A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras
aa r X i v : . [ m a t h . R A ] S e p A POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRASOF INFINITE-DIMENSIONAL LIE ALGEBRAS
OMAR LE ´ON S ´ANCHEZ AND SUSAN J. SIERRA
Abstract.
We consider when the symmetric algebra of an infinite-dimensionalLie algebra, equipped with the natural Poisson bracket, satisfies the ascendingchain condition (ACC) on Poisson ideals. We define a combinatorial condi-tion on a graded Lie algebra which we call
Dicksonian because it is related toDickson’s lemma on finite subsets of N k . Our main result is: Theorem. If g is a Dicksonian graded Lie algebra over a field of characteristiczero, then the symmetric algebra S ( g ) satisfies the ACC on radical Poissonideals.As an application, we establish this ACC for the symmetric algebra of anygraded simple Lie algebra of polynomial growth, and for the symmetric algebraof the Virasoro algebra. We also derive some consequences connected to thePoisson primitive spectrum of finitely Poisson-generated algebras. Contents
1. Introduction 22. A review on radical conservative systems 42.1. Applications to radical Poisson ideals 53. Dicksonian Lie algebras 64. Elimination algorithms in S ( g ) 84.1. Proof of main theorem 145. Examples 155.1. The Witt algebra 155.2. Cartan algebras 155.3. Special Cartan algebras 175.4. Hamiltonian Cartan algebras 195.5. Contact Cartan algebras 215.6. Loop algebras 235.7. Simple graded Lie algebras of polynomial growth 256. On the Poisson Dixmier-Moeglin Equivalence 25References 29 Date : September 3, 2020.2010
Mathematics Subject Classification.
Key words and phrases. graded Lie algebra, symmetric algebra, Poisson algebra, ascendingchain conditions, Poisson spectrum. Introduction
The paper deals with, and is motivated by, Ascending Chain Conditions onPoisson ideals in certain classes of Poisson algebras. More precisely, let k be afield of characteristic zero. We study noetherianity of certain systems of Poissonideals in the symmetric algebra S ( g ) of a Z -graded Lie k -algebra g . This problemis related to the following general question: for which Lie algebras g is the systemof two-sided ideals of the enveloping algebra U ( g ) noetherian? For instance, in [16,Conjecture 1.3] the following was conjectured: Conjecture 1.1.
Let W + be the positive Witt algebra, which has basis { e n : n ∈ Z ≥ } with(1.1) [ e n , e m ] = ( m − n ) e n + m . The system of two-sided ideals of U ( W + ) satisfies the ACC.Via the associated graded construction and using [14, Proposition 1.6.8], Conjec-ture 1.1 would follow if we knew that the symmetric algebra S ( W + ) equipped withits natural Poisson structure had the ACC on Poisson ideals. While this remainsopen, in [16, Corollary 2.17] it is established that U ( W + ) satisfies the ACC on(two-sided) ideals whose associated graded Poisson ideal in S ( W + ) is radical. Thisis a consequence of [16, Theorem 2.15] which states that the symmetric algebra S ( W + ) satisfies the ACC on radical Poisson ideals.As pointed out in [16, Remark 2.18], techniques from differential algebra can beuseful in the study of the Poisson ideal structure of S ( W + ). In this paper we exploitthis idea to prove a general Poisson basis theorem for symmetric algebras S ( g ) wherethe Lie algebra g is graded and satisfies a certain “combinatorial noetherianity”condition, which we call Dicksonian as it relates to Dickson’s lemma on subsets of N k . More precisely, taking our cue from techniques involved in the differential basistheorem of Kolchin [11, Chapter III, § Theorem 1.2. (Theorem 4.9)
Let k be a field of characteristic zero and g a gradedLie k -algebra. If g is Dicksonian, then the Poisson algebra S ( g ) has ACC on radicalPoisson ideals. In Section 3 we explain what we mean by a Lie algebra being
Dicksonian . Wealso provide, in Lemma 3.4, sufficient conditions that guarantee this property for g . One easily checks that the positive Witt algebra W + satisfies these conditions,and thus S ( W + ) has ACC on radical Poisson ideals, as already pointed out in [16,Theorem 2.15]. One can also easily check that the full Witt algebra W (which hasbasis { e n : n ∈ Z } and Lie bracket (1.1)) satisfies these conditions, and thus thesymmetric algebra S ( W ) also has ACC on radical Poisson ideals. To our knowledgethis does not appear elsewhere. Remark . We note that the conclusion of Theorem 1.2 cannot generally bestrengthened to the ACC on the whole system of Poisson ideals. Consider thefollowing example. Let D be the Lie algebra generated as a k -vector space by x , x , . . . , and y , with bracket[ x i , x j ] = 0 and [ y, x i ] = x i +1 . We can equip D with the Z -grading where x i has degree i and y has degree 1. Onereadily checks that D has ACC on Lie ideals and, moreover, it is Dicksonian. Thus POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 3 S ( D ) has ACC on radical Poisson ideals. However, S ( D ) does not have ACC onarbitrary Poisson ideals. For example, the chain[ x ] ⊆ [ x , x ] ⊆ [ x , x , x ] ⊆ · · · is strictly increasing. Here [ S ] denotes the Poisson ideal generated by S ⊆ D .We note that in S ( W + ) the chain [ e ] ⊆ [ e , e ] ⊆ [ e , e , e ] ⊆ · · · does stabilise,by [16, Corollary 4.8]. This illustrates the delicacy of the noetherianity questionswe discuss here.We further note that the conclusion of Theorem 1.2 cannot be achieved withoutassuming some suitable chain condition on the ideals of the Lie algebra g . Lemma 1.4.
Let k be a field and let g be a Lie k -algebra. If g is not noetherian,then S ( g ) does not have ACC on radical Poisson ideals.Proof. As g is not noetherian, there exists a strictly increasing chain of Lie-ideals I ⊂ I ⊂ · · · . Let P i be the ideal of S ( g ) generated by I i for i = 1 , , . . . . Then,as I i is a Lie-ideal, it can easily be checked that P i is a Poisson ideal of S ( g ). Also,as the P i ’s are generated by linear terms, they are all prime. Finally, one readilychecks that the chain P ⊂ P ⊂ · · · is strictly increasing (indeed any element in I i +1 \ I i is not in P i ). (cid:3) Examples of non-noetherian Lie k -algebras include free Lie algebras over k inat least two generators. For instance, in [1, Theorem 1.1], it is shown that if I is a nontrivial Lie-ideal of a free Lie algebra then [ I, I ] is not finitely generatedas a Lie-ideal. These examples show that even when the Lie algebra g is finitelygenerated it is not generally the case that S ( g ) has ACC on radical Poisson ideals.Our long-term goal is to apply Theorem 1.2 to a wide class of Poisson algebras:for instance, to all symmetric algebras of noetherian graded Lie algebras. We arethus far not aware of a counter-example, although we caution that noetherian Liealgebras can be quite wild; see [7]. A somewhat more accessible short-term goal isto restrict ourselves to Lie algebras with well-behaved growth. In fact, we expect: Conjecture 1.5.
Let k be an algebraically closed field of characteristic zero and g a graded Lie k -algebra of polynomial growth (also called finite growth). If g hasACC on Lie ideals, then g is Dicksonian. (As a consequence of Theorem 1.2, S ( g )would have ACC on radical Poisson ideals).A reasonable place to start towards proving Conjecture 1.5, is the case when g isa simple graded Lie algebra of polynomial growth. When the field k is algebraicallyclosed, these Lie algebras have been classified by Mathieu [13]; they are either finitedimensional, loop algebras, Cartan algebras, or the Witt algebra. In Section 5, weverify the conjecture for all these classes of Lie algebras, and then prove: Corollary 1.6. (Corollary 5.14)
Let k be a field of characteristic zero. If g is asimple graded Lie k -algebra of polynomial growth, then the symmetric algebra S ( g ) has ACC on radical Poisson ideals. We also prove that the Virasoro algebra and several related Lie algebras areDicksonian, and thus their symmetric algebras have ACC on radical Poisson ideals.In our final section, Section 6, we consider the Poisson version of the Dixmier-Moeglin equivalence relating primitive, rational, and locally closed prime ideals of
OMAR LE ´ON S ´ANCHEZ AND SUSAN J. SIERRA enveloping algebras of finite-dimensional Lie algebras. We make several remarkson the Poisson primitive spectrum of Poisson algebras of countable vector spacedimension; in particular, those that are finitely Poisson-generated. For instance,we prove in Theorem 6.3 that the notions of Poisson-primitive and Poisson-rationalcoincide in this setting. We then pay a closer look at S ( W + ) in this context. Acknowledgements:
We thank Rekha Biswal and Alexey Petukhov for usefuldiscussions on twisted loop algebras and Poisson primitive ideals, respectively.
Notation:
For us, N = { , , , . . . } .2. A review on radical conservative systems
In this section we review some standard results on radical divisible conservativesystems in arbitrary commutative rings. We follow closely [11, Chapter 0, § §
9] (where the omitted proofs appear). We then specialize these to the context ofPoisson algebras where the main result for us is Theorem 2.7.We fix a commutative ring R with unit. Recall that a conservative system C of R is a collection of ideals of R with the following two properties:(CS1) the intersection of elements in C is in C , and(CS2) the union of a chain (totally ordered set by inclusion) from C is again in C .For example, the collection of all ideals of R is a conservative system; as is, onthe other extreme, the collection consisting just of R .Let C be a conservative system of R . We refer to the elements of C as C -ideals.Given an arbitrary subset A of R , we denote by ( A ) C the intersection of all the C -ideals containing A . Note that ( A ) C is in C by condition (CS1). Hence, we callit the C -ideal C -generated by A . If a C -ideal I is of the form (Σ) C for some finiteset Σ, we say that I is finitely C -generated .Recall that given an ideal I of R and s ∈ R , the division of I by s is the ideal I : s defined by { r ∈ R : rs ∈ I } . Definition 2.1. (i) The conservative system C is called divisible if for all I ∈ C and s ∈ R , wehave I : s ∈ C .(ii) The conservative system C is called radical if all of its elements are radicalideals.(iii) A conservative system C is said to be noetherian if it satisfies the ACC on C -ideals (equivalently, every C -ideal is finitely C -generated). Example . The collection of all radical ideals of R is a radical divisible conser-vative system.The following two lemmas are two of the main ingredients for the proof of The-orem 2.5 below. Lemma 2.3. [11, § Suppose C is a radical divisible conservativesystem of R . If T and S are arbitrary subsets of R , then ( T · S ) C = ( T ) C ∩ ( S ) C . Here T · S denotes the set of products of the form ts ∈ R with t ∈ T and s ∈ S . POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 5
Lemma 2.4. [11, § Suppose C is a radical divisible conservativesystem of R . If C is not noetherian, then there is a C -ideal that is maximal (withrespect to inclusion) among the C -ideals that are not finitely C -generated and, moreimportantly, any such C -ideal is prime. We now prove an algebraic “basis theorem”, which we will use in the Poissoncontext later on.
Theorem 2.5.
