Ideals in the Enveloping Algebra of the Positive Witt Algebra

Let W+ be the positive Witt algebra, which has a C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}$\end{document}-basis {en:n∈Z≥1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{e_{n}: n \in \mathcal {Z}_{\geq 1}\}$\end{document}, with Lie bracket [ei,ej] = (j − i)ei+j. We study the two-sided ideal structure of the universal enveloping algebra U(W+) of W+. We show that if I is a (two-sided) ideal of U(W+) generated by quadratic expressions in the ei, then U(W+)/I has finite Gelfand-Kirillov dimension, and that such ideals satisfy the ascending chain condition. We conjecture that analogous facts hold for arbitrary ideals of U(W+), and verify a version of these conjectures for radical Poisson ideals of the symmetric algebra S(W+).


Introduction
Let k be a field of characteristic zero, and let W + be the positive Witt algebra, which has a k-basis {e n : n ∈ Z ≥1 }, with Lie bracket [e i , e j ] = (j − i)e i+j .
(1.1) This paper studies the two-sided ideal structure of U(W + ).
It follows that U(W + ) satisfies the ascending chain condition on ideals whose associated graded ideal is radical, see Corollary 2.17.
We then turn to studying the GK-dimension of quotients of U(W + ) more directly. For a Poisson algebra A, we define the Poisson Gelfand-Kirillov dimension PGKdim A, which measures the growth of A as a Poisson algebra. We show (Theorem 3.19) that the GKdimension of a quotient R of U(W + ) is equal to the Poisson GK-dimension of the associated quotient of S(W + ). 1 We further show: Theorem 1. 6 If K is a nontrivial radical Poisson ideal of S(W + ), then PGKdim S(W + )/K = GKdim S(W + )/K, which we have seen previously is finite.
Therefore, if I is an ideal of U(W + ) whose associated graded ideal is radical, then GKdim U(W + )/I < ∞, and thus Conjectures 1.2 and 1.3 both hold for ideals whose associated graded ideal is radical.
We then turn our attention to quadratic elements in the symmetric algebra, i.e. elements of S 2 (W + ). Through explicit computations, we show that S 2 (W + ) is a noetherian W + -module (Theorem 4.2), and as a consequence that S(W + ) satisfies the ascending chain condition on Poisson ideals generated by quadratic elements. Finally, we show: Theorem 1.7 (Corollary 4.14) If I is an ideal of U(W + ) that contains a quadratic expression in the e i , then U(W + )/I has finite GK-dimension.
Recall that W + is a subalgebra of the (full) Witt algebra W , which has a k-basis {e n : n ∈ Z} and Lie bracket defined by Eq. 1.1. Recall also that W is obtained from the Virasoro algebra V (which we do not define) by setting the central charge equal to zero. We conjecture that analogues of Conjectures 1.2 and 1.3 and Theorem 1.7 hold for U(W ) and U(V ). These questions will be the subject of future work.
The organisation of the paper is as follows. Section 2, where we prove Theorems 1.4 and 1.5, focuses on quotients of S(W + ) by radical Poisson ideals. In Section 3 we define the Poisson Gelfand-Kirillov dimension of a Poisson algebra, give some of its properties, and prove Theorem 1. 6. In Section 4 we study the structure of S 2 (W + ) and prove Theorem 1.7. This proof involves computer calculations which are discussed in an Appendix.

Poisson Ideals
We begin by collecting some basic properties of Poisson algebras, and then move to deriving consequences for S(W + ). We note that all Poisson algebras in this paper are commutative as algebras.
Our convention is that N is equal to the set of nonnegative integers, and Z ≥1 is the set of positive integers.

Operations on Ideals
Since we will be working with the non-noetherian ring S(W + ) ∼ = k[x 1 , x 2 , . . . ], we recall some basic concepts in commutative algebra which do not depend on the ascending chain condition.
Throughout the next two subsections A is a Poisson k-algebra, I is a Poisson ideal of A, and a, b, c are elements of A.
Recall that (I : b) := {a ∈ A : ab ∈ I }, and note I ⊆ (I : b). Also recall that an ideal I is radical if I = √ I := {a ∈ A : a n ∈ I for some n ∈ N}. Proof First we show that ( We fix a ∈ (I : b ∞ ) and n ∈ N such that ab n ∈ I . We have that (ab) n = a n−1 ab n ∈ I . Hence ab ∈ I and a ∈ (I : b).
Next, we wish to show that √ (I : b) = (I : b). We fix a ∈ √ (I : b) and n ∈ N such that a n ∈ (I : b). We have (ab) n = a n bb n−1 ∈ I and therefore ab ∈ I . Hence a ∈ (I : b).
An intersection of a collection of radical ideals is clearly a radical ideal and thus if I is radical so is (I+b).
For the final statement, if a ∈ (I : b ∞ ) = (I : b) then ab ∈ I and b ∈ (I : a ∞ ). Proof Let a ∈ (I+b) ∩ (I : b). Then a ∈ (I : b ∞ ) and therefore from the definition of (I+b) we have a ∈ (I : a ∞ ). Hence a ∈ √ I . The final statement holds since

If b ∈ A then we define
Although the Lasker-Noether primary decomposition theorem does not hold if A is not noetherian, Lemma 2.3 can provide a useful analogue.

