 n. Hence M S is generated as an A e S -module by any non-zero element, and so M S is a simple Poisson A S -module as required.
Remark 3.11. When p ∈ Spec(Z), we adopt the usual convention of writing A p and Z p for the localisations A S\p and Z S\p . When z ∈ Z \ {0} is not nilpotent, we write A z and Z z for the localisations at the multiplicative set {z i | i 0}.

A Poincaré-Birkhoff-Witt theorem for the enveloping algebra
Our present goal is to describe the associated graded algebra gr A e with respect to the PBW filtration (3.6) in the case where Z is a regular Poisson algebra, that is, when Spec(Z) is a smooth affine variety. Let Ω := Ω Z/k denote the Z-module of Kähler differentials for Z; see [20,Chapter II,§ 8] for an overview. The relations in the enveloping algebra imply that there is a natural map A ⊗ Z S Z (Ω) → gr A e which is surjective. The PBW theorem for Poisson orders takes the following form.
Theorem 3.12. Suppose that Z is a regular, affine Poisson algebra over an algebraically closed field k. The following hold.
(1) The natural surjective algebra homomorphism gr A e is an isomorphism.
(2) There is an isomorphism of (A, Z e )-bimodules (3) There is an isomorphism of (Z e , A)-bimodules The proof will occupy the rest of the current subsection. The approach is modelled on that of [34] where a similar result was proven for enveloping algebras of Lie-Rinehardt algebras. We first prove the theorem in the case where Ω is a finitely generated free Z-module and then use localisation of Poisson orders to deduce the theorem in the case where Ω is locally free, that is, projective. By [20,Theorem 8.15], we know that Ω is a projective Z-module if and only if Z is regular, from which we will conclude the theorem.
Suppose that Ω is a free Z-module of finite type, so there exist z 1 , . . . , z n ∈ Z such that d(z 1 ), . . . , d(z n ) is a basis for Ω. Therefore the symmetric algebra S Z (Ω) is free over Z and the ordered products d(z I ) := d(z i1 ) · · · d(z im ) with i 1 · · · i m provide a basis. When I is a sequence 1 i 1 · · · i m n we write |I| = m and write j I if j i 1 .

Lemma 3.13. Let Ω be a free Z-module with a finite basis. There is a Poisson
whenever j I.
Proof. It was first observed by Huebschmann that when Z is a Poisson algebra, Ω carries a natural Lie algebra structure and that (Z, Ω) is a Lie-Rinehardt algebra; see [22,Theorem 3.11]. Therefore we may apply the first part of the proof of [34, Theorem 3.1] to deduce that S Z (Ω) carries a Poisson Z-module structure satisfying (3.11), provided that Ω is free. Using Lemma 2.3 we see that A ⊗ Z S Z (Ω) carries the required Poisson A-module structure.
Proof of Theorem 3.12. We start by proving the statement of part (1) of the theorem; however, for the moment we replace the hypothesis that Z is regular with the assumption that Ω is a free Z-module of finite rank. We adopt the notation introduced preceding Lemma 3.13 so that z 1 , . . . , z n is a basis for Ω over Z, and we write We have A = F 0 A e ∼ = F 0 A e /F −1 A e ⊆ gr A e and so gr A e is a left A-module. We need to show that the set spans a free left A-submodule of gr A e . Observe that the Poisson A e -module structure defined in Lemma 3.13 makes T := A ⊗ Z S Z (Ω) into a filtered A e -module, and so gr(T ) ∼ = T is a graded gr A e -module. We denote the operation gr A e ⊗ k T → T by u ⊗ a → u · a. Thanks to (3.11) the map ψ : gr A e → T defined by u → u · (1 ⊗ 1) sends δ(z I ) to 1 ⊗ d(z 1 ) i1 · · · d(z n ) in for I = (i 1 , . . . , i n ). Since ψ is A-equivariant and the image of (3.12) is A-linearly independent, we deduce that (3.12) is A-linearly independent, as claimed. This proves part (1) in the case where Ω is a free Z-module. Now we suppose that Z is regular. Then it follows from [20,Theorem 8.15] that Ω is a locally free Z-module in the sense that there is a function r : Spec Z → N 0 such that as Z-modules, for all p ∈ Spec Z. By the previous paragraph, we deduce that the natural is an isomorphism. This shows that there is a commutative diagram of algebra homomorphisms:

We point out that the natural map
this is a special case of the very general statement that a Z-module M embeds in the product of the localisations over Spec Z. We deduce from the diagram that the natural map gr A e is an injection, hence an isomorphism as required. We now prove (2). There is a surjective homomorphism of (A, Z e )-bimodules Here, we view A as a subalgebra of A e as explained in Remark 3.8 and Z e → A e is the map described in Lemma 3.1. The kernel of φ is an A-linear dependence between the ordered monomials δ(z I ) in A e but by part (1) we know that all such dependences are trivial, whence (2). Part (3) follows by a symmetrical argument.
We now list some results which follow easily from Theorem 3.12. We thank the referee for pointing out the proof of freeness in part (iv) of the following result.
Corollary 3.14. Suppose that Z is regular and affine. Then the following hold.

(i) The natural map Z e → A e from Lemma 3.1 is an inclusion. (ii) If A is a free Z-module, then A e is a free (left and right) Z e -module. (iii) If A is a projective Z-module, then A e is a projective (left and right) Z e -module. (iv) A e is a free (left and right) A-module, hence A e is projective and faithfully flat over A.
Proof. The PBW theorems for Z e and A e show that the map Z e → A e is injective on the level of associated graded algebras, proving (i). Part (ii) follows from parts (2) and (3) of the PBW theorem, whilst (iii) is an application of Hom-tensor duality. Part (iv) requires slightly more work, and we begin by showing that A e is a countable direct sum of projective A-module. As we noted many times previously, when Z is regular and affine, we have Ω finitely generated and projective. Write S Z (Ω) = k 0 S k Z (Ω) for the Z-module decomposition into symmetric powers. Since projective modules of finite type are retracts of finite rank free modules, and since symmetric powers S k Z preserve retracts and free modules of finite rank, we see that S k Z (Ω) is projective of finite type for all k 0. If S k Z (Ω) ⊕ Q k = Z n(k) for Z-modules {Q k | k 0} and integers {n(k) ∈ N | k 0}, then we see splits for all k 0, which implies that A e is a direct sum of projective (left) A-modules, hence projective. In this last deduction we have used the fact that F 0 A e ∼ = A is a projective A-module.
A symmetrical argument shows that A e is projective also as a right A-module.
We have actually shown that A e is a direct sum of countably many projective A-modules. It follows that if I ⊆ A is a two-sided ideal, then A e /IA e is not finitely generated as an A-module.
In the language of [2], we have that A e is an ℵ 0 -big projective A-module. By Lemma 3.2 we know that A is noetherian, so A e satisfies the hypotheses of [2, Corollary 3.2] and A e is a free left A-module; by symmetry, it is also free as a right A-module. Faithful flatness follows immediately.

Poisson primitive ideals versus annihilators of simple Poisson A-modules
In this section, we shall prove parts (b) and (c) of the second main theorem which relate the Poisson primitive ideals of a Poisson order to the annihilators of simple modules. For the rest of the paper, the ground field will be the complex numbers C.

Poisson primitive ideals are annihilators
Let Z be a complex affine Poisson algebra and let A be a Poisson order over Z. We write  Proof. The first part follows from part (iv) of Corollary 3.14 and [4, Chapter I, § 3, No. 5, Proposition 8]. For I ∈ I P (A), we have φ(I) := IZ e = Z e I ∈ I(A e ), thanks to relation (3.4) in A e . Furthermore, when J ∈ I(A e ), we see that the derivation ad(δ(z)) stabilises ψ(J) := J ∩ A ⊆ A e for all z ∈ Z. Using (3.4) again, the latter assertion is equivalent to saying that ψ(J) ∈ I P (A).
We now prove part (c) of the second main theorem. Proof. Since I is primitive, there is a maximal left ideal L ⊆ A such that I = Ann A (A/L ). We consider the left ideal A e L ∈ I l (A e ) containing A e I and observe that, by Zorn's lemma, there is a maximal left ideal L ∈ I l (A e ) containing A e L and the quotient A e /L is a simple left A e -module. Since L is a proper ideal of A e , it follows that L ∩ A is a proper left ideal of A. By part (ii) of Lemma 4.1 we have L = A e L ∩ A ⊆ L ∩ A and so the maximality of L implies that (4.1) The annihilator Ann A e (A e /L) is the largest two-sided ideal contained in L, and we claim that Ann A e (A e /L) ∩ A = P(I). If we can show that P(I) (1) ⊆ Ann A (A e /L) (2) ⊆ I, then the claim will follow, since we know that P(I) is the largest Poisson ideal contained in I,