Let C be a radical divisible conservative system of R . Assume that (*) if P is a prime C -ideal, then there is a finite Σ ⊂ P and s ∈ R \ P suchthat P = (Σ) C : s .Then, C is noetherian.Proof. The proof can be deduced from arguments in [11, Chapter 0, § C is not noetherian. Then, by Lemma 2.4, thereis a maximal C -ideal M , with respect to inclusion, among the C -ideals that are notfinitely C -generated, and M is prime. By assumption (*), there is a finite set Σ ⊂ M and s ∈ R \ M such that M = (Σ) C : s . It follows that s · M ⊆ (Σ) C , Furthermore,as s / ∈ M , by choice of M there is a finite Φ ⊂ M such that ( s, M ) C = ( s, Φ) C .Thus, using Lemma 2.3, we get M = M ∩ ( s, M ) C = M ∩ ( s, Φ) C = ( s · M, Φ) C = (Σ , Φ) C . This contradicts the fact that M is not finitely C -generated. (cid:3) We conclude this review on conservative systems with a decomposition-type the-orem for radical ideals. If C is a conservative system of R and I is a radical C -ideal,by a C -component of I we mean a minimal element of the set, ordered by inclusion,of prime C -ideals that contain I . We have the following Proposition 2.6. [11, § § Assume C is a radicaldivisible conservative system of R and let I be a C -ideal. Then, the following hold: (i) I is the intersection of its C -components, (ii) If I is the intersection of finitely many prime ideals none of which containsthe other, then these prime ideals are in C and are the C -components of I . (iii) If C is noetherian, I is the intersection of a finite set of prime C -ideals noneof which contains the other. This finite set is unique, being precisely theset of C -components of I . Applications to radical Poisson ideals.
We now specialize some of theresults above to Poisson algebras (the rest of the translations are left to the inter-ested reader). Let ( A, {− , −} ) be a Poisson algebra over a field k ; note we makeno assumption on the characteristic at this point. Let C Poi be the collection of allradical Poisson ideals of A . Then C Poi is a radical divisible conservative system of A . Indeed, all conditions are more or less clearly satisfied; we only provide detailson divisibility. Let I ∈ C Poi and s ∈ A . We prove that I : s ∈ C Poi (i.e., I : s is aradical Poisson ideal of A ). Radicality easily follows. Now let g ∈ A , and suppose f ∈ I : s . Then sf and s f are in I ; and so { g, s f } ∈ I (as I is Poisson). But { g, s f } = { g, s } sf + s { g, f } , and so s { g, f } ∈ I . Hence, { g, f } ∈ I : s (as I isradical), and so I : s is Poisson.For Σ any subset of A , we let { Σ } denote the radical Poisson ideal of A generatedby Σ. Note that { Σ } = (Σ) C Poi . Here are the relevant specializations:
OMAR LE ´ON S ´ANCHEZ AND SUSAN J. SIERRA
Theorem 2.7.
Let ( A, {− , −} ) be a Poisson algebra over a field k , and let C Poi bethe system of radical Poisson ideals. (i)
Assume that for every prime Poisson ideal P of A there is a finite set Σ ⊂ P and s ∈ A \ P such that P = { Σ } : s . Then, C Poi is noetherian. (ii)
Let I be a C Poi -ideal (i.e., a radical Poisson ideal). Then, I is the intersec-tion of prime Poisson ideals. Furthermore, if the system C Poi is noetherian,then I is the intersection of a finite set of prime Poisson ideals none ofwhich contains the other (this set is unique and its elements are called thePoisson-components of I ). (cid:3) Below we will use part (i) of this theorem to prove noetherianity of the system C Poi for Poisson algebras of the form S ( g ) where g is a Dicksonian graded Liealgebra over a field k of characteristic zero.3. Dicksonian Lie algebras
Our goal in this section is to define the combinatorial condition used in Theo-rem 1.2, which is named for Dickson’s lemma on subsets of N k (see Theorem 5.3).We consider orderings on a Lie algebra g and the information they give us on Lieideals of g . We will then define the concept of a leading-Dicksonian sequence : asequence of (pairs of) elements of g satisfying a certain chain condition. A Liealgebra is Dicksonian if it has no infinite leading-Dicksonian sequence. We will seein Section 4 that Dicksonian Lie algebras have an elimination algorithm, which al-lows us to derive striking consequences for radical Poisson ideals of their symmetricalgebras.Throughout this section we assume that g is a ( Z -)graded Lie algebra over a field k of characteristic zero. Namely, g is a Lie k -algebra equipped with a decomposition g = L n ∈ Z g n such that [ g n , g m ] ⊆ g n + m and the homogeneous components g n havefinite dimension. We refer the reader to [13, §
1] for basic facts on graded Liealgebras.Let M be a k -basis of g consisting of homogeneous elements equipped with a totalorder ( M , < ) that is compatible with the grading (i.e., larger in degree implies largerin the order < ). As the g n are finite-dimensional, M has the order type of a subsetof Z . We let M + and M − denote the elements of M of positive and negative degree(with respect to the grading of g ), respectively. Furthermore, for nonzero e ∈ g , welet ℓ + ( e ), respectively ℓ − ( e ), denote the largest, respectively smallest, element of M with respect to < that appears in e when written as a k -linear combination ofelements of M . We call ℓ + ( e ) the upper-leader and ℓ − ( e ) the lower-leader of e . Asconvention, we set ℓ ± (0) = 0. Definition 3.1.
Let I ± be the set of (nonempty) finite tuples of elements from M ± . For any i = ( M , . . . , M n ) ∈ I ± we let D ± i be the operator on M given by M D ± i ( M ) := ℓ ± ([[[ M, M ] , M ] , · · · , M n ]) . For M ∈ M , we set L + ( M ) to be the set of elements of M of the form D + i ( M ) , for i ∈ I + , such that D + i ( N ) < D + i ( M ) for all N < M whenever D + i ( N ) is nonzero.Similarly, L − ( M ) denotes the set of elements of M of the form D − i ( M ) , POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 7 for i ∈ I − , such that D − i ( N ) > D − i ( M ) for all N > M whenever D − i ( N ) is nonzero.We summarise the notation in Table 1. Table 1.
Notation for operations on an ordered basis M of gM ± elements of M of positive (negative) degree ℓ ± ( e ) largest (smallest) element of M occurring in e ∈ g I ± finite tuples from M ± D ± i ( M ) ℓ ± ([[[ M, M ] , M ] , · · · , M n ]) where i = ( M , . . . , M n ) ∈ I ± L + ( M ) { D + i ( M ) : D + i ( N ) < D + i ( M ) if i ∈ I + , N < M, D + i ( N ) = 0 }L − ( M ) similar Definition 3.2. (i) A sequence of distinct pairs(( M i , N i )) ni =1 from M with n ≤ ω and M i ≤ N i is said to be leading-Dicksonian (withrespect to the order < of M ) if M j / ∈ L − ( M i ) and N j / ∈ L + ( N i ) for all i < j .(ii) We say that g is Dicksonian if there is a basis of homogeneous elementswith an order compatible with the grading such that, with respect to thisorder, there is no infinite leading-Dicksonian sequence.
Example . The (full) Witt algebra W is Dicksonian. Indeed, choosing the stan-dard k -basis ( e n : n ∈ Z ), for which [ e i , e j ] = ( j − i ) e i + j , one sees that the order in W given by e i < e j iff i < j has the desired properties. For example, L + ( e ) = { e i : i > } , and if n = 1 then L + ( e n ) = { e i : i > n } . The positive Witt algebra W + = span k ( e i : i ≥ , and the Cartan algebra W = span k ( e i : i ≥ − Lemma 3.4.
Suppose g is a graded Lie algebra with an ordered basis ( M , < ) where M consists of homogeneous elements and the order is compatible with the grading.Assume that the Lie bracket of two basis elements is a scalar multiple of a basiselement and that the following condition holds ( † ) if M , M , M ∈ M ± are such that [ M , M ] and [ M , M ] are nonzero and M < M , then ℓ ± ([ M , M ]) < ℓ ± ([ M , M ]) .If g + and g − have ACC on graded Lie ideals, then g has no infinite leading-Dicksonian sequence. Note in ( † ) that here [ M , M ] and [ M , M ] are scalar multiples of a basis element,and ℓ ± simply extracts this element. OMAR LE ´ON S ´ANCHEZ AND SUSAN J. SIERRA
Proof.
Towards a contradiction, let (( M i , N i )) ∞ i =1 be an infinite leading-Dicksoniansequence. Note that there are either infinitely many M i ’s in M − or infinitely N i ’sin M + . Without loss of generality assume the latter, we will show that g + has astrictly ascending chain of graded Lie ideals, contradicting our assumption.We thus have an infinite sequence ( N i ) ∞ i =1 of homogenous elements of g + withthe property that N j / ∈ L + ( N i ) for all i < j . It suffices to show that this lattercondition implies that N j is not in the Lie ideal generated by N , . . . , N j − in g + .Suppose towards a contradiction that N j is in this Lie ideal. By our assumptionthat the bracket of basis elements yields a scalar multiple of a basis element, wemust have that N j = D + i ( N i ) for some i ∈ I + and i < j . This yields, by condition( † ), that N j ∈ L + ( N i ), a contradiction. The result follows. (cid:3) Example . (1) Consider the loop algebra c sl := sl ( k )[ t, t − ]. Letting e, f, h be the stan-dard basis of sl ( k ), let M = { et i , f t j , ht k : i, j, k ∈ Z } . Give e, f, h, t degrees 1 , − , ,
3, respectively, and order elements of M by degree. ByLemma 3.4, c sl is Dicksonian.(2) It also follows from Lemma 3.4 that the Lie algebra D of Remark 1.3 isDicksonian. Remark . If g has a basis M so that for all M ∈ M we have(3.1) L + ( M ) ∪ L − ( M ) is cofinite in M , then g is easily seen to be Dicksonian; noting that M has order type of a subset of Z . This is true even if (3.1) holds for all but finitely many M ∈ M . This gives analternate proof that W , W + , and W are Dicksonian. Example . Let
Vir be the
Virasoro algebra , which has basis { e n : n ∈ Z } ∪ { z } and Lie bracket[ e n , e m ] = ( m − n ) e n + m + n − n δ n + m, z, [ e n , z ] = 0 . Ordering the basis by · · · < e − < e − < z < e < e < e < . . . , one sees that (3.1) is satisfied for all basis elements except z . Thus by Remark 3.6, Vir is Dicksonian. 4.
