aa r X i v : . [ m a t h . R A ] D ec ORE EXTENSIONS AND POISSON ALGEBRAS
DAVID A. JORDAN
Abstract.
For a derivation δ of a commutative Noetherian C -algebra A , a homeomorphismis established between the prime spectrum of the Ore extension A [ z ; δ ] and the Poissonprime spectrum of the polynomial algebra A [ z ] endowed with the Poisson bracket such that { A, A } = 0 and { z, a } = δ ( a ) for all a ∈ A . Introduction
The best known example of a simple Poisson algebra is the coordinate ring of the sym-plectic plane, that is the polynomial algebra C [ z, x ] with { z, x } = 1. This corresponds tothe best known example of a simple Ore extension A [ z ; δ ], namely the Weyl algebra A ( C ),generated by x and z subject to the relation zx − xz = 1. Here A = C [ x ] and δ = d/dx . Thefirst known example of a Poisson bracket on C [ x, y, z ] for which C [ x, y, z ] is a simple Poissonalgebra, due to Farkas [6, Example following Lemma 15], is such that { x, y } = 0 and thehamiltonian { z, −} acts on C [ x, y ] as the derivation δ = ∂ x + (1 − xy ) ∂ y , where ∂ x and ∂ y arethe partial derivatives. In the first known example, due to Bergman, see [3], of a derivation δ for which the Ore extension C [ x, y ][ z ; δ ] is simple the derivation δ is ∂ x + (1 + xy ) ∂ y . Theproofs of simplicity in [6] and [3] both remain valid for the common generalization where δ = ∂ x + (1 + λxy ) ∂ y for some λ ∈ C ∗ , giving rise to corresponding families of simple Poissonalgebras and simple Ore extensions. Unlike the case of the symplectic plane and the Weylalgebra, this correspondence does not appear to have been noted. These examples of simplePoisson algebras with corresponding simple Ore extensions are special cases of a generalsituation. Given any non-zero derivation δ of a commutative C -algebra A , there is a Poissonbracket on the polynomial algebra A [ z ] such that { A, A } = 0 and { z, a } = δ ( a ) for all a ∈ A .We shall show that if A is Noetherian then the Poisson prime spectrum of A [ z ] is homeo-morphic to the prime spectrum of A [ z ; δ ]. This fits into the philosophy of [11], in that A [ z ]is the commutative fibre version of the semiclassical limit of the family of noncommutativealgebras R α := A [ h ][ z ; hδ ] / ( h − α ) A [ h ][ z ; hδ ], where α ∈ C ∗ and the derivation δ is extendedto the polynomial algebra A [ h ] by setting δ ( h ) = 0. Note that R α ≃ A [ z ; αδ ].In addition to Bergman’s example, there are many known examples of simple derivations of C [ x, y ], for example see [2, 3, 4, 12, 16, 21, 22]. All such examples give rise to Poisson bracketsfor which C [ x, y, z ] is a simple Poisson algebra. In [7], Goodearl and Warfield illustratedtheir study of Krull dimension in Ore extensions with some non-simple Ore extensions of Mathematics Subject Classification.
Primary 17B63; Secondary 16S36, 13N15, 16W25, 16S80.
Key words and phrases.
Poisson algebra, Poisson prime ideal, Ore extension, simple derivation. C [ x, y ] with interesting prime spectra. In the final section we shall transfer these and someother known examples to the Poisson setting and also answer a question from [7] on Oreextensions by constructing an accessible example of a derivation of C [ x, y ] giving rise to aPoisson bracket on B := C [ x, y, z ] for which the height two prime ideal yB + zB is Poissonbut no height one prime ideal is Poisson.2. Background on Poisson algebras
Our base field will always be C though the results are valid over any field of characteristic0. In Remark 3.7 algebraic closure is pertinent. We denote the prime spectrum of a not-necessarily commutative ring by Spec R . Definition 2.1. A Poisson algebra is C -algebra A with a Poisson bracket, that is a bilinearproduct {− , −} : A × A → A such that A is a Lie algebra under {− , −} and, for all a ∈ A ,the hamiltonian ham( a ) := { a, −} is a C -derivation of A .The following definitions and the claims made for them are well-known. One comprehen-sive reference is [10, Lemma 1.1 and thereabouts]. Definitions 2.2.