Annihilators are Poisson rational
In order to prove part (b) of the second main theorem, we make a more detailed study of the torsion subset of a simple module.  as desired, proving (ii). Set I := Ann Z (M ). We have I ⊆ P(T (M )) and we now prove that this is an equality. According to [8, 3.3

.2], we have
According to (4.4), this set is equal to We claim that (4.6) is equal to {z ∈ Z | z∇(x 1 ) · · · ∇(x n )m 0 = 0 for all x 1 , . . . , x n ∈ Z, n 0}, (4.7) where ∇ : Z → End C (M ) is the structure map of the module M . To prove they are equal, we define . , x n ∈ Z, k n 0}; J k := {z ∈ Z | z∇(x 1 ) · · · ∇(x n )m 0 = 0 for all x 1 , . . . , x n ∈ Z, k n 0}, and we show that I k = J k for all k 0. The case k = 0 is trivial and so we prove the case k > 0 by induction. By part (i) of Lemma 3.3 and part (iii) of Lemma 3.5, we have The right-hand side of (4.8) is a sum of expressions of the form where {j 1 , . . . , j n } = {1, . . . , n} and 0 p n, and there is a unique summand in (4.8) with p = 0, in which case (j 1 , . . . , j n ) = (n, n − 1, . . . , 1). If z∇(x 1 ) · · · ∇(x n )m 0 = 0 for all x 1 , . . . , x n ∈ Z, k n 0, then it follows immediately that (4.8) vanishes, whence J k ⊆ I k . Conversely, if z ∈ I k , then z ∈ J k−1 by the inductive hypothesis, and so we deduce z∇(x k ) · · · ∇(x 1 )m 0 = 0 for all x 1 , . . . , x k ∈ Z from our description of the summands occurring in (4.8). This shows that I k ⊆ J k . Since (4.6) is equal to k 0 I k and (4.7) is equal to k 0 J k , we have proven that P(T (M )) is given by (4.7). It follows that this ideal annihilates Z e m 0 . By Lemma 3.4 we see that M = A e m 0 = AZ e m 0 . If z ∈ P(T (M )), then since z is central in A we have zM = A(zZ e m 0 ) = 0 and we have shown that P(T (M )) = I. This proves (iii).
We are ready to prove part (b) of the second main theorem. Let ab −1 ∈ C P Q(A/I) (defined in (1.5)) with b ∈ Z/Z ∩ I. We claim that ab −1 has a representative such that b / ∈ T (M ). To see this, suppose that b ∈ T (M ) and consider the ideal It is not hard to see that: This shows that I is rational and completes the proof.
Remark 4.5. (i) It seems credible that the hypothesis Z is regular can be removed from Theorem 4.2; see the remarks in § 1.7 for a suggested approach in some special cases.
(ii) The hypothesis that Z is affine is necessary in the statement of Theorem 4.4, as the following example shows. We are grateful to Sei-Qwon Oh for explaining this to us, and permitting us to reproduce it here. Let

The weak Poisson-Dixmier-Moeglin equivalence
Once again the ground field is C. In this section we prove (a) and the equivalence of (i), (ii) and (iii) from the second main theorem.