Elimination algorithms in S ( g )In this section we prove our main result, Theorem 1.2. One of the key ingredientsis an Elimination Algorithm result for S ( g ) that we prove in Theorem 4.5 below.We take our cue/presentation from the elimination theory of differential polynomialrings; see for instance [11, Chapter I]. Assumptions:
Throughout this section we assume that g is a graded Lie algebraover a field k of characteristic zero and M is a k -basis consisting of homogeneouselements equipped with an order < compatible with the grading. Define M + and M − as in Section 3. Recall that the symmetric algebra S ( g ) is the polynomial ringin the formal variables M over k (i.e., S ( g ) = k [ M ]) and it carries a natural Poissonbracket, that we denote by {− , −} , induced from the Lie bracket on g . POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 9
Let f be a nonconstant element in S ( g ); i.e., a nonconstant polynomial in k [ M ].We define the upper leader of f , denoted by ℓ + ( f ), to be the largest element in M (according to the fixed order < ) that appears (nontrivially) in f . Then, f can bewritten in the form f = d X i =0 g i ( ℓ + ( f )) i with the g i ’s in k [ M ] having leaders strictly smaller than ℓ + ( f ), and g d nonzero. Wedefine the upper-degree of f , denoted d + ,f , to be this d . We call g d the upper-initialof f , denoted i + ,f . The upper-separant of f is defined to be s + ,f = ∂f∂ℓ + ( f ) . The upper-rank of f is defined as rk + ( f ) = ( ℓ + ( f ) , d + ,f ). We can compare elementsfrom S ( g ) lexicographically by upper-rank. Note that s + ,f and i + ,f both have lowerupper-rank than f .In a similar fashion one defines the lower-leader of f , denoted by ℓ − ( f ), as thesmallest element of M that appears in f . The notions of lower-degree d − ,f , lowerinitial i − ,f , lower-separant s − ,f , and lower-rank rk − ( f ) are defined similarly. Definition 4.1.
Recall that I ± denotes the set of (nonempty) finite tuples ofelements from M ± . For any i = ( M , . . . , M n ) ∈ I ± we let D i be the operator on S ( g ) given by f D i ( f ) := {{{ f, M } , M } , · · · , M n } . Note that D i ( f ) is always in the Poisson ideal [ f ] generated by f .For f ∈ S ( g ) we let s f = s + ,f · s − ,f . For Λ a finite subset of S ( g ) we let ℓ + (Λ) = max { ℓ + ( f ) : f ∈ Λ } and ℓ − (Λ) = min { ℓ − ( f ) : f ∈ Λ } .We summarise this notation in Table 2. Table 2.
Notation for operations on S ( g ) ℓ ± ( f ) upper- (lower-) leader of f ∈ S ( g ) d ± ,f uppper- (lower-) degree of fi ± ,f upper- (lower-) initial of fs ± f upper- (lower-) separant of f , ∂f∂ℓ ± ( f ) rk ± ( f ) upper- (lower-) rank of f , ( ℓ ± ( f ) , d ± ,f ) D i ( f ) {{{ f, M } , M } , · · · , M n } for i = ( M , . . . , M n ) ∈ I ± s f s + ,f · s − ,f ℓ + (Λ) max { ℓ + ( f ) : f ∈ Λ } ℓ − (Λ) min { ℓ − ( f ) : f ∈ Λ } .The content of the next lemma is that if the sets L ± ( e ) are large enough, onecan control the largest/smallest element of M appearing in some D i ( f ). Lemma 4.2.
Let f ∈ S ( g ) be nonconstant and i ∈ I ± . If D ± i ( ℓ ± ( f )) ∈ L ± ( ℓ ± ( f )) ,then ℓ ± ( D i ( f )) = D ± i ( ℓ ± ( f )) and D i ( f ) = αs ± ,f D ± i ( ℓ ± ( f )) + h ± for some α ∈ k ∗ and h + , h − ∈ S ( g ) , with ℓ − ( f ) ≤ ℓ − ( h + ) ≤ ℓ + ( h + ) < D + i ( ℓ + ( f )) for i ∈ I + and D − i ( ℓ − ( f )) < ℓ − ( h − ) ≤ ℓ + ( h − ) ≤ ℓ + ( f ) for i ∈ I − . Proof.
We establish the result with the positive indices (the case with negativeindices is analogous). Write f = P di =0 g i ( ℓ + ( f )) i with ℓ + ( g i ) < ℓ + ( f ). Then, forany M ∈ M + , using the fact that {− , M } : S ( g ) → S ( g ) is a derivation we get D M ( f ) := { f, M } = s + ,f { ℓ + ( f ) , M } + d X i =0 { g i , M } ( ℓ + ( f )) i . If D + M ( ℓ + ( f )) = ℓ + ( { ℓ + ( f ) , M } ) ∈ L + ( ℓ + ( f )), then, by definition, ℓ + ( { g i , M } ) < D + M ( ℓ + ( f )) . Also, since the order < is compatible with the grading we get ℓ + ( f ) < D + M ( ℓ + ( f )),which also implies that ℓ + ( s + ,f ) < D + M ( ℓ + ( f )). Thus, the upper-leader of { f, M } is D + M ( ℓ + ( f )), and D M ( f ) = αs + ,f D + M ( ℓ + ( f )) + h + where α is the coefficient of D + M ( ℓ + ( f )) when writing { ℓ + ( f ) , M } in terms of thebasis M , and h + = P di =0 { g i , M } ( ℓ + ( f )) i − s + ,f (cid:0) { ℓ + ( f ) , M } − αD + M ( ℓ + ( f )) (cid:1) .Finally, note that all terms of h + are of the form e or appear in { e, M } for some e ∈ M that appears in f . Since the order < on M is compatible with the gradingand M ∈ M + , we get that all these terms are larger or equal to ℓ − ( f ) showing that ℓ − ( f ) ≤ ℓ − ( h + ).We have thus establish the result for the case when i is the 1-tuple M . For longerlength tuples simply iterate this process. (cid:3) Definition 4.3. (i) Let f, g ∈ S ( g ) be nonconstant (i.e., not in k ). We say that g is partiallyreduced with respect to f if no element in L ± ( ℓ ± ( f )) appears (nontrivially)in g . If in addition ℓ + ( f ) appears in g only with degree < d + ,f , we say that g is reduced with respect to f .(ii) Suppose Λ = ( f i ) ni =1 is a sequence of nonconstant elements of S ( g ) with n ≤ ω . If g is in S ( g ), we say that g is (partially) reduced with respect to Λif g is (partially) reduced with respect to every element in Λ (by conventionconstant elements are reduced). Furthermore, we say that Λ is (partially)reduced if every f j is (partially) reduced with respect to f i for all i < j . Remark . Note that in a reduced sequence of elements from S ( g ) distinct ele-ments have distinct upper-rank (where recall that rk + ( f ) = ( ℓ + ( f ) , d + ,f )).The following Elimination Algorithm is one of the key ingredients of the proofof Theorem 1.2. Theorem 4.5.
Let Λ be a finite sequence of nonconstant elements of S ( g ) and g ∈ S ( g ) . POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 11 (1)
There exists g ∈ S ( g ) partially reduced with respect to Λ and integers r f ≥ for each f ∈ Λ such that Y f ∈ Λ s r f f g ≡ g mod[Λ] and min { ℓ − ( g ) , ℓ − (Λ) } ≤ ℓ − ( g ) ≤ ℓ + ( g ) ≤ max { ℓ + ( g ) , ℓ + (Λ) } . (2) Furthermore, if Λ is reduced, then there also exists g ∈ S ( g ) reduced withrespect to Λ and integers m f , n f ≥ for each f ∈ Λ such that Y f ∈ Λ i m f + ,f s n f f g ≡ g mod[Λ] and ℓ − ( g ) ≤ ℓ − ( g ) ≤ ℓ + ( g ) ≤ ℓ + ( g ) . Proof. (1) If no element in L + ( ℓ + ( f )), for f ∈ Λ, appears in g then we let g + , = g and r + ,f = 0. Otherwise, let M be the largest element of M that appears in g and isin L + ( ℓ + ( f )) for some f ∈ Λ. Then M = D + i ( ℓ + ( f )) for some i = ( M , . . . , M n ) ∈I + . By Lemma 4.2, we have D i ( f ) = − αs + ,f D + i ( ℓ + ( f )) + h + for some α ∈ k ∗ and h + ∈ S ( g ) with ℓ − ( f ) ≤ ℓ − ( h + ) ≤ ℓ + ( h + ) < M .Now write g = P rj =0 h j M j where h j is free of M (and so the largest element of M appearing in h j that is in L + ( ℓ + ( p )) for some p ∈ Λ is strictly less than M ).Then, g = P rj =0 h j ( D + i ( ℓ + ( f ))) j and so(4.1) s r + ,f g = r X j =0 h j s r − j + ,f ( s + ,f D + i ( ℓ + ( f ))) j ≡ r X j =0 h j s r − j + ,f ( α − h + ) j mod[Λ] . Since ℓ + ( s + ,f ) ≤ ℓ + ( f ) < M and ℓ + ( h + ) < M , the largest element from M appearing in g ′ + := P rj =0 h j s r − jf ( α − h + ) j that is in L + ( ℓ + ( p )) for some p ∈ Λ isstrictly less than M ≤ ℓ + ( g ). Furthermore, since ℓ − ( f ) ≤ ℓ − ( h + ), we havemin { ℓ − ( g ) , ℓ − ( f ) } ≤ ℓ − ( g ′ + ) ≤ ℓ + ( g ′ + ) ≤ ℓ + ( g ) ≤ max { ℓ + ( g ) , ℓ + ( f ) } . Repeat the above process on g ′ + until we reach g + , ∈ S ( g ) such that no elementin L + ( ℓ + ( f )), for f ∈ Λ, appears in it (this process eventually terminates as Λ isfinite). Note that, by (4.1), Y f ∈ Λ s r + ,f + ,f g ≡ g + , mod[Λ] , for some r + ,f , andmin { ℓ − ( g ) , ℓ − (Λ) } ≤ ℓ − ( g + , ) ≤ ℓ + ( g + , ) ≤ max { ℓ + ( g ) , ℓ + (Λ) } . Now, if no element in L − ( ℓ − ( f )), for f ∈ Λ, appears in g + , then we let g = g + , and r f = r + ,f . Otherwise, we perform the counterpart (i.e., negative-indices) of theabove process to g + , . Again the process will eventually terminate, as Λ is finite,and yields the desired g . (2) Let Λ = ( f , . . . , f s ) and now we assume that it is reduced. Since distinctelements in Λ have distinct upper-rank, there are k , . . . , k s ∈ { , . . . , s } such that rk + ( f k ) > rk + ( f k ) > · · · > rk + ( f k s ) . If ℓ + ( f k ) does not appear in g then we let g , = g . Otherwise, assume ℓ + ( f k )appears in g with degree r ≥ d := d + ,f k . Write g = P rj =0 h j ( ℓ + ( f k )) j with h j free of ℓ + ( f k ). Then, in(4.2) i + ,f k g − h r ( ℓ + ( f k )) r − d f k ℓ + ( f k ) appears with degree ≤ r −
1. Repeating this process yields ˜ g , reducedwith respect to f k . Since Λ is reduced and g is partially reduced with respectto Λ k := ( f , . . . , f k ), ˜ g , is partially reduced with respect to Λ k . However, theabove process (4.2) might yield ˜ g , that is not partially reduced with respect toΛ ∗ k := { f k +1 , . . . , f s } (as f k is not necessarily partially reduced with respect toΛ ∗ k ). Note that since g is partially reduced with respect to Λ, the only way thatthe above process could produce an element which is not partially reduced withrespect to Λ ∗ k is if in f k appears an element from L + ( ℓ + f i ) for some i > k . Infact, any element from L + ( ℓ + ( f i )), for some i > k , that appears in ˜ g , is < ℓ + ( f k )as ℓ + ( f k ) cannot be in any ℓ + ( f i ) since it appears in g .Now perform the algorithm from part (1) to ˜ g , with f = f k . Notice that onthe right-handed term in (4.1) the upper-leaders of s + ,f and h + will be < ℓ + ( f k )and in the h j ’s this basis term will appear with degree < d + ,f k . Thus, the outputof the algorithm from part (1) is g , with degree in ℓ + ( f k ) strictly less than d + ,f k .In other words, g , is partially reduced with respect to Λ and reduced with respectto f k .Perform the same process with f k and g , to obtain g , partially reduced withrespect to Λ and reduced with respect to f k . Note that this g , will also bereduced with respect to f k . Indeed, if ℓ + ( f k ) = ℓ + ( f k ) then it is clear since rk + ( f k ) > rk + ( f k ); otherwise, ℓ + ( f ) > ℓ + ( f k ) and in this case the degree of ℓ + ( f k ) is not increased in the algorithm of part (1) applied to g , , meaning thatits degree in g , is < d + ,f k and so g , is reduced with respect to f k . Repeatingthe above process yields the desired g . (cid:3) The Elimination Algorithm yields the following useful corollary:
Corollary 4.6.