Let ∆ be a set of derivations of a commutative C -algebra A . The ∆- centre , Z ∆ ( A ), of A is { a ∈ A : δ ( a ) = 0 for all δ ∈ ∆ } .An ideal I of A is a ∆ -ideal if δ ( I ) ⊆ I for all δ ∈ ∆ and a ∆-ideal P of A is ∆- prime if,for all ∆-ideals I and J of A , IJ ⊆ P implies I ⊆ P or J ⊆ P . If ∆ = { δ } is a singletonthen, in these and subsequent definitions, we replace ∆ by δ rather than by { δ } .To say that A is ∆- simple means that 0 is the only proper ∆-ideal I of A . A derivation δ of A is said to be simple if A is δ -simple.The ∆- core of an ideal I of A , denoted ( I : ∆), is the largest ∆-ideal of A contained in I .If P is a prime ideal of A then ( P : ∆) is prime, see [10, Lemma 1.1(a)].If I is a ∆-ideal of A then each derivation δ ∈ ∆ induces a derivation δ of A/I such that δ ( a + I ) = δ ( a ) + I for all a ∈ A . If I is a ∆-ideal and is also prime then δ extends to thequotient field Q ( R/I ) by the quotient rule, δ ( as − ) = ( sδ ( a ) − aδ ( s )) s − .A ∆-ideal P of A is ∆- primitive if P = ( M : ∆) for some maximal ideal M of A . Every∆-primitive ideal is ∆-prime.If A is a Poisson algebra and ∆ = { ham( b ) : b ∈ A } then we replace the prefix ∆- by theword Poisson. In particular an ideal I of a Poisson algebra is a Poisson ideal if { i, a } ∈ I forall a ∈ A and i ∈ I and A is a simple Poisson algebra if and only if the only Poisson idealsof A are 0 and A . The Poisson centre of A and the Poisson core of a Poisson ideal I of A will be denoted by PZ( A ) and P ( I ) respectively. Lemma 2.3.
Let A be a commutative Noetherian C -algebra and let ∆ be a set of derivationsof A . If P is a ∆ -prime ideal of A then P is prime.Proof. See [10, Lemma 1.1 (d)]. (cid:3)
Definitions 2.4.
Let ∆ be a set of derivations of a commutative Noetherian C -algebra A .The ∆-prime spectrum of A , denoted ∆-Spec( A ), is the set of all ∆-prime ideals of A with RE EXTENSIONS AND POISSON ALGEBRAS 3 the topology induced from the Zariski topology in Spec( A ). The Poisson spectrum of A will be denoted by P . Spec( A ). Thus a closed set in P . Spec( A ) has the form V ( I ) := { P ∈ P . Spec( A ) : P ⊇ I } for some ideal I of A . As is observed in [11, § I by thePoisson ideal that it generates, I can be assumed to be a Poisson ideal. Definition 2.5.
Let A be a Poisson algebra and I be a Poisson ideal of A . If the inducedPoisson bracket on A/I is zero, we say that I is residually null . This is equivalent to sayingthat I contains all elements of the form { a, b } where a, b ∈ A , or that I contains all suchelements where a, b ∈ G for some generating set G for A . The set of residually null Poissonprime ideals of A is clearly closed in P . Spec( A ). Definitions 2.6.
By a
Poisson maximal ideal we mean a maximal ideal that is also Poissonwhereas by a maximal Poisson ideal we mean a Poisson ideal that is maximal in the latticeof Poisson ideals. These notions are not equivalent. Any Poisson maximal ideal is maximalPoisson but the converse is false as can be seen by considering the ideal 0 in any simplePoisson algebra that is not simple as an associative algebra, such as C [ y, z ] with { y, z } = 1. Definitions 2.7.
A G- domain is a commutative integral domain R such that the intersectionof the non-zero prime ideals is non-zero, in other words 0 is locally closed in Spec R . See [20,Theorems 19 and 20 and the intermediate text]. With A and ∆ as in Definitions 2.2, let P be a ∆-prime ideal of A . We shall say that P is ∆-G if it is locally closed in ∆-Spec( A ). Tosay that A is ∆-G means that 0 is a ∆-G ideal of A .If P is a ∆-ideal and prime, in particular if A is Noetherian and P is ∆-prime, we saythat P is ∆- rational if Z ∆ ( Q ( A/P )) = C , where ∆ is the set of derivations of the quotientfield Q ( A/P ) induced, via
R/P , by derivations belonging to ∆.3.