Δ-ideals in Δ-algebras
It will be convenient to work in a slightly more general context than the setting of Poisson orders over affine algebras: we do not need to assume that the derivations H(Z) arise from a Poisson structure in order to state and prove that Poisson weakly locally closed, Poisson primitive and Poisson rational ideals all coincide. We proceed by stating all of the notations needed, which will remain fixed throughout the current section.
Let A be a finitely generated semiprime noetherian C-algebra which is a finite module over some central subalgebra Z. By the Artin-Tate lemma, it follows immediately that Z is an affine algebra. The centre of A will be written C(A) for the current section. We continue to denote the primitive and prime spectra of A by Prim(A) and Spec(A), endowed with their Jacobson topologies (cf. § 2.4).
We fix for the entire section an arbitrary subset Δ ⊆ Der C (A) such that Δ(Z) ⊆ Z, and we remark that we do not need to assume that Δ is a Lie algebra, or even a vector space in what follows. When I is any subset of A, we write Δ(I) ⊆ I whenever δ(I) ⊆ I for all δ ∈ Δ. We say that an ideal I of A is a Δ-ideal if Δ(I) ⊆ I. For every ideal I ⊆ A, we consider the Δ-core of I, denoted as P Δ (I), which is the unique maximal two-sided Δ-ideal of A contained in I. It is easy to see that such an ideal exists and is unique since it coincides with the sum of all Δ-ideals contained in I. The Δ-primitive ideals of A are the ideals An ideal is called Δ-prime if whenever J, K are Δ-ideals satisfying JK ⊆ I we have J ⊆ I or K ⊆ I. The Δ-spectrum of A is the space of all Δ-prime ideals, equipped with the Jacobson topology, denoted by Spec Δ (A).
Lemma 5.1. The following hold.
(2) If I is a Δ-ideal and I 1 , . . . , I n are the minimal prime ideals over I, then I 1 , . . . , I n are Δ-ideals.
(  The following result was first proven for Poisson algebras in [32]. Proof. Let I be Δ-locally closed. By [30, Lemma 9.1.2(ii), Corollary 9.1.8(i)] we know that A is a Jacobson ring, and so I = s∈S I s for some index set S where each I s is a primitive ideal of A. From part (4) of Lemma 5.1, we deduce that I = s∈S P Δ (I s ). Now by Lemma 5.2 it follows that I = I s for some s ∈ S, hence I is Δ-primitive.
We now prove (2), so suppose that I = P Δ (J) is Δ-primitive, and that J ⊆ A is primitive.
Since J is primitive, Dixmier's lemma tells us that J 0 is a maximal ideal of Z. Since A is a finite module over Z, it follows that A/J 0 A is finite-dimensional over C, thus C is the only subfield of A/J 0 A. Hence once we have proven the existence of such an embedding, the lemma will be complete.
After replacing A by A/I we can assume that P Δ (J) = 0 and show that C Δ (Q(A)) → A/J 0 A. Since A is prime and finite over its centre, we have Q(A) ∼ = A ⊗ Z Q(Z) (cf. § 1.4) and this isomorphism is Δ-equivariant. Now if a ⊗ z −1 ∈ C Δ (A ⊗ Z Q(Z))), then by (1.4) we have a ⊗ z −1 = δ(a) ⊗ δ(z) −1 for all δ ∈ Δ. Since J contains no non-zero Δ-ideals, we can use this observation repeatedly to find a representative a 1 ⊗ z −1 1 of a ⊗ z −1 such that z 1 / ∈ J. In other words, we have a ⊗ z −1 ∈ A ⊗ Z Z J0 where Z J0 denotes the localisation of Z at the prime J 0 Z. We have and so there is a map The composition is necessarily an embedding since C Δ (Q(A)) is a field.