Suppose that the basis M of g satisfies (3.1) . Then, for any nonzeroprime Poisson ideal P of S ( g ) , there is h ∈ S ( g ) \ P such that the localisation ( S ( g ) /P ) h is an affine k -algebra.Proof. Condition (3.1) is equivalent to the hypothesis that for each M ≤ N ∈ M the k -subspace of g spanned by L − ( M ) ∪ L + ( N ) has finite codimension. Let f bea nonzero element of P such that s f is not in P (for instance, choose f ∈ P ofminimal total degree). By part (1) of the Elimination Algorithm, for any g ∈ S ( g )there is g partially reduced with respect to f and an integer r ≥ s rf g ≡ g mod [ f ] . By our assumption, g lives in the affine k -algebra generated by the finite set M \ ( L − ( ℓ − ( f )) ∪ L + ( ℓ + ( f ))). Thus, setting h = s f , we get that ( S ( g ) /P ) h is afinitely generated k -algebra. (cid:3) POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 13
Under the hypothesis of Corollary 4.6 g is automatically Dicksonian by Re-mark 3.6. Note that S ( Vir ) / ( z ) ∼ = S ( W ) is not contained in any affine k -algebra,so for the proof of Corollary 4.6 we need (3.1) to hold for all M ∈ M .The following lemma is used to relate the Dicksonian condition to chain condi-tions in symmetric algebras. Lemma 4.7. If g is Dicksonian, then every reduced sequence of S ( g ) is finite.Proof. Suppose there is an infinite reduced sequence Λ = ( f , f , · · · ) in S ( g ). Weclaim that an infinite subsequence of ( ℓ − ( f i ) , ℓ + ( f i )) ∞ i =1 is leading-Dicksonian. Thissequence of pairs clearly satisfies ℓ − ( f i ) ≤ ℓ + ( f i ). Also, by definition of reducedsequence, it satisfies ℓ − ( f j ) / ∈ L − ( ℓ − ( f i )) and ℓ + ( f j ) / ∈ L + ( ℓ + ( f i ))for all i < j . Thus, the only condition from the definition of leading-Dicksonianthat is missing is that the elements of ( ℓ − ( f i ) , ℓ + ( f i )) ∞ i =1 are distinct. Now, becausein the definition of reduced sequence we require that ℓ + ( f i ) appears in f j only withdegree strictly less than d + ,f , equality of ℓ + ( f j ) and ℓ + ( f i ) with i < j can onlyhappen finitely many times. Thus there will be a subsequence with the desiredproperties, contradicting our assumption. (cid:3) Recall that if I is a radical ideal then the division ideal I : s is again radical, andif I is radical and Poisson then the same is true of I : s . Further, if I is Poissonthen the ideal I : s ∞ = { f ∈ S ( g ) : f s n ∈ I for some n ≥ } is again Poisson, by a similar argument to the proof that C Poi is a radical conser-vative system (see § S ( g ), we let [Λ] denote the Poisson ideal generatedby Λ in S ( g ) and set i + s Λ = Q f ∈ Λ i + ,f s f , where recall that s f = s + ,f · s − ,f . InTheorem 4.8 below we will be looking at Poisson ideals of the form [Λ] : i + s ∞ Λ .The above Elimination Algorithm yields the following fundamental “basis theo-rem” for prime Poisson ideals: Theorem 4.8.
Assume g is Dicksonian. If P is a nonzero prime Poisson ideal of S ( g ) , then there is an reduced set Λ (which is hence finite by Lemma 4.7) containedin P such that i + s Λ / ∈ P and P = [Λ] : i + s ∞ Λ . Proof.
Let f be a nonzero element of P of minimal total degree. Since the upper-initial i + ,f , the upper-separant s + ,f and the lower-separant s − ,f are nonzero oftotal degree smaller than that of f , none of them is in P . Let Λ be the singletonsequence ( f ). As P is prime it follows that i + s Λ / ∈ P , and so[Λ ] : i + s ∞ Λ ⊆ P. If P = [Λ ] : i + s ∞ Λ we are done. Otherwise, there is g ∈ P but not in [Λ ] : i + s ∞ Λ .By Theorem 4.5, there is g which is reduced with respect to Λ and integers m f , n f ≥ i m f + ,f s n f f g ≡ g mod[Λ ] . So g is in P and is nonzero (otherwise g would be in [Λ ] : i + s ∞ Λ ). So we can choose f of minimal total degree among the nonzero elements in P that are reduced withrespect to Λ . Then, the upper initial and upper and lower separants of f are notin P (as they are all nonzero reduced with respect to Λ and of total degree smallerthan that of f ). Let Λ be the sequence ( f , f ). Then Λ is reduced and, as P isprime, i + s Λ / ∈ P . So we have [Λ ] : i + s ∞ Λ ⊆ P. If [Λ ] : i + s ∞ Λ is not equal to P , we can continue this process and find f suchthat the sequence Λ = ( f , f , f ) is reduced and i + s Λ / ∈ P . This process musteventually stop since reduced sequences are finite (by Lemma 4.7). Thus, thisprocess yields a (finite) reduced sequence Λ such that i + s Λ is not in P and P = [Λ] : i + s ∞ Λ . (cid:3) Proof of main theorem.
We can now easily prove Theorem 1.2. We restateit for the reader’s convenience.
Theorem 4.9.
Let k be a field of characteristic zero and g a graded Lie k -algebra.If g is Dicksonian, then the Poisson algebra S ( g ) has ACC on radical Poisson ideals.Proof. By Theorem 4.8, for every prime Poisson ideal P of S ( g ) there is a finitereduced sequence Λ ⊂ P such that i + s Λ / ∈ P and(4.3) P = [Λ] : i + s ∞ Λ . Setting s = i + s Λ we see that [Λ] : i + s ∞ Λ ⊆ { Λ } : s where { Λ } denotes the radical Poisson ideal generated by Λ in S ( g ). Since P isprime Poisson and s / ∈ P , we get { Λ } : s ⊆ P . By (4.3), we actually have P = { Λ } : s. The result now follows immediately from Theorem 2.7(i). (cid:3)
We finish this section by noting that our Poisson basis theorem applies to a widecollection of Poisson algebras.
Corollary 4.10.
Let A be a Poisson algebra over a field k of characteristic zeroand N ⊆ A a Poisson-generating set (i.e., N generates A as a Poisson algebra). Ifthe Lie k -algebra generated by N is Dicksonian (with respect to some Z -grading andsome ordered basis of homogeneous elements) then A has ACC on radical Poissonideals.Proof. Theorem 1.2 establishes the ACC (on radical Poisson ideals) for the Poissonalgebra S ( g ) where g is the Lie algebra generated by N . But A is a factor of thesymmetric algebra S ( g ) by a Poisson ideal, and hence we also have the ACC onradical Poisson ideals for A . (cid:3) POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 15 Examples
In this section we provide a wide family of examples of graded Lie algebraswhich are Dicksnonian. Throughout this section k is a field of characteristic zero.At the end, we prove that all simple graded Lie algebras of polynomial growth areDicksonian. Consequently, by Theorem 1.2, their symmetric algebras have ACCon radical Poisson ideals (namely, we prove Corollary 1.6) and, by Theorem 2.7(ii),every radical Poisson ideal is a finite intersection of prime Poisson ideals.5.1. The Witt algebra.
In Section 3 we saw that the Witt algebra W , the pos-itive Witt algebra W + , the Cartan algebra W , and the Virasoro algebra Vir areDicksonian (with respect to the natural choice of ordered basis).We thus get the following consequences from Theorems 1.2 and 2.7(ii).
Corollary 5.1.
Let g be one of W , W + , W , or Vir . Then S ( g ) has ACC onradical Poisson ideals and every radical Poisson ideal is a finite intersection ofprime Poisson ideals. Cartan algebras.
Let n ≥
2. In this section we prove that the Cartan algebra W n has a basis with an order < such that W n has no infinite leading-Dicksoniansequence. Theorem 1.2 then tells us that S ( W n ) has ACC on radical Poisson ideals.Let k be a field of characteristic zero. Recall that the Cartan algebra W n over k is the Lie k -algebra of derivations on k [ x , · · · , x n ]. We will use multi-indexnotation; that is, for i = ( i , . . . , i n ) ∈ N n we write x i = ( x i , . . . , x i n ) . By | i | we mean i + · · · + i n . Also, when we write i ≤ j with i, j ∈ N n we mean i ≤ j , . . . , i n ≤ j n (i.e., i < j means i is less than j in the product order of N n ).For k ∈ { , . . . , n } , we let 1 k denote the n -tuple with a 1 in the k -entry and zeroeselsewhere.We choose as M the natural basis for W n ; that is, { x i ∂ k : i ∈ N n and k ∈ { , . . . , n }} . where ∂ k = ∂∂x k . We equip this basis with the following (total) ordering: x i ∂ k < x j ∂ ℓ ⇔ ( | i | , k, i n , . . . , i ) < lex ( | j | , ℓ, j n , . . . , j ) . So this ordering is compatible with the natural Z -grading of W n .We now check that, with respect to this order, W n is Dicksonian. First we needa lemma. A word on notation. In the lemma below we write 0 = (0 , , . . . , k = 1 we set ( i , . . . , i k − ) = 0 for convenience of exposition. Lemma 5.2.