Compatibility with Poisson Structure
We now show that the constructions above preserve the Poisson structure of A.
Proof Fix a ∈ A and n ∈ N such that ab n ∈ I . It is enough to show that for any c ∈ A we have {a, c} ∈ (I : b ∞ ). We have The terms {ab n+1 , c} and (n + 1)ab n {b, c} belong to I and thus {a, c}b n+1 ∈ I .
We immediately obtain:

Corollary 2.5 If I is a Poisson ideal then the algebra (A/I )[b −1 ] is Poisson with respect
to the Poisson bracket defined as follows:

The natural maps
Corollary 2.5 is a special case of a more general result: that if A is a Poisson algebra and C is a multiplicatively closed set in A then AC −1 has a natural Poisson structure compatible with that on A.
Let P be a minimal prime of the commutative algebra A, let C = A \ P , and let Q = {x ∈ A : xc = 0 for some c ∈ C} be the kernel of the natural map A → AC −1 . If xyc = 0 where y, c ∈ C, then yc ∈ C and so x ∈ Q. Thus if xy ∈ Q and y ∈ P , then x ∈ Q. However, even if A is a quotient of S(W + ), we do not know if Q must be primary. Note that if A is in addition noetherian, then P C −1 is the unique minimal prime of the noetherian ring AC −1 and so is nilpotent. Thus if x ∈ P , we have some x n ∈ Q and Q is in addition P -primary.
Lemma 2.6 (see also [5,Lemma 1.8 Proof We fix a, b ∈ A and n ∈ N such that a n ∈ I . It is enough to show that {a, b} ∈ √ I . We will prove that 1 ∈ (I : {a, b} ∞ ) (this statement is equivalent to the previous one). Assume to the contrary that 1 ∈ (I : {a, b} ∞ ).
We have that a n ∈ I ⊆ (I : {a, b} ∞ ). Let m be the minimal nonnegative integer such that a m ∈ (I : {a, b} ∞ ). Since  It is well known [4, Corollary 2.12] that any radical ideal I of A is an intersection of prime ideals and thus of primes minimal over I -this follows from Zorn's Lemma and does not require A to be noetherian. If I has finitely many minimal primes p 1 , . . . , p m then √ I = p 1 ∩· · ·∩p m . Conversely, if √ I = p 1 ∩· · ·∩p m is an irredundant intersection then the p j are precisely the minimal primes of I , as if I ⊆ q for some prime q then some p j ⊆ q. Proof Without loss of generality I = √ I is radical. Let p be a minimal prime over I . Let I p be the sum of all Poisson ideals contained in p. Clearly I p is the maximal Poisson ideal contained in p. To complete the proof it is enough to show that I p is prime.
Certainly I p ⊆ p. Since √ I p is Poisson by Lemma 2.6, I p is a radical ideal. Let x, y ∈ A be such that xy ∈ I p . We will show that either x ∈ I p or y ∈ I p . By definition, y ∈ (I : x), and by Lemma 2.1, x ∈ (I+x). By Corollary 2.7, I = (I : x) ∩ (I+x), and both (I : x) and (I+x) are Poisson ideals. Since I ⊆ p, either (I : x) ⊆ p or (I+x) ⊆ p. Thus either y ∈ (I : x) ⊆ I p or x ∈ (I+x) ⊆ I p .