Semiclassical limits of Ore extensions
Let A denote a commutative C -algebra that is also a domain and let D be the polynomialalgebra A [ h ]. Let δ be a derivation of A and extend δ to D by setting δ ( h ) = 0. Then hδ is a derivation of D and we can form the Ore extension (or skew polynomial ring or ringof formal differential operators) T := D [ z ; hδ ] in which elements have the form P n d i z i , d i ∈ D , and zd − dz = hδ ( d ) for all d ∈ D . Note that hz = zh and h is a central non-unitregular element of T such that T /hT is isomorphic to the commutative polynomial algebra B := A [ z ] and T / ( h − T is isomorphic to the Ore extension R := A [ z ; δ ]. If α ∈ C ∗ ,then T / ( h − α ) T ≃ A [ z ; αδ ] ≃ A [ z ; δ ], where the final isomorphism maps z to αz . In thissituation, there is a well-defined Poisson bracket on B such that { u, v } = h − [ u, v ]for all u = u + hT and v = v + hT ∈ B . With this bracket, B is the semiclassical limit ofthe family A [ z ; αδ ], α ∈ C ∗ , as in [11, 2.1], T is a quantization of the Poisson algebra B inthe sense of [1, Chapter III.5] and R is a deformation of B in the sense of [18]. A familiarexample is obtained by taking A = C [ x ] and δ = d/dx . Here R is the Weyl algebra A ( C ), DAVID A. JORDAN with generators x and z subject to the relation zx − xz = 1, and the semiclassical limit B is C [ x, z ] with { z, x } = 1, that is the coordinate ring of the symplectic plane.To emphasise the role of the single derivation δ , the Poisson bracket on B will sometimesbe written {− , −} δ . Thus { a, b } δ = 0 and { z, a } δ = δ ( a ) for all a, b ∈ A . In the terminologyof [23], B is a Poisson polynomial ring over A for which the Poisson bracket on A and thederivation α are both zero. Lemma 3.1.
Let A be a commutative C -algebra with a derivation δ and let B = A [ z ] equippedwith the Poisson bracket {− , −} δ . (i) For all a, b ∈ A and all m, n ∈ N , { az m , bz n } = ( maδ ( b ) − nbδ ( a )) z m + n − . (ii) Let Q be a δ -ideal of A . Then QB is a Poisson ideal of B and there is an isomorphismof Poisson algebras, θ Q : B/QB → ( A/Q )[ z ] given by θ Q n X i =0 a i z i ! + QB ! = n X i =0 ( a i + Q ) z i , where the Poisson bracket on A/Q [ z ] is {− , −} δ .Proof. (i) is routine using the fact that the hamiltonians are derivations. The first statementin (ii) is immediate from (i) and the second statement is straightforward. (cid:3) Lemma 3.2.
Let A be a commutative Noetherian C -algebra that is also a domain, let δ bea non-zero derivation of A and let P be a non-zero Poisson prime ideal of B := C [ z ] for thePoisson bracket {− , −} δ . Let Q = P ∩ A . (i) Q is a non-zero δ -prime ideal of A . (ii) If δ ( A ) Q then P = QB .Proof. (i) Let p = P ni =0 a i z i , with each a i ∈ A , be an element of minimal degree n in z among non-zero elements of P . Let a ∈ A be such that δ ( a ) = 0. Then (ham a )( p ) = − P ni =0 iδ ( a ) a i z i − ∈ P . This contradicts the minimality of n unless n = 0. Thus n = 0 and Q = 0.As P is a Poisson ideal of B , Q is a δ -ideal of A . Let I and J be δ -ideals of A such that IJ ⊆ Q . By Lemma 3.1(i), IB and J B are Poisson ideals of B . Clearly IBJ B ⊆ P so either IB ⊆ P , whence I ⊆ P ∩ A , or J B ⊆ P , whence J ⊆ P ∩ A . Thus P ∩ A is δ -prime.(ii) By Lemma 2.3, A/Q is a domain. Suppose that δ ( A ) Q , so that the induced Poissonbracket on the domain A/Q is non-zero. Clearly QB ⊆ P . If QB = P then θ Q ( P/BQ ) is anon-zero Poisson ideal of (
A/Q )[ z ] intersecting A/Q in 0. This is impossible by (i) appliedto
A/Q so QB = P . (cid:3) The situation is similar for the prime spectrum of the Ore extension R = C [ z ; δ ]. Let A be a commutative C -algebra with a derivation δ and let Q be a δ -ideal of A . By [9, §
1, finalparagraph], QR is an ideal of R and there is an isomorphism ψ Q : R/QR → A/Q [ z ; δ ] givenby ψ Q n X i =0 a i z i ! + QR ! = n X i =0 ( a i + Q ) z i . RE EXTENSIONS AND POISSON ALGEBRAS 5
Lemma 3.3.