The Δ-rational ideals are the Δ-primitive ideals
Now we suppose that S ⊆ Z is a multiplicative subset, so that the localisation ZS −1 may be defined. Notice that S is also a multiplicative subset of A and the ring AS  The lemma leads directly to a crucial proposition, which is probably well known.
Proposition 5.5. The following hold.
(1) Extension and contraction of ideals define inverse bijections between the sets Spec S (A) and Spec(AS −1 ).
(3) When AS −1 is countably generated, these bijections also preserve the set of all primitive ideals.
Proof. It is a fact, easily checked using part (i) of the previous lemma, that contraction through a central extension of rings preserves prime ideals. For the reader's convenience, we check that extension sends Spec S (A) to Spec(AS −1 ). Pick P ∈ Spec S (A) and suppose that IJ ⊆ P e . Then I c J c ⊆ P ec = P by parts (ii) and (iii) of the previous lemma, and so we may assume I c ⊆ P by primality. Using part (i) of that same lemma once again I = I ce ⊆ P e and so ideals in Spec S (A) extend to Spec(AS −1 ) as claimed. Now apply all three parts of the previous lemma to deduce that extension and contraction are inverse bijections on prime ideals, proving (1) of the current proposition. The fact that the Δ-ideals are preserved is an immediate consequence of the Leibniz rule for derivations. The statement regarding primitive ideals requires slightly more work, as we now explain.
Suppose that M is a simple A-module with I = Ann A (M ) satisfying I ∩ S = ∅. We claim that MS −1 := M ⊗ Z ZS −1 is a simple non-zero AS −1 -module. The kernel of map M → M [S −1 ] consists of m ∈ M such that sm = 0 for some s ∈ S. If such an m = 0 exists, then M = Am and so sM = 0, meaning s ∈ I ∩ S. Since this is not the case, MS −1 is nonzero and we conclude that it is also simple over AS −1 , by essentially the same argument as we used in the proof of Proposition 3.10. What is more, the reader can easily verify that Ann A (M ) = A ∩ Ann AS −1 (MS −1 ). This shows that every primitive ideal in Spec S (A) is equal to J ∩ A for some J ∈ Prim(AS −1 ).
We claim that whenever M is a simple AS −1 -module, there exists a simple A-submodule N ⊆ M . To show that such a simple A-submodule N ⊆ M exists, we observe that AS −1 is a countably generated C-algebra and hence it satisfies the endomorphism property [30, Proposition 9.1.7]. Since A is a finite module over Z, it follows that AS −1 is a finite ZS −1 -module, say certain elements a 1 , . . . , a t ∈ AS −1 . Therefore, for any 0 = m ∈ M , Recall that we say that A is an essential extension of Z, provided that every non-zero ideal of A intersects Z non-trivially.
Lemma 5.6 [11,Theorem 6.3.8]. If A is a prime C-algebra and a finite extension of a central subalgebra Z, then A is an essential extension of Z.
Proof. We suppose that J is a Δ-prime ideal, that C Δ (Q(A/J)) = C and we aim to find a primitive ideal I ⊆ A with P Δ (I) = J. After replacing A by A/J and replacing Z by Z/Z ∩ J, we see that it is sufficient to suppose that C Δ (Q(A)) = C and find a primitive ideal I ⊆ A with P Δ (I) = (0). By part (3) of Lemma 5.1, we may adopt the hypothesis that A is prime. Since C Δ (Q(Z)) → C Δ (Q(A)), it follows that C Δ (Q(Z)) = C.
Let M be the set of a minimal non-zero Δ-prime ideals of A. We claim that M is countable. First of all notice that, since A is a finite extension of Z, there are finitely many prime ideals of A lying over each prime ideal of Z. For the reader's convenience we sketch the proof of this fact. If p ∈ Spec(Z) is any prime ideal, then the ideal pA is not necessarily prime, but since A is a noetherian ring, there are finitely many prime ideals P 1 , . . . , P m of A over pA. Suppose that Q ∈ Spec(A) is any ideal with Q ∩ Z = p. Then it follows that Q contains one of the minimal primes P 1 , . . . , P m . We may suppose P 1 ⊆ Q. Since A/P 1 is an essential extension of Z/P 1 ∩ Z, we conclude from Lemma 5.6 that either Q = P 1 or the image of Q in A/P 1 intersects Z/P 1 ∩ Z non-trivially. By assumption Q ∩ Z = p = P 1 ∩ Z and so the latter does not hold, hence we may conclude Q = P 1 . This proves that there are finitely many primes of A lying over p. Now in order to prove that M is countable, it suffices to show that Z contains only countably many minimal non-zero Δ-prime ideals. This follows from the argument given in [3, Theorem 3.2], using the fact that C Δ (Q(Z)) = C. Now we may enumerate M = {P 1 , P 2 , P 3 , . . .} and write p i := P i ∩ Z for all i = 1, 2, 3, . . .. By assumption A is prime and so thanks to Lemma 5.6, we see that A is an essential extension of Z. In particular {p 1 , p 2 , p 3 , . . .} are all non-zero. Choose non-zero elements {s 1 , s 2 , s 3 , . . .} with s i ∈ p i and let S be the multiplicative subset of S generated by {s 1 , s 2 , s 3 , . . .}. Note that AS −1 is countably generated, so we are in a position to apply every conclusion of Proposition 5.5.
Let I ∈ Spec(AS −1 ) be any primitive ideal. Suppose for a moment that I contains some nonzero Δ-prime ideal. Then it must contain a minimal non-zero Δ-prime ideal, which we may denote by K. It follows from Proposition 5.5 that K ∩ A is a non-zero Δ-prime ideal which intersects S trivially. This is impossible, since S contains a non-zero element of every non-zero Δ-prime ideal of A. We may conclude that P Δ (I) = (0). Now apply part (4) of Lemma 5.1 to see that P Δ (I ∩ A) = P Δ (I) ∩ A = (0). Thanks to the last part of Proposition 5.5, we see that I ∩ A is a primitive ideal of A and so we have shown that (0) is Δ-primitive, as required.