Let i ∈ N n and k ∈ { , . . . , n } . (i) If ( i , . . . , i k − ) = 0 , then L + ( x i ∂ k ) contains { x r ∂ k : r > i } . (ii) If ( i , . . . , i k − ) = 0 , then L + ( x i ∂ k ) contains { x r ∂ k : r > i, ( r , . . . , r k − ) = 0 , and r k = 2 i k − } . If in addition i k = 1 and i = 1 k , then L + ( x i ∂ k ) contains { x r ∂ k : r > i, ( r , . . . , r k − ) = 0 } Proof. (i) Assume ( i , . . . , i k − ) = 0, say i m = 0 with m ∈ { , . . . , k − } . Then,for any j ∈ N n , [ x i ∂ k , x j ∂ m ] = j k x i + j − k ∂ m − i m x i + j − m ∂ k . So, since i m = 0, for any r > i if we set j = r − i + 1 m we get ℓ + ([ x i ∂ k , x j ∂ m ]) = x r ∂ k . Moreover, if x u ∂ ℓ < x i ∂ k , one can easily see that ℓ + ([ x u ∂ ℓ , x j ∂ m ]) < ℓ + ([ x i ∂ k , x j ∂ m ]) . Thus, L + ( x i ∂ k ) contains all elements of the form x r ∂ k for r > i .(ii) Assume ( i , . . . , i k − ) = 0. Let j ∈ N n such that ( j , . . . , j k − ) = 0. Then(5.1) [ x i ∂ k , x j ∂ k ] = ( j k − i k ) x i + j − k ∂ k while for any ℓ < k and u ∈ N n (5.2) [ x u ∂ ℓ , x j ∂ k ] = − u k x u + j − k ∂ ℓ . For any r > i with ( r , . . . , r k − ) = 0 and r k = 2 i k −
1, if we set j = r − i + 1 k thenby (5.1) we get ℓ + ([ x i ∂ k , x j ∂ k ]) = x r ∂ k and by (5.2), for any x u ∂ ℓ < x i ∂ k , we see that ℓ + ([ x u ∂ ℓ , x j ∂ k ]) < ℓ + ([ x i ∂ k , x j ∂ k ]) . Thus, L + ( x i ∂ k ) contains all elements of the desired form.Now, if in addition i k = 1 and i = 1 k , we must show that L + ( x i ∂ k ) also containsall elements of the form x r ∂ k where r > i , ( r , . . . , r k − ) = 0 and r k = 1. Note thatsince i = 1 k , we must have that k < n and there is m > k such that i m = 0. Let j = r − i + 1 m , which is in N k since r > i in the product order. Then ℓ + ([ x i ∂ k , x j ∂ m ]) = x r ∂ k . Moreover, for any x u ∂ ℓ < x i ∂ k , we see that ℓ + ([ x u ∂ ℓ , x j ∂ m ]) < ℓ + ([ x i ∂ k , x j ∂ m ]) . Thus, L + ( x i ∂ k ) contains such x r ∂ k . (cid:3) We now recall Dickson’s lemma, as we will make use of it. Let N = N n ×{ , . . . , n } . We equip N with the following ordering: for ( i, k ) , ( j, ℓ ) ∈ N , we set( i, k ) < ( j, ℓ ) if and only if k = ℓ and i < j (the latter denotes the product order of N n ). A sequence of elements ( a i ) from N is called Dicksonian if a j a i for j > i . Theorem 5.3 (Dickson’s lemma [6]) . Every Dicksonian sequence of N is finite. Proposition 5.4.
With respect to the above ordering on M , the Cartan algebra W n is Dicksonian.Proof. Assume towards a contradiction that there is an infinite leading-Dicksoniansequence (( M i , N i )) ∞ i =1 in W n . From the sequence of N i ’s we can find an infinitesequence ( a i ) of elements in M + such that a j / ∈ L + ( a i ) for i < j . To each a i = x j ∂ k ∈ M + we associate the element b i := ( j, k ) ∈ N . This gives us an infinitesequence ( b i ) of N . By Lemma 5.2, there is an infinite subsequence of ( b i ) whichis Dicksonian, but this contradicts Dickson’s lemma, Theorem 5.3. (cid:3) POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 17
The proof of Proposition 5.4 is the origin of our use of the term
Dicksonian todescribe our key condition on Lie algebras.5.3.
Special Cartan algebras.
Let n ≥
2. Recall that the special Cartan algebra S n is the Lie subalgebra of W n given by elements of the form(5.3) p ∂ + · · · + p n ∂ n such that ∂ ( p ) + · · · + ∂ n ( p n ) = 0where p i ∈ k [ x , . . . , x n ].In this section we prove that S n is Dicksonian and hence, by Theorem 1.2, thesymmetric algebra S ( S n ) has ACC on radical Poisson ideals.Induce the grading on S n from the natural grading on W n . We seek an orderedhomogeneous basis of S n . Let N be the subset of S n consisting of elements of theform x i ∂ , such that i ∈ N n and i = 0 , together with the elements of the form i k x i − k ∂ − i x i − ∂ k , with 2 ≤ k ≤ n, i ∈ N n and i = 0 . Lemma 5.5.
The set N is a k -basis for S n .Proof. A straightforward computation shows that N is k -linearly independent andcontained in S n . To see that N k -spans S n , note that ∂ defines a surjectiveendomorphism of k [ x , . . . , x n ] whose kernel is k [ x , . . . , x n ]. Thus ∂ is split by themap ∂ − : k [ x , . . . , x n ] → x k [ x , . . . , x n ] defined by extending x i x x i i +1 linearly.Solutions to (5.3) are thus given by p ∈ ∂ − ( − ∂ ( p ) − · · · − ∂ n ( p n )) + k [ x , . . . , x n ] , and all such solutions are clearly in the k -span of N . (cid:3) We now equip N with the (total) order < induced from the order we defined onour basis M of W + . That is, let ℓ + , M denote the leading term of an element of N with respect to M and define M < N N if and only if ℓ + , M ( M ) < M ℓ + , M ( N ).Explicitly, let i, j ∈ N n and k, l ∈ { , . . . , n } . When i = j = 0 we set x i ∂ < x j ∂ ⇔ ( | i | , i n , . . . , i ) < lex ( | j | , j n , . . . , j ) , when i = 0 and j = 0 x i ∂ < j k x j − k ∂ − j x j − ∂ k ⇔ ( | i | , < lex ( | j | − , k ) , and when i = 0 and j = 0 i k x i − k ∂ − i x i − ∂ k < j l x j − l ∂ − j x j − ∂ l ⇔ ( | i | , k, i n , . . . , i ) < lex ( | j | , l, j n , . . . , j ) . We now prove a lemma which can be thought of as the S n analogue of Lemma 5.2. Lemma 5.6.
Let i ∈ N n and k ∈ { , . . . , n } . (i) If i = 0 , then L + ( x i ∂ ) contains { x r ∂ : r > i and r = 0 } . (ii) If i = 1 , then L + ( i k x i − k ∂ − i x i − ∂ k ) contains { r k x r − k ∂ − r x r − ∂ k : r > i and n X j =2 ( r j − i j ) = 1 } . If i ≥ , then L + ( i k x i − k ∂ − i x i − ∂ k ) contains { r k x r − k ∂ − r x r − ∂ k : r > i, n X j =2 ( r j − i j ) = 1 , and r k ( r − i + 2) + 1 = i } . Proof. (i) Assume i = 0 and r > i with r = 0. If j = r − i + 1 + 1 n , then[ x i ∂ , j n x j − n ∂ − j x j − ∂ n ] = ( i n + j n ) x r ∂ . Since j n >
0, it follows that ℓ + ([ x i ∂ , j n x j − n ∂ − j x j − ∂ n ]) = x r ∂ . Moreover, if x u ∂ < x i ∂ , one easily checks that ℓ + ([ x u ∂ , j n x j − n ∂ − j x j − ∂ n ]) < ℓ + ([ x i ∂ , j n x j − n ∂ − j x j − ∂ n ]) , and if u k x u − k ∂ − u x u − ∂ k < x i ∂ one also easily checks that ℓ + ([ u k x u − k ∂ − u x u − ∂ k , j n x j − n ∂ − j x j − ∂ n ]) < ℓ + ([ x i ∂ , j n x j − n ∂ − j x j − ∂ n ]]) . Thus, L + ( x i ∂ ) contains all elements of the form x r ∂ for r > i with r = 0.(ii) Let i ∈ N n and r > i .Set r ′ = ( r , i , . . . , i n ). If we let j = ( r − i + 1) · + 1 k , then[ i k x i − k ∂ − i x i − ∂ k , j k x j − k ∂ − j x j − ∂ k ]equals(5.4) ( i k ( r − i + 1) − i ) · (cid:16) r ′ k x r ′ − k ∂ − r ′ x r ′ − ∂ k (cid:17) . On the other hand, if i ≥ r ′′ = ( i − , r , . . . , r n ). If we let j = (0 , r − i , . . . , r n − i n ), we get(5.5) [ i k x i − k ∂ − i x i − ∂ k , x j ∂ ] = ( − i ) (cid:16) r ′′ k x r ′′ − k ∂ − r ′′ x r ′′ − ∂ k (cid:17) . Note that x j ∂ ∈ N + as long as P nj =2 ( r j − i j ) > i = 1, from (5.4) and since 2 i k = 1 we see that ℓ + ([ i k x i − k ∂ − i x i − ∂ k , x · ∂ − x +1 k ∂ k ]) = i k x i +1 − k ∂ − ( i + 1) x i ∂ k and if N < i k x i − k ∂ − i x i − ∂ k then ℓ + ([ N, x · ∂ − x +1 k ∂ k ]) < ℓ + ([ i k x i − k ∂ − i x i − ∂ k , x · ∂ − x +1 k ∂ k ]) . So i k x i +1 − k ∂ − ( i + 1) x i ∂ k is in L + ( i k x i − k ∂ − i x i − ∂ k ). By a similar argument, using now (5.5) one checksthat r k x (1 ,r ,...,r k − ,...,r n ) ∂ − x (0 ,r ,...,r n ) ∂ k POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 19 is in L + ( i k x i − k ∂ − i x i − ∂ k ). Finally, another application of (5.4) yields that r k x r − k ∂ − r x r − ∂ k is in L + ( i k x i − k ∂ − i x i − ∂ k ), as desired.Now, in the case i ≥ r k x ( i − ,r ,...,r k − ,...,r n ) ∂ − ( i − x ( i − ,r ,...,r n ) ∂ k is in L + ( i k x i − k ∂ − i x i − ∂ k ). And finally, using (5.4) one checks that r k x r − k ∂ − r x r − ∂ k is in L + ( i k x i − k ∂ − i x i − ∂ k ) as long as r k ( r − i + 2) + 1 = i , as desired. (cid:3) Proposition 5.7.
With respect to the above ordering on N , the special Cartanalgebra S n is Dicksonian.Proof. The proof is almost identical to the proof of Proposition 5.4 (but usingLemma 5.6 rather than Lemma 5.2). Namely, to a given infinite leading-Dicksoniansequence of S n one naturally associates an infinite sequence ( b i ) of N (recall from § N n × { , . . . , n } equipped with the natural productorder). This is done as follows: for i ∈ N n and k ∈ { , . . . , n } ; when i = 0 x i ∂ ( i, i = 0 i k x i − k ∂ − i x i − ∂ k ( i − , k ) . Now by Lemma 5.6 there is an infinite subsequence of ( b i ) which is Dicksonian,contradicting Dickson’s lemma, Theorem 5.3. (cid:3) Hamiltonian Cartan algebras.