Radical Ideals in S(W + )
The positive Witt algebra is the Lie algebra W + with basis e i (i ∈ Z ≥1 ) and Lie bracket [e i , e j ] = (j − i)e i+j . The symmetric algebra of W + is denoted by S(W + ). Our convention is that the image of e i in S(W + ) is denoted by x i .
We now specialise to studying the Poisson structure on S(W + ) induced by the Lie bracket on W + . In this section, we will show that S(W + ) satisfies the ascending chain condition on radical Poisson ideals and that proper quotients by Poisson ideals have finite Gelfand-Kirillov dimension. Our first step is to show that any nontrivial quotient of S(W + ) by a radical ideal embeds into a finitely generated Poisson algebra.
As with any symmetric algebra, S(W + ) carries a natural grading, which we refer to the order gradation and denote by o. We have o(x i ) = 1 for all i, and o({x i , f }) ≤ o(f ) for all i and for all f ∈ S(W + ). On U(W + ), there is an order filtration, which we also denote by o, with o(e i ) = 1 for all i. Recall that S(W + ) = gr o U(W + ) is the associated graded ring of the order filtration on U(W + ).
In addition, W + is a graded Lie algebra if we give e i degree i, and this extends to a graded structures on U(W + ) and S(W + ), which we refer to as the degree gradation. We denote the degree gradation by d,  To prove Lemma 2.9 we need several auxiliary facts. Let f ∈ I be a nonzero element of minimal order. We pick the smallest number n such that f ∈ k[x 1 , . . . , x n ]. The following lemma is straightforward.
where q is a polynomial of order ≤ o.
We now prove Lemma 2.9. This proves part (a).
We now prove part (b). Let p, x 1 , . . . , x 2n+2 be as in the proof of part (a). Primality of I implies that (I : p) = I so the natural map φ : S(W + )/I → S(W + )/I [p −1 ] is injective. Let B be the subalgebra of S(W + )/I generated by x 1 , . . . , x 2n+2 . It is easy to check that p, B as above satisfy the conclusions of part (b).
Since the maps involved are homomorphisms of Poisson algebras, (c) also holds. Lemma 2.9 has the following important consequence:

Corollary 2.11
Let I be a Poisson ideal of S(W + ). Then I has finitely many minimal primes: that is, there exist prime ideals p 1 , . . . , p n of S(W + ) such that √ I = p 1 ∩ · · · ∩ p n . Further, the p i are Poisson ideals.
Proof Thanks to Lemma 2.9 either I = (0) or there is an embedding of S(W + )/ √ I into a reduced finitely generated commutative algebra A. For such an algebra A we have p 1 ∩ · · · ∩ p n = (0) for some finite set of prime ideals p 1 , . . . , p n of A. The ideals φ −1 (p 1 ), . . . , φ −1 (p n ) are prime in S(W + ) and we have the desired equality The last sentence is Lemma 2.8.
We wish to show that an ascending chain of radical Poisson ideals in S(W + ) stabilises. To do this, we recall two definitions of dimension. The Krull dimension of a commutative ring A, which we write Kdim A, is the supremum over all strictly ascending chains of prime ideals of A of the length of the chain minus one.
The Gelfand-Kirillov dimension (or GK-dimension) of A is written GKdim A and defined as the supremum over all finite-dimensional subspaces V of A of lim log n dim k V n (see also Section 3 and [6]).
The following facts are well known.

Proposition 2.12
Let A be a commutative k-algebra.
(d) Assume that A is a finitely generated domain and let p ∈ A\0. Let A be an algebra with .
Proof (a) is [6, Proposition 3.16] and (b) is [6,Theorem 4.5]. For (c), we have Part (d) follows from the fact that For (e), note that GKdimA is equal to the maximum of the Gelfand-Kirillov dimensions of the finitely generated subalgebras of A. It is immediate from the definitions that Therefore (e) follows from (b).
We now derive some more consequences of Lemma 2.9.
where the first inequality is Proposition 2.12(a) and the second comes from the embedding Put where the P j,i are the finitely many minimal primes over I j . We have We induct on codim I 1 . If codim I 1 = 0 then the P j,i are maximal ideals. Since (by primality) each P j +1,i ⊇ I j contains some P j,i , we have P j +1,i = P j,i and so {P j, * } ⊇ {P j +1, * } and n j +1 ≤ n j . For j 0 all n j are equal and thus all I j are equal.
So now assume that any ascending chain that begins with a radical Poisson ideal of codimension k must be finite, and suppose that codim I 1 = k +1. Without loss of generality, all I j have codimension k + 1. Reorder the P j,i so that they have codimension k + 1 for i ≤ j and codimension ≤ k for j < i ≤ n j . Now each P j +1,i contains some P j,i and for dimension reasons if i ≤ j +1 then we must have i ≤ j and P j +1,i = P j,i . Thus j +1 ≤ j and we may assume without loss of generality that all j are equal to some and for i ≤ that all P j,i are equal.
Let J = P 1,1 ∩· · ·∩P 1, and K j = ∩ i> P j,i , so I j = J ∩K j . As all the P j,i are minimal over I j , for fixed j the P j,i are mutually incomparable (i.e. P j,i 1 ⊆ P j,i 2 if i 1 = i 2 ). Let < i ≤ n j +1 . By primality, P j +1,i does not contain J . As P j +1,i ⊇ I j = J ∩ K j , we have P j +1,i ⊇ K j and thus K j +1 ⊇ K j . Since codim K j ≤ k, by induction the K j stabilise and thus the chain I j = J ∩ K j stabilises.
Proof of Theorem 2.15 Any ascending chain of radical Poisson ideals of S(W + ) satisfies the assumptions of Lemma 2.16 thanks to Corollary 2.1 and Corollary 2.14. Therefore Theorem 2.15 follows from Lemma 2.16.
is a Poisson ideal of S(W + ). By [8, Proposition 1.6.8], if S(W + ) satisfies the ascending chain condition (ACC) on Poisson ideals, it would follow that U(W + ) has ACC on ideals. We cannot prove this at the moment, but the argument above does give:

Corollary 2.17
The algebra U(W + ) satisfies the ascending chain condition on ideals whose associated graded ideals are radical.
We do not know what conditions on an ideal I of U(W + ) guarantee that the associated graded ideal is radical. However, it is known that if I is the kernel of one of the homomorphisms from U(W + ) to an Artin-Schelter regular algebra considered in [12], then the associated graded ideal of I with respect to the order filtration is prime. Note in this case that I is completely prime.

Remark 2.18
Some results of this section can also be deduced from differential algebra (see [5] and [7]). Differential algebra (as a branch of mathematics) considers commutative algebras with derivation(s) and the ideals of such algebras which are stable under the derivation(s). Now, the adjoint action of W + on itself defines an action of W + on S(W + ) by derivations such that The Poisson ideals of S(W + ) are the ideals of S(W + ) which are stable under all of the above derivations (equivalently under the derivations induced by e 1 and e 2 ). Thus it is quite natural to connect results on the Poisson structure of S(W + ) with the results of differential algebra. Consider S(W + ) as a differential algebra with respect to the derivation ∂ 1 defined by e 1 . By the above, any Poisson ideal I of S(W + ) is a differential ideal with respect to ∂ 1 . It is easy to check that (S(W + ), ∂ 1 ) is generated by x 1 , x 2 as a differential algebra. It follows from the Ritt-Raudenbush basis theorem [5, Theorem 7.1] that any chain of radical ∂ 1 -differential ideals of S(W + ) stabilises, and thus any chain of radical Poisson ideals of S(W + ) stabilises.
Note also that it can be deduced from [7, Lemma 1.8] that if I is a prime differential ideal of (S(W + ), ∂ 1 ) then there is f ∈ I such that I is the minimal prime differential ideal containing f . We thank Omar Leon Sanchez for calling our attention to this result.
Overall, this shows that differential algebra can be helpful in the study of Poisson ideals of S(W + ).

Growth of (Poisson) Algebras
In this section we first define the Poisson GK-dimension of a Poisson algebra, and then show that this can be used to compute the GK-dimension of an almost-commutative filtered ring under appropriate conditions. Finally, we give applications of our general results to U(W + ).

Poisson GK-dimension
In this subsection we define and give general results on Poisson GK-dimension. The techniques here are standard, but since the terminology is new we give the proofs in a fairly high level of detail.
We begin with definitions. We work over the fixed ground field k, and write dim V for dim k V if V is a k-vector space. We first recall some standard definitions from [6, Chapter 1].
By [6, Lemma 1.1], the growth G(d V ) does not depend on the choice of the generating subspace V , and we refer to it as the growth of R, written G(R).
The Gelfand-Kirillov dimension or GK-dimension of R is: where V is a finite-dimensional subspace of R which generates R as an algebra. (The last equality is [6, Lemma 2.1].) For a not necessarily finitely generated algebra R, we define GKdim(R) = sup R GKdim(R ), where the supremum is taken over all finitely generated subalgebras R of R.
Our first task is to define the Poisson GK-dimension of a Poisson algebra.

Definition 3.2 Let
A be a Poisson algebra over k. Let V be a subspace of A. We inductively define the subspaces V {n} as follows: We wish to show G(pd V ) does not depend on V as long as V generates A as a Poisson algebra. We first show: Proof We prove the lemma by induction on a. By definition, the lemma holds for a = 1 and for any b. Suppose now that the lemma holds for all a ≤ c and for any b. Then The inclusion (1) ⊆ V {c+b+1} is immediate by induction. We have: All of these are contained in V {c+b+1} by induction.