Let A be a commutative C -algebra that is also a domain, let δ be a non-zeroderivation of A and let P be a non-zero prime ideal of R := A [ z ; δ ] . Let Q = P ∩ A . (i) Q is a non-zero δ -prime ideal of A . (ii) If δ ( A ) Q then P = QR .Proof. (i) By [14, Lemma 1.3], Q is δ -prime and, by [15, Lemma 1], Q = 0.(ii) By Lemma 2.3 with ∆ = { δ } , A/Q is a domain. Suppose that δ ( A ) * Q so that theinduced derivation δ on the domain A/Q is non-zero. The ideal QR is prime by [14, Lemma1.3] and QR ⊆ P . If QB = P then φ Q ( P/RQ ) is a non-zero ideal of (
A/Q )[ z ; δ ] intersecting A/Q in 0. This is impossible by (i) applied to
A/Q so QR = P . (cid:3) Corollary 3.4.
Let A be a commutative C -algebra that is a domain and let δ be a non-zeroderivation of A . Let R = A [ z ; δ ] and let B be the Poisson algebra A [ z ] with the Poissonbracket {− , −} δ . Then B is Poisson simple if and only if R is simple if and only if δ issimple.Proof. It follows from Lemmas 3.2 and 3.3 that if δ is simple then B is Poisson simple and R is simple. On the other hand, if J is a non-zero δ -ideal of A then J R is a non-zeroproper ideal of R , by [14, Lemma 1.3], and J B is a non-zero proper Poisson ideal of B byLemma 3.1(ii). (cid:3) We now aim to generalize Corollary 3.4 to establish a homeomorphism between Spec R and P . Spec B . On each side, we shall partition the spectrum into two types of prime ideal. Notation 3.5.
Let A be a commutative C -algebra and domain with a non-zero derivation δ ,let R = A [ z ; δ ] and let B = A [ z ] equipped with the Poisson bracket {− , −} δ . Let J = δ ( A ) A ,which is a δ -ideal of A , and let S = ( A/J )[ z ]. Then(i) J B = { B, B } B is a residually null Poisson ideal of B and is contained in all residuallynull Poisson ideals of B .(ii) θ J : B/J B → S is an isomorphism of C -algebras. The Poisson brackets are both 0.(iii) J R is an ideal of R such that R/J R is commutative and
J R is contained in all ideals I of R such that R/I is commutative.(iv) ψ R : R/J R → S is an isomorphism of commutative C -algebras. The induced deriva-tion δ on A/J is 0.Let P . Spec ( B ) = { P ∈ P . Spec B : P is residually null } = { P ∈ P . Spec B : J B ⊆ P } and let P . Spec ( B ) = P . Spec B \ P . Spec ( B ). By analogy, letSpec ( R ) = { P ∈ Spec R : R/P is commutative } = { P ∈ Spec R : J R ⊆ P } and let Spec ( R ) = Spec R \ Spec ( R ). Note that P . Spec ( B ) and Spec ( R ) are closed inP . Spec B and Spec R respectively. Also P . Spec ( B ) is homeomorphic to Spec( B/J B ) andSpec ( R ) is homeomorphic to Spec( R/J R ). DAVID A. JORDAN
Let κ be the isomorphism ψ − J θ J : B/J B → R/J R . Thus κ n X i =0 a i z i ! + J B ! = n X i =0 a i z i ! + J R.
Then κ induces a homeomorphism between Spec( R/J R ) and Spec(
B/J B ) and there is ahomeomorphism Γ : P . Spec ( B ) → Spec ( R ) such that Γ ( P ) /J R = ψ ( P/J B ) for all P ∈ P . Spec ( B ). Theorem 3.6.