Let n = 2 m with m a positive integer.Recall that the Hamiltonian Cartan algebra H n is the Lie subalgebra of W n givenby elements of the form D H ( p ) := m X ℓ =1 ( ∂ m + ℓ ( p ) ∂ ℓ − ∂ ℓ ( p ) ∂ m + ℓ )for p ∈ k [ x , . . . , x n ]. In fact D H : k [ x , . . . , x n ] → W n as defined above is a k -linearmapping with kernel k . One can easily derive that for p, q ∈ k [ x , . . . , x n ] we have[ D H ( p ) , D H ( q )] = D H ( h ) where(5.6) h = m X ℓ =1 ( ∂ m + ℓ ( p ) ∂ ℓ ( q ) − ∂ ℓ ( p ) ∂ m + ℓ ( q )) = D H ( p )( q ) . In other words, [ D H ( p ) , D H ( q )] = D H ( { p, q } ) where { p, q } = D H ( p )( q ).In this section we prove that H n is Dicksonian. Hence, by Theorem 1.2, thesymmetric algebra S ( H n ) has ACC on radical Poisson ideals. Again, we seek an ordered homogeneous basis M for H n . Let M = { D H ( x i ) : i ∈ N n and i = } ⊂ H n . The set M is clearly a k -basis for H n . From (5.6), wesee that(5.7) [ D H ( x i ) , D H ( x j )] = D H m X ℓ =1 ( i m + ℓ j ℓ − i ℓ j m + ℓ ) x i + j − ℓ − m + ℓ ! . We equip M with the following (total) order. Given nonzero i, j ∈ N n , we set D H ( x i ) < D H ( x j ) ⇔ ( | i | , i n , . . . , i ) < lex ( | j | , j n , . . . , j ) . We now prove a lemma which can be thought of as the H n analogue of Lem-mas 5.2 and 5.6. Lemma 5.8.
Let i ∈ N n and ≤ ℓ ≤ m . (1) If i ℓ = 2 i m + ℓ , then L + ( D H ( x i )) contains { D H ( x i + r · ℓ ) : r ≥ } . (2) If i ℓ = 2 i m + ℓ and i ℓ = 0 , then L + ( D H ( x i )) contains { D H ( x i + r · ℓ ) : r ≥ } . Similar results hold when we swap ℓ for m + ℓ in (1) and (2) .Proof. Let r ≥
1. Setting j = ( r + 1) · ℓ + 1 m + ℓ , from (5.7), we get[ D H ( x i ) , D H ( x j )] = ( i m + ℓ ( r + 1) − i ℓ ) D H ( x i + r · ℓ ) . So when i m + ℓ ( r + 1) = i ℓ , we see that(5.8) ℓ + ([ D H ( x i ) , D H ( x j )]) = D H ( x i + r · ℓ )and furthermore (5.7) also yields that when D H ( x v ) < D H ( x i )(5.9) ℓ + ([ D H ( x v ) , D H ( x j )]) < ℓ + ([ D H ( x i ) , D H ( x j )]) . Hence, D H ( x i + r · ℓ ) ∈ L + ( D H ( x i )).We now prove (1). Assume i ℓ = 2 i m + ℓ . From what we have shown in (5.8), wemay assume i m + ℓ ( r + 1) = i ℓ , letting u = 2 · ℓ + 1 m + ℓ , from (5.7) we see that[ D H ( x i ) , D H ( x u )] = (2 i m + ℓ − i ℓ ) D H ( x i +1 ℓ ) . Note that the coefficient above is nonzero (as we are assuming i ℓ = 2 i m + ℓ ). Itfollows, using again (5.7) as in (5.9), that D H ( x i +1 ℓ ) ∈ L + ( D H ( x i )). If r = 1 weare done. On the other hand, if r >
1, letting j ′ = r · ℓ + 1 m + ℓ we see that[ D H ( x i +1 ℓ ) , D H ( x j ′ )] = ( i m + ℓ r − ( i ℓ + 1)) D H ( x i + r · ℓ ) . Since we are assuming i m + ℓ ( r + 1) = i ℓ , we get i m + ℓ r = i ℓ + 1, so ℓ + ([ D H ( x i +1 ℓ ) , D H ( x j ′ )]) = D H ( x i + r · ℓ )and again using (5.7) as in (5.9) we get D H ( x i + r · ℓ ) ∈ L + ( D H ( x i )), as desired.We now prove (2). Assume i ℓ = 2 i m + ℓ and i ℓ = 0. These assumptions yield that i m + ℓ ( r +1) = i ℓ . Hence from (5.8) and (5.9) we get that D H ( x i + r · ℓ ) ∈ L + ( D H ( x i )),as desired. (cid:3) Proposition 5.9.
With respect to the above ordering on M , the Hamiltonian Car-tan algebra H n has no infinite leading-Dicksonian sequence. POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 21
Proof.
The proof is almost identical to the proofs of Proposition 5.4 and 5.7. Weassociate to each element D H ( x i ) in the basis the nonzero tuple i in N n . Now useLemma 5.8 to contradict Dickson’s lemma. (cid:3) Contact Cartan algebras.
Let n = 2 m + 1 with m a positive integer. The contact Cartan algebra K n is the Lie subalgebra of W n given by elements of theform D K ( p ) := m X ℓ =1 ( ∂ m + ℓ ( p ) ∂ ℓ − ∂ ℓ ( p ) ∂ m + ℓ ) + m X ℓ =1 x ℓ ∂ n ( p ) ∂ ℓ + p − m X ℓ =1 x ℓ ∂ ℓ ( p ) ! ∂ n for p ∈ k [ x , . . . , x n ]. In fact D K : k [ x , . . . , x n ] → W n as defined above is aninjective k -linear mapping. For p, q ∈ k [ x , . . . , x n ] we have [ D K ( p ) , D K ( q )] = D K ( h p, q i ) where(5.10) h p, q i = D K ( p )( q ) − ∂ n ( p ) q. See [8, § K n has a basis M (of homogeneous elements) with an order (com-patible with the natural grading) such that there is no infinite leading-Dicksoniansequence. Hence, by Theorem 1.2, the symmetric algebra S ( K n ) has ACC onradical Poisson ideals.Let M = { D K ( x i ) : i ∈ N n } ⊂ K n . The set M is clearly a k -basis for K n .From (5.10), we see that(5.11) [ D K ( x i ) , D K ( x j )] = D K ( D K ( x i )( x j ) − ∂ n ( x i ) x j ) . The inside term can be computed as D K ( x i )( x j ) − ∂ n ( x i ) x j = m X ℓ =1 ( i m + ℓ j ℓ − i ℓ j m + ℓ ) x i + j − ℓ − m + ℓ + m X ℓ =1 ( i n j ℓ − i ℓ j n ) x i + j − n + 2( j n − i n ) x i + j − n . For i ∈ N n , we set | i | K = P mℓ =1 i ℓ + 2 i n . Recall that in the natural grading of K n the element D K ( x i ) is homogeneous of degree | i | K −
2. Thus we equip M withthe following (total) order. Given i, j ∈ N n , we set D K ( x i ) < D K ( x j ) ⇔ ( | i | K , i n , . . . , i ) < lex ( | j | K , j n , . . . , j ) . We now prove a lemma which can be thought of as the H n analogue of Lem-mas 5.2, 5.6 and 5.8. Lemma 5.10.
Let i ∈ N n . (1) Let ≤ ℓ ≤ m . If i ℓ = 2 i m + ℓ , then L + ( D K ( x i )) contains { D K ( x i + r · ℓ ) : r ≥ } . If i ℓ = 2 i m + ℓ and i ℓ = 0 , then L + ( D K ( x i )) contains { D K ( x i + r · ℓ ) : r ≥ } . Similar results hold when we swap ℓ for m + ℓ . (2) If | i | 6 = 2 , then L + ( D K ( x i )) contains { D K ( x i + r · n ) : r ≥ } . If | i | = 2 , then L + ( D K ( x i )) contains { D K ( x i + r · n ) : r ≥ } . Proof. (1) Let r ≥
1. Setting j = ( r + 1) · ℓ + 1 m + ℓ , from (5.11), we get[ D K ( x i ) , D K ( x j )] =( i m + ℓ ( r + 1) − i ℓ ) D K ( x i + r · ℓ ) + i n ( r − D K ( x i +( r +1) · ℓ +1 m + ℓ − n ) . Thus, due to the nature of the order in the basis M , when i m + ℓ ( r + 1) = i ℓ , we seethat ℓ + ([ D K ( x i ) , D K ( x j )]) = D K ( x i + r · ℓ )and, using the two equalities above, one easily checks that for D K ( x v ) < D K ( x i )(5.12) ℓ + ([ D K ( x v ) , D K ( x j )]) < ℓ + ([ D K ( x i ) , D K ( x j )]) . Hence, D K ( x i + r · ℓ ) ∈ L + ( D K ( x i )). The rest of the argument follows the samelines as the proof of Lemma 5.8.(2) Let r ≥
1. Setting j = ( r + 1) · n , from (5.11), we get(5.13) [ D K ( x i ) , D K ( x j )] = ( r + 1)(2 − m X ℓ =1 i ℓ ) − i n ! D K ( x i + r · n ) . Thus, when the coefficient on the right-hand-term is nonzero, we see that(5.14) ℓ + ([ D K ( x i ) , D K ( x j )]) = D K ( x i + r · n )and from these formulas one also readily checks that inequalities of the form (5.12)still hold. Hence, D K ( x i + r · n ) ∈ L + ( D K ( x i )).Now let us assume | i | 6 = 2, where recall that | i | = P nℓ =1 i ℓ . From what we haveshown in (5.14), we may assume that(5.15) ( r + 1)(2 − m X ℓ =1 i ℓ ) = 2 i n . Letting u = 2 · n , from (5.13) we see that[ D H ( x i ) , D H ( x u )] = − n X ℓ =1 i ℓ ) ! D H ( x i +1 n ) . Note that the coefficient above is nonzero (as we are assuming | i | 6 = 2). It thenfollows, after using these formulas to check that inequalities of the form (5.12) stillholds, that D K ( x i +1 n ) ∈ L + ( D K ( x i )). If r = 1 we are done. On the other hand, if r >
1, letting j ′ = r · n we see that[ D K ( x i +1 n ) , D K ( x j ′ )] = r (2 − m X ℓ =1 i ℓ ) − i n + 1) ! D K ( x i + r · n ) . POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 23
Since we are assuming (5.15), we get that the above coefficient is nonzero, so ℓ + ([ D K ( x i +1 n ) , D K ( x j ′ )]) = D K ( x i + r · n )and again one checks using these formulas that inequalities of the form (5.12) hold.Hence, D K ( x i + r · n ) ∈ L + ( D K ( x i )) as desired.Now let us assume r ≥ | i | = 2. This assumption yields that( r + 1)(2 − m X ℓ =1 i ℓ ) = 2 i n . Hence from (5.14) we get that D K ( x i + r · n ) ∈ L + ( D K ( x i )), as desired. (cid:3) Proposition 5.11.