Proposition 3.4 For any finite-dimensional spaces V , W which generate A as a Poisson algebra, we have
Proof Since V and W generate A as a Poisson algebra, there are positive integers s, t so that

Definition 3.5 If
A is generated as a Poisson algebra by some finite-dimensional subspace V , we define the Poisson GK-dimension of A to be By Proposition 3.4, this does not depend on the generating space V chosen.
For an arbitrary Poisson algebra we define where the supremum is taken over all finitely generated Poisson subalgebras A of A.
For Poisson algebras with a sufficiently nice filtration, we can compute Poisson GKdimension from the growth of the filtration. Let A be an algebra, and let k = A(0) ⊆ A(1) ⊆ · · · be a filtration of A; recall this means that the A(i) are subspaces of A so that

Lemma 3.6 Let A be a finitely generated Poisson algebra, discretely, finitely, and exhaustively filtered by
) and the first inequality follows. Now suppose that Eq. 3.7 holds for k. Clearly A(k) generates A as a Poisson algebra. By the first paragraph and Eq. 3.7, G(dim A(n)) = G(pd A(k) ). The final statement follows.
If the filtration A(n) on A satisfies (3.7) for some k, we say that A has good growth with respect to the filtration. To end the subsection, we note that if A is a finitely generated Poisson algebra, then the Poisson GK-dimension of A is also the GK-dimension of A as a module over a certain ring of differential operators. We refer to [6,Chapter 5] for definitions, see also Proposition 3.9.
If v ∈ A, define ∂ v := {v, −}. This is a derivation of A.

Proposition 3.9 Let A be generated as a Poisson algebra by a finite-dimensional subspace
considered as a subalgebra of the ring D(A) of differential operators on A. Note that A has a natural left D-module structure. Then PGKdim A = GKdim D A.
Proof We write the action of D on A as D · A. Inside D, our convention is that Without loss of generality, we may assume that 1 ∈ V . Let W = V + ∂ V ⊆ D. We claim that W n ⊇ V {n} for all n. To see this assume that it holds for n. Then Since A = V {n} and D = A ∂ V we have that W generates D as a k-algebra.
Note that for any X ⊆ A we have W · X = V X + {V , X}, and so W n · k = V {n} . Thus

Relating GK-dimension and Poisson GK-dimension
Let R be a finitely and discretely filtered ring so that the associated graded ring gr R is finitely generated. It is standard that GKdim R = GKdim(gr R); see [6, Proposition 6.6]. We wish to use a similar technique to understand the GK-dimension of quotients of U(W + ). Unfortunately, gr U(W + ) = S(W + ) is not finitely generated as an algebra; however, it is finitely generated as a Poisson algebra, and we will show that we can relate the GKdimension of (a quotient of) U(W + ) and the Poisson GK-dimension of the associated graded ring.

Definition 3.10
Let R be a finitely generated k-algebra together with a filtration that is discrete, finite, and exhaustive. (Recall that these terms were defined before Lemma 3.6.) Then R is almost commutative with respect to this filtration if [r i , r j ] ∈ R(i + j − 1) for all i, j ≥ 0 and r i ∈ R(i), r j ∈ R(j ).
In this subsection we consider an algebra R that is almost commutative with respect to a discrete, finite, exhaustive filtration as in Eq. 3.11. Let Since Let A(n) = gr R(n) = n k=0 A k . Since the filtration is discrete, dim A(n) = dim R(n). We have Lemma 3.13 Let R be an algebra that is almost commutative with respect to a discrete, finite, exhaustive filtration as above. For any subsets X, Y ⊆ R we have Proof We can reduce the statement to the case dim X = dim Y = 1, which is given by Eq. 3.12.
Our main result on Poisson GK-dimension is the following: Proposition 3.14 Let R be an algebra that is almost commutative with respect to the discrete, finite, exhaustive filtration (3.11), and let A = gr R with A(n) = gr R(n). Then

GKdim(R) ≥ PGKdim(A).
If A has good growth with respect to the filtration {A(n)}, that is if Eq. 3.7 holds for some k, then GKdim(R) = PGKdim(A) = lim log n dim A(n) Proof Let V be a finite-dimensional subspace of A, with 1 ∈ V . Choose a finitedimensional subspace W of R, with 1 ∈ W , so that gr W ⊇ V . We claim that V {n} ⊆ gr W n for all n. The claim is true for n = 1; assume that it holds for n. Then Since [W, W n ] ⊆ W n+1 , the claim is proved.
Taking the supremum over all V and W , we obtain that PGKdim A ≤ GKdim R. Assume now that Eq. 3.7 holds for k, and let V = A(k) and W = R(k). We claim that W generates R as an algebra; in fact, we claim that R(n) ⊆ W n for all n. This is clearly true for n ≤ k. Let r ∈ R(n) \ R(n − 1). We have gr r ∈ V {n} ⊆ gr W n and so there is But by Lemma 3.6, lim log n dim R(n) = lim log n dim A(n) = PGKdim A, completing the proof.