Let A be a Noetherian C -algebra that is a domain and let δ be a non-zeroderivation of A . Let R = A [ z ; δ ] and let B be the Poisson algebra A [ z ] with the Poissonbracket {− , −} δ . There is a homeomorphism between Spec R and P . Spec B .Proof. We have seen in 3.5 that there is a homeomorphism Γ : P . Spec ( B ) → Spec ( R )such that Γ ( P ) /J R = κ ( P/J B ) for all P ∈ P . Spec ( B ). We aim to extend this to ahomeomorphism Γ : P . Spec( B ) → Spec( R ).By Lemma 3.2, every element of P . Spec B has the form QB for a δ -prime ideal Q of A such that J Q and, by Lemma 3.3, every element of Spec R has the form QR forsuch an ideal Q . Define Γ : P . Spec B → Spec R by setting Γ ( QB ) = QR . Then Γ isbijective and Γ and Γ − preserve inclusions. Combine Γ and Γ by defining a bijectionΓ : P . Spec B → Spec R byΓ( P ) = ( Γ ( P ) if P ∈ P . Spec ( B ) , Γ ( P ) if P ∈ P . Spec ( B ) . Inclusions within P . Spec B and Spec R and within P . Spec B and Spec R are preservedby Γ and Γ − . There are no inclusions P ′ ⊆ P with P ′ ∈ P . Spec B and P ∈ P . Spec B orwith P ′ ∈ Spec R and P ∈ Spec R . Let P ′ = QB ∈ P . Spec B and P ∈ P . Spec B . Then QB ⊆ P ⇔ QB + J BJ B ⊆ PJ B ⇔ κ (cid:18) QB + J BJ B (cid:19) ⊆ κ (cid:18) PJ B (cid:19) = Γ( P ) J R ⇔ QR + J RJ R ⊆ Γ( P ) J R ⇔ Γ( QB ) = QR ⊆ Γ( P ) . Thus both Γ and Γ − preserve inclusions. By [11, Lemma 9.4], Γ is a homeomorphism. (cid:3) Remark . For many affine algebras, particularly enveloping algebras and quantum al-gebras, there are prime ideals that are not completely prime and there is an establishedhomeomorphism between the completely prime part of the spectrum of a deformation andthe Poisson prime spectrum of a corresponding semiclassical limit. Some such algebras arediscussed in [17], where a common theme is that the incompletely prime ideals are annihila-tors of finite-dimensional simple modules of dimension d > d -dimensional simple RE EXTENSIONS AND POISSON ALGEBRAS 7
Poisson module. In the context of this paper, this issue is not present on either side. Onthe Ore side, Sigurdsson [24] shows that all prime ideals of A [ z ; δ ] are completely prime. Onthe Poisson side, by [17, Theorem 1], a d -dimensional simple Poisson module over an affinePoisson algebra corresponds to a d -dimensional simple Lie module for the Lie algebra M/M for some maximal Poisson ideal M . In the context of the present paper, M/M is alwayssoluble and, by [5, Corollary 1.3.13], every finite-dimensional simple Lie module for M/M has dimension one. 4. Primitivity
The purpose of this section is to show that, for a commutative affine domain A withderivation δ , the Ore extension A [ z ; δ ] is primitive if and only if A [ z ] is Poisson primitive,for the Poisson bracket {− , −} δ , and that, under the homeomorphism Γ of Theorem 3.6,Poisson primitive ideals of A [ z ] correspond to primitive ideals of A [ z ; δ ].It follows from [15, Theorems 1,2], where A is not necessarily affine, that if δ = 0 and A is either δ -primitive or δ -G then A [ z ; δ ] is primitive. The converse, in the Noetheriancase, was established in [8, Theorem 3.7]. The logical independence, in the general case,of the two conditions, δ -primitive and δ -G, was shown in [15] by means of the non-affineexamples A = C [[ y ]] with δ = y /. dy , which is δ -G but not δ -primitive, and A = C ( t )[ y ]with δ = t /. dt + y /. dy , which is δ -primitive but not δ -G. If A is affine and δ -G then A is δ -primitive by [10, Proposition 1.2]. It would be interesting to know whether there is anaffine δ -primitive C -algebra A which is not δ -G. Such an example would have consequencesfor the Poisson Dixmier-Moeglin equivalence as it would give rise to a Poisson bracket on A [ z ] for which 0 is Poisson primitive, and hence Poisson rational, but not locally closed.In the Poisson setting we have analogues of [15, Theorems 1,2]. Theorem 4.1.