The Δ-weakly locally closed ideals are the Δ-rational ideals
We now go on to prove that Δ-rational ideals of A enjoy a property which is strictly weaker than being Δ-locally closed. We say that an ideal I ⊆ A is Δ-weakly locally closed if the following set is finite  Proof. After replacing A with A/I we may show that (0) is Δ-rational if and only if it is Δ-weakly locally closed. We begin by supposing that (0) is not Δ-rational and show that it is not Δ-weakly locally closed. Recall that we identify Q(A) with A ⊗ Z Q(Z), and we identify A and (Z \ {0}) −1 with subsets of Q(A). Suppose that there exists some nonconstant az −1 ∈ C Δ (Q(A)) with 0 = z ∈ Z. We consider the localisation Z z := Z[z −1 ] and . For all c ∈ C we observe that az −1 − c is central in A z and we consider the ideals I c := (az −1 − c)A z .
We claim that {I c | c ∈ C} are generically proper ideals.
Combined with the fact that az −1 − c is not a zero divisor, we conclude that (az −1 − c) −1 is central in A z . Writing C(A z ) for the centre of A z , we conclude that I c is proper if and only if J c := (az −1 − c)C(A z ) is proper. The Artin-Tate lemma implies that C(A z ) is an affine algebra and so the ideals {J c | c ∈ C} are generically proper, which confirms the claim at the beginning of this paragraph.
Since A is a noetherian ring so is A z and so by [30, 4.1.11], there are finitely many minimal prime ideals over I c each of which has height one. If some prime ideal contains both I c and I c for some c, c ∈ C, then it contains c − c and so these prime ideals are all distinct. Now we have found infinitely many Δ-prime ideals of A z , all of height one. Finally we apply parts (1) and (2) of Proposition 5.5 to see that (0) is not Δ-weakly locally closed, as desired.
Now we show that if (0) is Δ-rational, it is Δ-weakly locally closed. First of all we observe that (0) is a Δ-rational ideal of Z, thanks to the identification Q(A) = A ⊗ Z Q(Z). Thanks to [3,Theorem 7.1] we know that (0) is Δ-weakly locally closed in Spec(Z). Suppose that p 1 , . . . , p l are the set of those minimal non-zero prime ideals of Z which are Δ-stable. Since there are finitely many prime ideals of A lying above each ideal p 1 , . . . , p l , it suffices to show that each one of the minimal non-zero prime ideals of A which is Δ-stable lies over one of p 1 , . . . , p l . Let P be a minimal non-zero prime of A which is Δ-stable and observe that P ∩ Z is a Δ-prime ideal. We must show that P ∩ Z is minimal amongst non-zero primes of Z. If not then there exists a prime q with q P ∩ Z. It then follows from the going down theorem [30, 13.8.14(iv)] that there is a prime Q of A with Q ∩ Z = q and Q P ; this contradicts the minimality of P and so we deduce that P ∩ Z is a minimal non-zero prime, as required.

Proof of the first theorem and some applications
In this section, we continue to take all vector spaces over C.