Consequences for Quotients of U(W + ) and S(W + )
We now apply our previous results to quotients of U(W + ) and S(W + ). First consider a quotient of S(W + ) by a radical Poisson ideal.

Theorem 3.19 Let K be a nonzero radical Poisson ideal of S(W + ) and let
To prove Theorem 3.15 we need the following result. Proof This is a direct consequence of the methods of [13], although this result does not seem to appear in the literature. For any subalgebra A of L, let A be the module of derivations of A. If Spec A is smooth and affine, then by [

D(L) generated by A and A : that is D(A) = A[ A ]. It follows that, D(L) = L[ L ], and that
GKdim where the supremum is taken over all finitely generated subalgebras A of L with Q(A) = L.
Since char k = 0, by generic smoothness we may enlarge A to obtain a finitely generated algebra A ⊆ L with Spec A smooth and Q(A ) = L. As A is projective, there is a finitely generated algebra A with so that A is free over A , and it suffices to prove that GKdim Clearly PGKdim A ≥ GKdim A, so it suffices to prove that GKdim D A ≤ Kdim A,which is GKdim A by Corollary 2.14.
We first assume that K is prime. By Lemma 2.9(b), there is some nonzerodivisor p ∈ A so that A → A = A[p −1 ] and A is a finitely generated algebra. As A is also Poisson, D also acts on A . Let For general K, by Corollary 2.11 we have K = p 1 ∩ · · · ∩ p m , where the p i are prime Poisson ideals and are therefore D-stable. Thus D acts on A i = A/p i ; let the operators ∂ 1,2 on A i be induced from the action of ∂ 1,2 on A. Let

But this is Kdim A by definition.
We note that the conclusion of Theorem 3.15 can fail for non-radical ideals. Indeed, let I = (x i x j : i, j ∈ Z ≥1 ) and A = S(W + )/I . It is easy to see that I is a Poisson ideal, and that GKdim A = 0 and PGKdim A = 1.
We now derive results for quotients of U(W + ). Recall that U(W + ) is both graded by degree and filtered by order of operators (as the enveloping algebra of a Lie algebra), and we write the degree and order of an element f respectively as d Our first result is that d-graded ideals of U(W + ) automatically give rise to Poisson ideals of S(W + ) so that the quotients have good growth in the sense of Eq. 3.7.

R(n) = (U (n) + I )/I ∼ = U(n)/I (n).
It is immediate that the R(n) give a discrete, finite, exhaustive filtration on R, which we will refer to as the do-filtration on R. Since U(W + ) is almost commutative with respect to the do-filtration, clearly R is almost commutative with respect to the do-filtration on R, and thus (a) holds.
As A = gr do (R), then Let y ∈ A(n) \ A(n − 1). We must show that y ∈ V {n} .
Notice that d(y) ≤ do(y) ≤ n. We can write y as a sum of monomials of the form e i 1 e i 2 · · · e i , where i 1 ≤ i 2 ≤ · · · ≤ i and i j ≤ n. To show that y ∈ V {n} , it suffices to show that e m ∈ V {m} for all m. This is true for m = 1, 2; and for m ≥ 3 we have for some λ ∈ k \ 0 that e m = λ{e 1 , e m−1 } ∈ {V , V {m−1} } by induction.

Remark 3.18
There is an alternate filtration on A defined via the d-grading: let and define a filtration F i A on A accordingly. Then the argument above shows that so A also has good growth with respect to the filtration Combining the previous proposition with earlier results, we obtain:

Remark 3.20
If R = j ∈N R j is any N-graded ring that also has a discrete finite exhaustive filtration with respect to which R is almost commutative, and so that each R(n) = j (R(n) ∩ R j ) is a graded vector space, then the argument above shows (by adding the two gradings on A := gr R) that if A is finitely generated as a Poisson algebra then A has good growth and therefore that GKdim R = PGKdim A.
Finally, we have:

Corollary 3.21 Let J be a nontrivial ideal of U(W + ) so that gr o (gr d (J )) is radical. Then
Proof This follows directly from Theorems 3.19 and 3.15.
We conjecture that the conclusion of Corollary 3.21 holds for any nontrivial ideal of U(W + ); see Conjecture 1.2. Note that by Theorem 3.19, it suffices to prove that PGKdim S(W + )/K < ∞ for any d-graded and o-graded Poisson ideal K of S(W + ).