With respect to the above ordering on M , the contact Cartanalgebra K n is Dicksonian.Proof. The proof is almost identical to the proofs of Proposition 5.4 and 5.7. We as-sociate to each element D K ( x i ) in the basis the tuple i in N n . Now use Lemma 5.10to contradict Dickson’s lemma. (cid:3) Loop algebras.
In this section we prove that loop algebras, their currentsubalgebras, and twisted loop algebras are Dicksonian. Hence, by Theorem 1.2,their symmetric algebras have ACC on radical Poisson ideals.In this subsection, let k be an algebraically closed field of characteristic zero. Theorem 5.12.
Let g be a simple finite-dimensional Lie algebra. The loop al-gebra g [ t, t − ] and the current subalgebra g [ t ] have no infinite leading-Dicksoniansequences.Proof. We give the proof for the loop algebra ˆ g := g [ t, t − ]. Recall that the Liebracket on ˆ g is given by [ gt i , ht j ] = [ g, h ] t i + j , where g, h ∈ g .We first fix notation. See [9] for terminology. Let h be a Cartan subalgebra of g .Let Φ ⊂ h ∗ be the set of roots of h acting on g , and fix a set ∆ ⊂ Φ of simple roots;note that ∆ is a basis for h ∗ . Let Φ + and Φ − be respectively the set of positiveand negative roots with respect to ∆, so Φ = Φ − ⊔ Φ + .Let g = h ⊕ L α ∈ Φ g α be the root space decomposition of g . This gives a gradingof g by Z ∆ since [ g α , g β ] ⊆ g α + β . By definition, if x ∈ g α and h ∈ h , then[ h, x ] = α ( h ) x .We first define a (strict, total) order ≺ on Φ. Fix an enumeration { δ , . . . , δ r } of∆. Let α, β ∈ Φ, and write α = P ri =1 a i δ i and β = P ri =1 b i δ i . Let ht( α ) := P a i .We say that α ≺ β if and only if(5.16) ht( α ) < ht( β ) , or ht( α ) = ht( β ) and ( a r , . . . , a ) < lex ( b r , . . . , b ) . (The reason for this definition is so that δ ≺ δ ≺ · · · ≺ δ r .) We make theconvention that ≺ extends to an order on Φ ∪ { } by defining all elements of Φ − to be ≺ + to be ≻ α ∈ Φ, fix 0 = x α ∈ g α . Let { h , . . . , h r } be the basis of h dual to ∆.Let B = { x α : α ∈ Φ } ∪ { h , . . . , h r } ; as each vector space g α is one-dimensional, B is a basis for g . It is not always true that the bracket of two basis elements is ascalar multiple of a basis element because of our choice of basis for h ; however, if α + β = 0 then [ x α , x β ] is a scalar multiple of a basis element. Now define an order < on B by: • x α < x β ⇐⇒ α ≺ β ; • For all α ∈ Φ − , β ∈ Φ + , and 1 ≤ i ≤ r , we have x α < h i < x β ; • h < h < · · · < h r .We observe that, by simplicity of g , there is some positive integer K so that forall x ∈ B , the elements { [ y , [ y , . . . , [ y K , x ] . . . ]] : y , . . . , y K ∈ B } span g .We now consider ˆ g , which is Z -graded by degree in t , and also Z × Z ∆-graded,in the obvious way. Define a triangular decomposition ˆ g = ˆ g − ⊕ ˆ g ⊕ ˆ g + via the Z -grading, so ˆ g = g . We extend B to a basis ˆ B := { xt n : x ∈ B , n ∈ Z } of ˆ g ,and define an order < on ˆ B by xt m < yt n ⇐⇒ ( m, x ) < lex ( n, y ). We refer to thecorresponding order on Z × (Φ ∪ { } ) as ≺ as well.Let ˆ B + = ˆ B ∩ ˆ g + and ˆ B − = ˆ B ∩ ˆ g − . Let I + be the set of finite tuples from ˆ B + and likewise let I − be the set of finite tuples from ˆ B − .Fix M = xt m ∈ ˆ B (where x ∈ B and m ∈ Z ), and let α be the Z × Z ∆-weightof M . We analyze the sets L − ( M ) , L + ( M ). Let N = yt n ∈ ˆ B satisfy N < M . Let β be the Z × Z ∆-weight of N .Now if x h then ˆ g α is one-dimensional, so N < M ⇐⇒ β ≺ α . Let i = ( M , . . . , M n ) ∈ I + . For 1 ≤ i ≤ n , write M i ∈ ˆ g α i , where α i ∈ Z × (Φ ∪ { } ).The Z × Z ∆-grading of ˆ g means that D + i ( M ) ∈ ˆ g α + P α i . Thus β + P α i ≺ α + P α i and if D + i ( M ) , D + i ( N ) = 0 we have D + i ( N ) < D + i ( M ). Thus L + ( M ) = { D + i ( M ) : i ∈ I + , D + i ( M ) = 0 } . By letting i be of the form ( y t s , y t, . . . , y K t ) we see that L + ( M ) ⊇ B t ≥ m + K . Likewise L − ( M ) ⊇ B t ≤ m − K .Now suppose that x = h j ∈ h , and consider i of the form ( y t s , y t, . . . , y K t, x δ j t ).We may suppose that D + i ( M ) = 0. If β = α then N = h i t m with i < j , and so D + i ( N ) = [ x δ j , h i ] = 0. If β ≺ α then as before D + i ( N ) < D + i ( M ). In any case D + i ( M ) ∈ L + ( M ); letting the y i vary we see that L + ( M ) ⊇ B t >m + K . Likewise, L − ( M ) ⊇ B t Let g be a finite-dimensional simple Lie algebra, let σ be an au-tomorphism of g of order m , and let η be a primitive m -th root of unity. Then L := L ( g , σ, m ) is Dicksonian.Proof. The proof is similar to the proof of Theorem 5.12, and we omit the details. (cid:3) POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 25 Simple graded Lie algebras of polynomial growth. We now prove Corol-lary 1.6. We restate it for the reader’s convenience. Corollary 5.14. Let k be a field of characteristic zero. If g is a simple gradedLie k -algebra of polynomial growth, then the symmetric algebra S ( g ) has ACC onradical Poisson ideals. When k is algebraically closed, simple graded Lie algebras of polynomial growthhave been classified by Mathieu as follows: Theorem 5.15 (Mathieu’s Classification [13]) . Assume k is an algebraically closedfield of characteristic zero. Let g be a Z -graded simple Lie k -algebra of polynomialgrowth. Then, g is one of the following (1) a finite dimensional simple Lie algebra; or (2) a (twisted or untwisted) loop algebra; or (3) a Cartan type algebra of the form W n , S n , H n , or K n ; or (4) the Witt algebra. Due to the above classification, Corollary 1.6 is an immediate consequence ofthe previous results in this section together with the following general result (whichallows us to assume that the base field k is algebraically closed). Recall that givena Poisson k -algebra ( A, {− , −} ) and a field extension L/k we can equip L ⊗ k A withthe canonical Poisson bracket where the Poisson structure on L is trivial and so A becomes naturally a Poisson L -algebra. Namely, { a ⊗ b, c ⊗ d } = ac ⊗ { b, d } , for a, c ∈ L and b, d ∈ A. Below we assume L ⊗ k A is equipped with this Poisson bracket. Lemma 5.16. Let ( A, {− , −} ) be a Poisson algebra over a field k of characteristiczero and let L be a field extension of k . If the Poisson L -algebra L ⊗ k A has ACCon radical Poisson ideals then the same holds in A .Proof. First note that, due to the nature of the Poisson bracket of L ⊗ k A , if I isa Poisson ideal of A then L ⊗ k I is a Poisson ideal of L ⊗ k A . It suffices to showthat if I and I are radical Poisson ideals of A with I properly contained in I ,then the same holds for L ⊗ k I and L ⊗ k I . Since k is of characteristic zero, theideals L ⊗ k I i are radical (recall that being a reduced ring is preserved under basechange over perfect fields). By faithful flatness ( k being a field), L ⊗ k I is properlycontained in L ⊗ k I . The result follows. (cid:3) On the Poisson Dixmier-Moeglin Equivalence In this final section we make some remarks on the Poisson Dixmier-Moeglinequivalence in the context of Poisson algebras of countable dimension (for instance,those that are finitely Poisson-generated). Let k be a field (in arbitrary characteris-tic unless stated otherwise) and ( A, {− , −} ) a Poisson k -algebra. Let P be a primePoisson ideal of A . We recall that P is Poisson-primitive if there is a maximalideal M of A such that P is the largest Poisson ideal contained in M (i.e., P isthe Poisson core of M ). On the other hand, P is said to be Poisson-locally closedif P is a locally closed point in the Poisson spectrum of A ; and P is said to bePoisson-rational if the Poisson centre of the field of fractions of A/P is algebraicover k . An algebra is said to satisfy the Poisson Dixmier-Moeglin equivalence (or PDME)if the notions of Poisson-primitive, Poisson-locally closed, and Poisson-rational,coincide. We refer the reader to the introduction of [3] for further details and recentdevelopments on the subject. Under mild assumptions, we prove in Theorem 6.3that one has the following implications:Poisson-locally closed = ⇒ Poisson-primitive ⇐⇒ Poisson-rational . To prove that Poisson-rational implies Poisson-primitive we will use the followingresult from [3, Lemma 3.1]. Lemma 6.1. Let k be a field and A an integral and commutative k -algebra equippedwith a collection of k -linear derivations ( δ j ) i ∈ J . Suppose that there is a finite-dimensional k -vector subspace V of A and a set S of ideals satisfying: (i) δ j ( I ) ⊆ I for all j ∈ J and I ∈ S , (ii) T S = (0) , and (iii) V ∩ I = (0) for all I ∈ S .Then there exists f in Frac ( A ) \ k with δ j ( f ) = 0 for all j ∈ J .Remark . (1) We point out that in [3, Lemma 3.1] the above statement appears in thecase when J is finite (namely, a finite collection of derivations). However,the proof does not use finiteness of J and could have been stated there forgeneral indexing set J .