Existence of the bijection
We begin with some topological observations about the space Prim(A) when A is finite over its centre.  . . , J t . Since A is prime and finite over the centre Z, we have that Z ⊆ A is an essential extension by Lemma 5.6, and so we can choose non-zero element i k ∈ I k ∩ Z for k = 1, . . . , s and j k ∈ J k ∩ Z for k = 1, . . . , t. Since A is prime, not one of these elements is nilpotent, and we may form the multiplicative subset S ⊆ Z \ {0} which they generate. The set O 1 ∩ O 2 = {J ∈ Prim(A) | I 1 , . . . , I s , J 1 , . . . , J t ⊂ J} contains the set {J ∈ Prim(A) | J ∩ S = 0} and according to Proposition 5.5, the latter is bijection with Prim(AS −1 ). Note that the hypotheses of that proposition are satisfied because A is a finite module over a finitely generated algebra, hence finitely generated. We know that Prim(AS −1 ) is non-empty by Zorn's lemma, which proves (1).
Suppose that Prim(A) = k∈K L k (6.1) decomposes as a disjoint union of locally closed subsets and L 1 , L 2 are two of these sets such that L 1 = L 2 = V (I) for some prime I. Then applying part (1) to the open subsets L 1 , L 2 of Prim(A/I), we see that L 1 ∩ L 2 = ∅, and so L 1 = L 2 since the decomposition (6.1) is disjoint.
Suppose that Z is a regular complex affine Poisson algebra, so that Z := Spec(Z) is smooth, and that the symplectic leaves of Z are locally closed in the Zariski topology. Thanks to the last part of the second main theorem, we know that the PDME holds for A. By the equivalence of (I) and (II) in the second main theorem, we know that the closures of the symplectic cores of Prim(A) are the sets of the form V (P(I)) = {J ∈ Prim(A) | P(I) ⊆ J}. Combining parts (b) and (c) and the equivalence of (ii) and (iii) from the second main theorem, we deduce that the closures of the symplectic cores are precisely the sets {V(M ) | M simple Poisson A-module}. Applying Lemma 6.1 we see that each symplectic core C(I) ⊆ Prim(A) is uniquely determined as the open core in its closure, which shows that the map from the first main theorem is a bijection, as claimed.
Remark 6.2. The same proof works if we replace the assumption that the leaves are algebraic with the assumption that the PDME holds for Z, which is a strictly weaker hypothesis, as may be seen upon comparing [17, Example 3.10(v)] and [15]. Another example of a Poisson algebra Z with non-algebraic symplectic leaves for which the PDME holds is the polynomial ring C[x 1 , x 2 , x 3 ] with brackets {x, y} = ax, {y, z} = −y, {x, y} = 0. According to [16,Example 3.8], the leaves are non-algebraic but the PDME holds [3] since the Gelfand-Kirillov (GK) dimension is 3.

The bijection is a homeomorphism
Retain the hypotheses of the previous subsection. Now that we have shown that the Poisson primitive ideals of A are the annihilators of simple modules, we may denote the set of such ideals P-Prim(A). Denote the bijection described in the first main theorem by φ : P-Prim(A) → Prim C (A). It remains to prove that φ is a homeomorphism. We must give a precise description of the topology on each of these two spaces and observe that φ sets up a bijection between closed subsets of its domain and codomain.
The where the right-hand vertical arrow is constructed in our first main theorem, and all other arrows are defined above. The composition of the horizontal maps are the constant maps, respectively, sending Prim U (g) to the annihilator of the trivial Poisson C[g * ]-module, and sending g * /G to the zero orbit.

Quantum groups and open problems
In the introduction, we proposed two applications of the first main theorem: a description of the annihilators of simple Poisson C[g * ]-modules when g is a complex algebraic group, and also annihilators of simple modules over the classical finite W -algebra. Both of these examples are Poisson algebras and so they do not use the full force of the first main theorem. We conclude by mentioning some famous examples where A is a Poisson order over a proper Poisson subalgebra Z, satisfying the hypotheses of the main theorem. Let q be a variable and consider the affine C[q]-algebra A generated by X 1 , . . . , X n subject to relations X i X j − qX j X i for i < j. This is the single parameter generic quantum affine space. When we specialise q → where is a primitive th root of unity for some > 1, we obtain a Poisson order. To be precise, the subalgebra Z 0 of A := A/(q − )A generated by {X i | i = 1, . . . , n} is central, known as the -centre. Following the observations of § 2.3, we see that Z 0 is a Poisson algebra and A is a Poisson order over Z 0 . There is a (k × ) naction on A rescaling the generators and it is not hard to see that there are only finitely many (k × ) n -stable Poisson prime ideals. It follows from the results of [15] that the PDME holds for Z 0 and so by Remark 6.2, our first main theorem applies to A . In particular, the space P-Prim(A ) of annihilators of simple Poisson A -modules is homeomorphic to the set Prim C (A ).
Two other natural examples to consider are the quantised enveloping algebras U q (g) where g is a complex semisimple Lie algebra, and their restricted Hopf duals O q (G); see [5] for more detail. Once again, after specialising the deformation parameter q to an th root of unity, we denote one of these algebras by A . Just as for quantum affine space, the th powers of the standard generators in either of these examples generate a central subalgebra Z 0 . In these cases the symplectic leaves are actually locally closed and so our main theorem applies here too.
Problem. For all of the families of algebras discussed above: Of course, Theorem 4.2 shows that such a module exists but the proof is non-constructive. We hope that by constructing modules more explicitly (for example, by generators and relations) as per Problem (2), it will become more apparent how we could hope to deform a simple A /mA -module over the closure of the symplectic core C(m), where m ∈ Prim(Z 0 ).