Quotients by Quadratic Elements
The results in the previous sections may be thought of as providing evidence that Conjecture 1.2 holds and thus that nontrivial ideals of U(W + ) and Poisson ideals of S(W + ) are large. If this is the case, it is natural to expect that U(W + ) satisfies the ascending chain condition on ideals: in other words, that Conjecture 1.3 holds. (Examples such as [1,Theorem 2.14] show that finite GK-dimension does not even imply the ascending chain condition on prime ideals, so we phrase this as an expectation, not a formal consequence.) In this section we study Conjecture 1.2 and Conjecture 1.3 for ideals containing elements of order two. We first prove that S 2 (W + ) is a noetherian representation of W + , from which it follows trivially that S(W + ) satisfies the ascending chain condition on Poisson ideals generated by elements of order two. (As a byproduct, we show that S 2 (W + ) is GK-2 critical.) As a consequence of our methods, we show that any quotient of U(W + ) by an ideal containing a nontrivial element of order one or two has finite GK-dimension.

Noetherianity of S 2 (W + )
Before proving that S 2 (W + ) is noetherian, we show that the adjoint representation of W + is noetherian. This is implied by the following lemma. We would thank Jacques Alev for the proof of this result.
We say that s is the length of x. If s > 1 then [e i 1 , x] is nonzero and has length < s. By induction, there is some e n in the Lie ideal generated by x. It is an easy computation that the Lie ideal generated by e n contains e ≥n+2 = {e j : j ≥ n + 2}.
By the above, any nontrivial ideal of W + has cofinite dimension, and thus W + is noetherian as a Lie algebra and as a W + -module. The main result of this subsection is: Proof Our strategy is to put a monomial order on S 2 (W + ) and then for a submodule M ≤ S 2 (W + ), describe the combinatorial structure of the set of leading terms of elements of M.
We first establish notation. A basis for so is a grading semigroup for S 2 (W + ) as a vector space. Define an order ≺ on by setting (i, j ) ≺ (k, ) if and only if either i + j < k + or i + j = k + and j < . Note that ≺ is a well-ordering, and that the smallest elements of are If f ∈ S 2 (W + ), let γ (f ) be the degree of the leading term of f with respect to the order on ; so Our convention going forward is that if X is a d-graded object, then X d = {x ∈ X : d(x) = d}.
We then have: and let π ij : S 2 (W + ) → V ij be the projection. There is some integer N > 1 so that for all N ≤ i ≤ j and for all d-homogeneous f ∈ S 2 (W + ) with γ (f ) = (i, j ), the linear map Assume Lemma 4.3 for the moment. For any n ∈ N, let and let S (n) = k · (x i x j : n ≤ i ≤ j with n = j). There is a chain of W + -modules
Since e k · f m = (m − k)f m+k , thus S (n)/S(n + 1) is isomorphic to a subrepresentation of W + and is noetherian.
where N is the constant given in Lemma 4.3. By the above, S 2 (W + )/S is noetherian, so it suffices to prove that S is noetherian. Since S is N-graded by degree, by [8, Proposition 1.6.7] it suffices to prove that any d-graded submodule is finitely generated.

Proof of Lemma 4.3
The proof is computational. We write e λ λ λ = e λ 1 . . . e λ k where λ λ λ = (λ 1 ≤ λ 2 ≤ · · · ≤ λ k ) is a partition. (In this proof, λ λ λ will be a partition of 6, but later we will use this notation for a general partition.) Thus, for example, e 114 = e 2 1 e 4 . We have: We rewrite these computations by defining vectors v 0 , . . . , v 6 in Z[i, j ] 11 so that v k consists of the coefficients of x i+k x j +6−k in the expressions above; in other words we have the matrix equation (4.4) defining an element of S 2 (W + ). Explicitly, Note that the v k depend on i and j .
Let q ab be the coefficient of x a x b in Eq. 4.4. If j − i > 6 then the elements x i x j +6 , . . . , x i+6 x j are distinct, and q ab may be read directly from Eq. 4.4. Slightly more generally, in fact, if j > i + k, then q i+k,j +6−k = αv k . First assume that j > i + 6. It follows from Eq. 4.5 that for p = λ λ λ 6 α λ λ λ e λ λ λ we have and B is the matrix with columns v 0 (i, j ), v 1 The statement of the proposition is equivalent to the statement that BC has rank 4, and for this, since C clearly has rank 4, it is sufficient that B has (full) rank 10; in other words, we claim that for i 0, the vectors in Eq. 4.7 are linearly independent. Let X be the locus in the rational (i, j )-plane Spec Q[i, j ] where the vectors (4.7) are linearly independent. If X = Spec Q[i, j ], then Supp X consists of finitely many curves and finitely many isolated points, by primary decomposition. Computing in Macaulay2 (see Appendix Routine A.1), we see that X = Spec Q[i, j ] and that these finitely many curves are the lines i = −1, i = 0, i = 1, j = −1, j = 1,and i = j − 3. Our assumption that j > i + 6 means that the condition i = j − 3 is vacuous. Thus for j − 6 > i > 1, we avoid all of these curves, and increasing i further we may avoid the finitely many isolated points in Supp X. Thus there is some N so that for j − 6 > i > N, the vectors (4.7) are linearly independent, and Lemma 4.3 holds. Note that we do not need to compute the 0-dimensional components of X unless we want to calculate N exactly. This is the general case. We now suppose that j − i is small. If j = i + 6 we must modify the final column of B, replacing (4.6) by where B 6 is the matrix whose columns are By the Macaulay2 computation in Appendix Routine A.2, this holds for i 0, using similar arguments to those in the proof of the general case.
If j = i + 5 or j = i + 4 then If j = i + 5, Eq. 4.6 is replaced by Again, it suffices to prove that B 5 has full rank for i 0. This follows from the computation in Routine A.3.
If j = i + 4 then Eq. 4.6 becomes This follows from the computation in Routine A.4. If j = i + 3 then Eq. 4.6 becomes This follows from Routine A.5.
This follows from Routine A.6. If j = i + 1 or j = i, then f = x i x j . We have , The result follows similarly from the computations in Routines A.7 and A.8.