Let δ be a non-zero derivation of a commutative C -algebra A . Then A [ z ] ,with the Poisson bracket {− , −} δ , is Poisson primitive if A is δ -primitive or δ -G.Proof. Suppose that A is δ -G and let I = 0 be the intersection of the non-zero δ -prime idealsof A . As A is a domain, the Jacobson radical Jac( A [ z ]) = 0, for example by [13, Theorem4]. Suppose that A [ z ] is not Poisson primitive and let M be a maximal ideal of A [ z ]. Then P ( M ) = 0 and, by Lemma 3.2(i), P ( M ) ∩ A is a non-zero δ -prime ideal of A . Therefore I ⊆ M for all maximal ideals M of A [ z ] so I ⊆ Jac( A [ z ]) = 0. This contradiction showsthat A [ z ] is Poisson primitive.Suppose that A is δ -primitive and let M be a maximal ideal of A containing no non-zero δ -ideal of A . Let N be any maximal ideal of A [ z ] containing M . Thus N ∩ A = M . Let P = P ( N ). Then P = 0 otherwise, by Lemma 3.2, P ∩ A is a non-zero δ -ideal of A containedin M . Thus A [ z ] is Poisson primitive. (cid:3) It would be interesting to know whether the converse is true in the Noetherian case. Asthe next result shows, it is true in the affine case.
Theorem 4.2.
Let δ be a non-zero derivation of a commutative affine C -algebra A . Then A [ z ] is Poisson primitive if and only if A is δ -primitive. DAVID A. JORDAN
Proof.
Suppose that A [ z ] is Poisson primitive and let M be a maximal ideal of A [ z ] containingno non-zero Poisson ideal of A [ z ]. Then M ∩ A contains no non-zero δ -ideal of A for if J is a non-zero δ -ideal of A contained in M ∩ A then, by Lemma 3.1(ii), J A [ z ] is a non-zeroPoisson ideal of A [ z ] contained in M . But, by [20, Theorem 27], A/ ( M ∩ A ) is a G-domain.As A is affine, it is a Hilbert ring, by [20, Theorem 31], so M ∩ A is a maximal ideal of A .Thus A is δ -primitive. The converse holds by Theorem 4.1. (cid:3) Corollary 4.3.
Let δ be a non-zero derivation of a commutative affine C -algebra A . Then A [ z ] , with the Poisson bracket {− , −} δ , is Poisson primitive if and only if A [ z ; δ ] is primitive.Proof. As we observed above, [10, Proposition 1.2] tells us that, in the affine case, if A is δ -primitive then A is δ -G. The result is then immediate from Theorem 4.2 and [8, Theorem3.7]. (cid:3) Corollary 4.4.
Let δ be a non-zero derivation of a commutative affine C -algebra A , let B = A [ z ] with the Poisson bracket {− , −} δ and let R = A [ z ; δ ] . (i) In P . Spec( B ) , the Poisson primitive ideals are the maximal elements of P . Spec B ,that is the Poisson ideals P of B such that B/P ≃ C , and the ideals of the form QB where Q is a δ -primitive ideal of A . (ii) In Spec( R ) , the primitive ideals are the maximal elements of Spec R , that is theideals P of B such that R/P ≃ C , and the ideals of the form QR where Q is a δ -primitive ideal of A . (iii) In the homeomorphism between P . Spec( B ) and Spec( R ) , the Poisson primitive idealsof B correspond to the primitive ideals of R .Proof. (i) Let P be a Poisson prime ideal of B . Suppose first that P ∈ P . Spec B . Thenthe Poisson bracket on B/P is 0 so P is Poisson primitive if and only if P is maximal if andonly if B/P ≃ C . Now suppose that P in P . Spec B . Then P = QB for some δ -prime ideal Q of A with δ ( A ) Q and, by Corollary 4.2 applied to A/Q , P is Poisson primitive if andonly if Q is δ -primitive.(ii) Let P be a prime ideal of R . Suppose first that P ∈ Spec B . Then R/P is commutativeso P is primitive if and only if P is maximal if and only if B/P ≃ C . Now suppose that P ∈ Spec R . Then P = QR for some δ -prime ideal Q of A with δ ( A ) Q and, byCorollaries 4.2 and 4.3 applied to A/Q , P is primitive if and only if Q is δ -primitive.(iii) This follows from (i) and (ii). (cid:3) Examples in C [ x, y, z ]Here we look at some examples where A = C [ x, y ], so that B = C [ x, y, z ], the polynomialalgebra in three indeterminates. For w = x, y or z , we denote the derivation ∂/∂w of B by ∂ w and, for a ∈ B , we write a w for ∂ w ( a ) and grad a for the triple ( a x , a y , a z ) ∈ B . Poissonbrackets on C [ x, y, z ] are the subject of [19]. Any such bracket is determined by the triple( f, g, h ) ∈ B such that { y, z } = f, { z, x } = g and { x, y } = h. RE EXTENSIONS AND POISSON ALGEBRAS 9