(2) We also note that the conclusion of the lemma can be strengthen to find f not algebraic over k . Indeed, if we let k alg be the relative algebraic closureof k in A , then we can view A as a k alg -algebra and the derivations ( δ j ) j ∈ J are k alg -linear. Letting V ′ be the span of V over k alg , we see that theconditions in the lemma still hold when replacing V for V ′ . Thus, we find f ∈ F rac ( A ) \ k alg as desired. Theorem 6.3. Let k be an uncountable field and ( A, {− , −} ) a Poisson k -algebrathat has countable dimension over k (for example, when A is finitely Poisson-generated over k ). Then, for a prime Poisson idealPoisson-locally closed = ⇒ Poisson-primitive ⇐⇒ Poisson-rational . Proof. By [2], the assumptions on k and A yield that A is a Jacobson ring. Hence,[17, Proposition 1.7(i)] yields that Poisson-locally closed implies Poisson-primitive.On the other hand, for P a Poisson-primitive ideal of A with corresponding max-imal ideal M , in the proof of [17, Proposition 1.10] an injective morphism from thePoisson centre of the field of fractions of A/P to End A ( A/M ) is constructed. By ourassumptions on k and A , A satisfies the Nullstellensatz; in particular, End A ( A/M )is algebraic over k . Thus, P is Poisson-rational.Finally, to show that Poisson-rational implies Poisson-primitive we adapt theargument from [3, Theorem 3.2] . Without loss of generality, we may assume that P = (0) is Poisson-rational (as we may replace A for A/P if necessary). Let S be the collection of all nonzero proper Poisson ideals of A . Since A has countable We thank Alexey Petukhov for pointing out the necessary adaptations. POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 27 dimension, there is a chain of finite-dimensional k -vector subspaces V ⊂ V ⊂ · · · such that A = [ n ≥ V n . Set S n = { Q ∈ S : Q ∩ V n = (0) } . Note that S = S n S n . We claim that each S n has nontrivial intersection. Towardsa contradiction, assume T S n = (0). Let ( a j ) j ∈ J be generators of A as a k -algebraand let δ j = { a j , −} be the Hamiltonian derivation associated to a j for each j ∈ J .As the ideals in S n are Poisson, they are also differential with respect to ( δ j ) j ∈ J .We can now apply Lemma 6.1, also see Remark 6.2(2), to get f ∈ F rac ( A ) \ k alg such that δ j ( f ) = 0 for all j ∈ J . But the latter equalities imply that f is in thePoisson centre of the fraction field of A , contradicting Poisson-rationality of (0).Now let L n = T S n for n ≥ 1. We have shown that L n = (0), so let f n bea nonzero element in L n . If we let T be the (countable) multiplicatively closedset generated by the f n ’s, we see that the localization B := T − A is a countablygenerated k -algebra. Hence B satisfies the Nullstellensatz (as k is uncountable). Ifwe let I be any maximal ideal of B and J := I ∩ A , then A/J embeds into B/I andthe latter is an algebraic extension of k (as B satisfies the Nullstellensatz). Hence, A/J is also an algebraic extension of k , thus a field, and so J is a maximal ideal of A . By construction, J does not contain any element in S (since S = S n S n ), andso (0) is the largest Poisson ideal in J . In other words, (0) is Poisson-primitive asdesired. (cid:3) Thus, in the cases of interest (finitely Poisson-generated complex Poisson alge-bras, for instance), the PDME reduces to showing that Poisson-primitive impliesPoisson-locally closed. It is shown in [3] that there are finitely generated Pois-son algebras that do not satisfy the PDME (namely, have Poisson-primitive idealsthat are not Poisson-locally closed). However, these examples are far from beingsymmetric algebras, and so one can ask whether the PDME holds for symmetricalgebras S ( g ) that are finitely Poisson-generated (and have ACC on radical Poissonideals). When the Lie algebra g is finite dimensional and k has characteristic zero, S ( g ) does satisfy the PDME; this appears in [12, Theorem 5.7] (and can be thoughtof as the Poisson analogue of Dixmier and Moeglin seminal work [5, 15] showingthat the enveloping algebra U ( g ) satisfies the classical DME). But in general theanswer is no; for instance, when g is the positive Witt algebra W + we have: Lemma 6.4. Assume k is of characteristic zero and let W + be the positive Wittalgebra. In S ( W + ) the zero ideal (0) is Poisson-rational but not Poisson-locallyclosed.Proof. Recall that W + = span k ( e i : i ≥ . If we let P i be the Poisson ideal of S ( W + ) generated by e i , we see that each P i is a nonzero prime Poisson-ideal and T i P i = (0). Thus (0) is not Poisson-locallyclosed.On the other hand, let f /g be in a nonzero element in the centre of F rac ( S ( W + )).We may assume that f and g have no common factors (recall that S ( W + ) is a UFD). Let e n be larger than any e i appearing in f and g . As f /g is in the centre, we have { f /g, e n } = 0 and this yields { f, e n } g = f { g, e n } . As f and g have no common factors, by the choice of e n , we must have { f, e n } = 0and { g, e n } = 0. But this can only happen if f and g are in k . Thus (0) isPoisson-rational. We note that this is also proved in [4, Lemma 3.3]. (cid:3) Remark . By Theorem 6.3, if k is uncountable of characteristic zero, then (0) isa Poisson primitive ideal of S ( W + ).Nonetheless, in the presence of ACC on radical Poisson ideals, one gets close toa PDME, as we point out in the lemma below. Remark . If A is a Poisson algebra with ACC on radical Poisson ideals, thena prime Poisson-ideal P is Poisson-locally closed iff there are finitely many (butat least one) prime Poisson ideals of Poisson-height one in A/P . Indeed, if P isPoisson-locally closed then the intersection of all nonzero prime Poisson ideals in A/P is nonzero; as this intersection is a nonzero radical Poisson ideal, the ACC andTheorem 2.7 yield that it must be a finite intersection of nonzero prime Poissonideals of Poisson-height one (i.e., the Poisson components). Thus there are finitelymany (and at least one) prime Poisson ideals of Poisson-height one in A/P . Theother direction is clear. Lemma 6.7. Assume A is a Poisson k -algebra of countable dimension over k thathas ACC on radical Poisson ideals. Let P be a prime Poisson ideal of A . If P is Poisson-rational then there are at most countably many prime Poisson ideals ofPoisson-height one in A/P .Proof. We may assume that P = (0). Let S denote the set of nonzero primePoisson ideals of A of Poisson-height one. We must show that S is countable. Let V ⊂ V ⊂ · · · be a chain of finite-dimensional k -subspaces such that A = S i V i .Set S i = { Q ∈ S : Q ∩ V i = (0) } . Note that S = S i S i , and so it suffices to show that S i is countable. We actuallyshow each S i is finite. The same argument as in the proof of Theorem 6.3 showsthat the intersection T S i is nonzero. As the latter is a nonzero radical Poissonideal, the ACC assumption and Theorem 2.7(ii), yield that it must be a finiteintersection of nonzero prime Poisson ideals of Poisson-height one (i.e., the Poissoncomponents). This finite collection is in fact formed by all the elements in S i , andso S i is finite. (cid:3) As a consequence of Lemmas 6.4 and 6.7, we see that in S ( W + ) there are eithercountably infinite many prime Poisson-ideals of Poisson-height one or there arenone. We currently do not know the answer to this, so we leave it as an openproblem: Question . Are there any prime Poisson ideals in S ( W + ) of Poisson-height one(equivalently, of finite Poisson-height)?The situation is quite different for nonzero Poisson ideals of S ( W + ). More pre-cisely, we conclude by showing that if P is a nonzero Poisson-rational ideal of S ( W + ) POISSON BASIS THEOREM FOR SYMMETRIC ALGEBRAS 29 then there are only finitely many (and at least one) prime Poisson ideals of heightone in S ( W + ) /P (the latter height being the classical algebraic one). Proposition 6.9. Assume k is of characteristic zero. Let P be a nonzero primePoisson ideal in S ( W + ) . If P is Poisson-rational, then there are finitely many (andat least one) prime Poisson ideals in S ( W + ) /P of (algebraic) height one.Proof. Note that for any e i , e j ∈ W + , the k -span of L + ( e i ) ∪ L − ( e j ) has finitecodimension in W + . Thus, by Corollary 4.6, there is h ∈ S ( W + ) \ P such that thelocalisation ( S ( W + ) /P ) h is a finitely generated k -algebra. Let S be the collectionof prime Poisson-ideals in S ( W + ) /P of height one that contain h and S those thatdo not contain h . Since T S is nonzero (as it contains h ), the ACC on radicalPoisson ideals implies that this intersection has finitely many Poisson components.Thus S is finite. On the other hand, each element of S yields a prime-Poisson idealof height one in the finitely generated algebra ( S ( W + ) /P ) h . By [3, Theorem7.1],the fact that P is Poisson-rational implies that this latter collection is finite, andso S is finite. The result follows. (cid:3) Remark . (i) We note that the proof of Proposition 6.9 works for any g with the propertythat the basis M of g satisfies (3.1), as then we can invoke Corollary 4.6. Inparticular, Proposition 6.9 applies also to the symmetric algebra of the Wittalgebra W , the (first) Cartan algebra W , and any (twisted or untwisted)loop algebra.(ii) We also note that the proposition does not imply that any nonzero Poisson-rational P must be Poisson-locally closed (as it only refers to algebraicheight, not Poisson-height). In fact, we currently do not know whether thisis the case or not. References [1] J. Abarbanel and S. Rosset, Some non-finitely presented Lie algebras, J. of Pure andApplied Algebra (1998), 105–112.[2] S. Amitsur, Algebras over infinite fields, Proceedings of the American MathematicalSociety (1956), 35–48.[3] J. Bell, S. Launois, O. Le´on S´anchez, and R. Moosa, Poisson algebras via model theoryand differential algebraic geometry, Journal of the Euro. Math. Soc. (2017), 2019–2049.[4] J.-M. Bois, Corps enveloppants des alg`ebres de type Witt, J. Algebra (2003), no.2, 669–700.[5] J. Dixmier, Id´eaux primitifs dans les alg`ebres enveloppantes, J. Algebra (1977), no.1, 96–112.[6] D. Figueira, S. Figueira, S. Schmitz and P. Schnoebelen, Ackermannian and primitive-recursive bounds with Dickson’s lemma, (2011), 269–278.[7] Be’eri Greenfeld, Growth of finitely generated simple Lie algebras, arXiv:2006.14504.[8] Naihong Hu, The graded modules for the graded contact Cartan algebras, Communi-cations in Algebra (1994), no. 11, 4475–4497.[9] James E. Humphreys, Introduction to Lie algebras and representation theory , GraduateTexts in Mathematics, 9, Springer-Verlag, New York-Berlin, 1978.[10] V. G. Kac. Infinite dimensional Lie algebras , Progress in Mathematics Vol. 44. Springer,1983.[11] E. Kolchin, Differential algebra and algebraic groups , Academic Press, 1973. [12] S. Launois and O. Le´on S´anchez. On the Dixmier-Moeglin equivalence for Poisson-Hopfalgebras, Advances in Mathematics (2019), 48–69.[13] O. Mathieu, Classification of simple graded Lie algebras of finite growth, Inventionesmathematicae (1992), 455–519.[14] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings , Graduate Stud-ies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001.[15] C. Moeglin, Id´eaux bilat`eres des alg`ebres enveloppantes, Bull. Soc. Math. France (1980), no. 2,143–186.[16] A. Petukhov and S. J. Sierra, Ideals in the enveloping algebra of the positive Wittalgebra, Algebras and Representation Theory (2020), 1569–1599.[17] Sei-Qwon Oh, Symplectic ideals of Poisson algebras and the Poisson structure associatedto quantum matrices, Communications in Algebra (1999), no. 5, 2163–2180. Omar Le´on S´anchez, University of Manchester, Department of Mathematics, Manch-ester, M13 9PL. E-mail address : [email protected] Susan J. Sierra, University of Edinburgh, School of Mathematics, Edinburgh EH93JZ. E-mail address ::