Quotients by quadratic elements
In this section we make more careful use of the computations in the previous subsection to show that if f is a nonzero homogeneous element of S 2 (W + ) and J is a Poisson ideal of S(W + ) containing f , then dim (S(W + )/J ) n has polynomial growth, see Proposition 4.13. It follows that if p is any order 2 element of U(W + ), then GKdim U(W + )/(p) < ∞, see Corollary 4.14.
We begin by establishing notation. Recall the terminology of Definition 3.1. If f : N → R + is a function with G(f ) ≤ P(d) for some d ∈ N, we say that f (n) = O(n d ) and that f has polynomial growth.
Clearly f 1 (n) = P k−1 (n) = O(n k−2 ) by Eq. 4.8. We have again as a consequence of Eq. 4.8. Finally, if = k then f 3 (n) = 0, so we may assume that > k. Then partitions counted by f 3 involve, for some b ≥ k, only the numbers Thus f 3 (n) is less than or equal to the number of ways to write In the equation above a 1 is determined by As each of the a i , b j , b ≤ n, we have that f 3 (n) ≤ n −1 . This proves the result.

Proposition 4.13
Let J be a Poisson ideal of S(W + ) that contains an element of order less than or equal to two. Then PGKdim(S(W + )/J ) < ∞.
Proof If J contains an element of order 1 the result is implied by Lemma 4.1. For the order 2 case, we will use Lemma 4.12.
Since J ∩ S 2 (W + ) is a U(W + )-submodule of S 2 (W + ), by Lemma 4.12 we have (k + ) + ⊆ γ (J ∩ S 2 (W + )) for some (k, ) ∈ . Thus for i ≥ k, j − i ≥ − k, there is some f ij ∈ J ∩ S ≤2 (W + ) with LT(f ij ) = x i x j . Since ≺ is a monomial ordering, for all partitions λ λ λ with x λ λ λ ∈ J (k, ) there is some f λ λ λ ∈ J with LT(f λ λ λ ) = x λ λ λ . Thus for any g ∈ S(W + ), by successively subtracting scalar multiples of the f λ λ λ we see that there is g ∈ S(W + ) so that g − g ∈ J and so that g is a sum of monomials not in J (k, ); further, d(g) ≤ d(f ).  We conjecture that Corollary 4.14 is true without restriction on o(f ), see Conjecture 1.3. Likewise, we conjecture that Proposition 4.13 holds for arbitrary Poisson ideals of S(W + ).
Recall that a module M is GK d-critical if GKdim M = d and the GK-dimension of any proper quotient of M is < d. That the adjoint representation of W + is GK 1-critical is Lemma 4.1. As in the proof of Proposition 4.13, for any g ∈ S 2 (W + ) there is g ∈ S 2 (W + ) so that g − g ∈ M, d(g ) ≤ d(g), and g involves only monomials of the form x i x j with i ≤ k or j ≤ . For fixed d, the number of such x i x j with i + j = d is ≤ k + . Thus dim N ≤d ≤ (k + )d and so grows at most linearly in d, and it follows that GKdim N ≤ 1 as desired.

Routine A.4
This routine is used for the case j = i + 4 of Lemma 4.3.