A triple F = ( f, g, h ) ∈ B is a Poisson triple if it does determine a Poisson bracket in thisway. By [19, Proposition 1.17(1)], a triple F = ( f, g, h ) ∈ B is a Poisson triple if and onlyif F. curl F = 0. Similar results are true for the rational function field Q ( B ) = C ( x, y, z ) andthe completion b B of B at any maximal ideal.For any a, b ∈ B , there is a Poisson bracket on B such that { y, z } = ba x , { z, x } = ba y and { x, y } = ba z . We call such a bracket exact if b = 1 and m-exact in general. A Poisson bracket on B is qm-exact , respectively cm-exact , if there exist a, b ∈ Q ( B ), resp a, b ∈ b B for some maximalideal of B , such that { y, z } = ba x ∈ B, { z, x } = ba y ∈ B and { x, y } = ba z ∈ B. In [19], it is shown that every Poisson bracket on B is cm-exact and the Poisson spectrum isdetermined for a qm-exact bracket with a = st − and b = t , s and t being coprime elementsof B . Taking t = 1, this includes the exact brackets.In the remainder of this section, we consider non-exact Poisson brackets on B = C [ x, y, z ]that extend the zero Poisson bracket on A = C [ x, y ], that is, we consider Poisson bracketson B with { x, y } = 0. Lemma 5.1.
Let f, g ∈ B and let F = ( f, g, . Then F is a Poisson triple if and only ifthere exist h ∈ B and f , g ∈ A such that f = hf and g = hg .Proof. Suppose that F is a Poisson triple. If g = 0 we can take h = 1, f = f and g = 0 sowe may assume that g = 0. As curl(( f, g, − g z , f z , g x − f y ), we have f g z = gf z . Hence ∂ z ( f /g ) = 0 and pf = qg for some p, q ∈ A . Let h be the highest common factor of f and g in B and let f , g ∈ B be such that f = hf and g = hg . Then pf = qg . If f / ∈ A then f has an irreducible factor u in B \ A and, as q ∈ A , u must divide g , contradictingthe choice of h . Thus f ∈ A and similarly g ∈ A .Conversely, suppose that F = ( hf , hg ,
0) where h ∈ B and f , g ∈ A . Then curl F hasthe form ( − g h z , f h z , ℓ ), where ℓ ∈ B , so F. curl F = 0 and hence F is a Poisson triple. (cid:3) The Poisson prime ideals of B for a Poisson triple F = ( hf , hg ,
0) are the prime idealscontaining h and the Poisson primes for the Poisson triple ( f , g ,
0) so it suffices to considerthe case where f, g ∈ A . Thus ham x = − g∂ z , ham y = f ∂ z and ham z = g∂ x − f ∂ y . Alsoham z ( A ) ⊆ A , so that ham z restricts to a derivation of A and the results of Section 3 applywith δ being the restriction to A of g∂ x − f ∂ y .If a ∈ A then the corresponding exact bracket on B has { y, z } = a x , { z, x } = a y and { x, y } = 0. The following theorem is a special case of [19, Theorem 3.8]. Theorem 5.2.
Let a ∈ A \{ } . The Poisson prime ideals of B under the exact bracketdetermined by a are , the residually null Poisson prime ideals and the height one primeideals uA , where u is an irreducible factor of a − λ for some λ ∈ C such that a − λ is anon-zero non-unit. Combining this with Theorem 3.6 and its proof, we obtain the following corollary.
Corollary 5.3.
Let a ∈ A \{ } , let δ be the derivation of A such that δ ( x ) = a y and δ ( y ) = − a x and let R = A [ x ; δ ] . Let J = a y A + a x A . Then (i) J R is an ideal of R and R/J R ≃ ( A/J )[ z ] . (ii) The prime ideals of R under the exact bracket determined by a are , the height oneprime ideals uR , where u is an irreducible factor of a − λ for some λ ∈ C such that a − λ is a non-zero non-unit and the prime ideals of the form π − ( Q ) where Q is aprime ideal of ( A/J )[ z ] and π is the composition R ։ R/J R ≃ ( A/J )[ z ] . Example 5.4.
Let a = x + y . Then { z, x } = 2 y, { y, z } = 2 x and { x, y } = 0. The residuallynull Poisson prime ideals of B are xB + yB and the maximal ideals that contain it. Theother Poisson prime ideals of B are 0, the height one prime ideals ( x + iy ) A , ( x − iy ) A and( x + y − λ ) A , where λ ∈ C ∗ . Note that those of the form ( x + y − λ ) A are maximalPoisson ideals.If δ = 2 y∂ x − x∂ y , so that δ ( x ) = 2 y and δ ( y ) = − x and R = A [ z ; δ ] then the primespectrum of R consists of 0, the height one prime ideals ( x + iy ) R , ( x − iy ) R and ( x + y − λ ) R ,where λ ∈ C ∗ , xR + yR and xR + yR + ( z − µ ) R , where µ ∈ C . For each λ ∈ C ∗ , the algebra R/ ( x + y − λ ) R is simple.In the remainder of the paper we consider non-exact Poisson brackets on B , beginningwith some for which B is Poisson simple. The following result of Shamsuddin, for which aproof may be found in [2, Proposition 3.2], is useful in identifying Poisson brackets for which B , or a localization of B , is Poisson simple. Proposition 5.5.
Let C be a commutative domain and let g = at + b ∈ C [ t ] , where a, b ∈ C .Suppose that there exists a derivation δ of C [ t ] such that δ ( C ) ⊆ C , C is δ | C -simple, δ ( t ) = g and, for all r ∈ C , δ ( r ) = ar + b . Then C [ t ] is δ -simple. Examples 5.6.
In the case where A = C [ x, y ] and B = C [ x, y, z ], there are many knownexamples of simple derivations δ = g∂ x − f ∂ y of A . For all of these, B is Poisson simple forthe Poisson bracket determined by the triple ( f, g, g = 1 sothat(5.1) { x, y } = 0 , { z, x } = 1 and { y, z } = f. In the best known example which is due to, but not published by, Bergman and is documentedin [3, § f = − (1 + xy ). The simplicity of δ follows easily from Proposition 5.5, with R = C [ x ] and t = y and the same argument works for f = − (1 + λxy ), λ ∈ C ∗ . When λ = − x, y and z are written − x , x and x respectively, this gives the first publishedexample, due to Farkas [6, Example following Lemma 15], of a Poisson bracket on B forwhich B is Poisson simple.Other examples of polynomials f ∈ A for which B is Poisson simple under the Poissonbracket in (5.1) include:(i) f = p ( x ) − y , where p ( x ) ∈ C [ x ] has odd degree. See [21, Theorem 6.2].(ii) f = − ( y m + ax n ), where m, n ∈ N , m ≥ a ∈ C \{ } . See [12, Theorem 1]which generalised an earlier result [22], for the case n = 1. RE EXTENSIONS AND POISSON ALGEBRAS 11
Example 5.7.
In contrast to the examples in Examples 5.6, B is also Poisson simple forthe Poisson bracket such that { x, y } = 0 , { z, x } = y and { y, z } = xy − , which has the property that, for all b ∈ B , { z, b } , { x, b } and { y, b } are not units. Clearly { x, b } = − y ∂ z ( b ) and { y, b } = ( xy − ∂ z ( b ) are never units. For { z, ( P a i z i ) } = P { z, a i } z i } to be a unit it is necessary that { z, a i } is a unit and it is shown in [16] that if a ∈ A then { z, a } = δ ( a ) is not a unit. Remark . The examples in 5.7 and 5.6(i) have analogues in the polynomial algebra C [ x , x , . . . , x n ] when n >
3. In these { x i , x j } = 0 for 1 ≤ i, j ≤ n − z is asimple derivation of C [ x , x , . . . , x n − ]. See [16, §
3] and [21, §
9] for details from the point ofview of Ore extensions.
Example 5.9.
Coutinho [3, 4] has used the theory of foliations to make a substantial contri-bution to the understanding of the simple derivations of C [ x, y ]. Let A be the subspace of C [ x, y ] consisting of polynomials of total degree at most 2 and let U be the set of unimodularrows ( a, b ) where a, b ∈ A × A are such that at least one of a and b has total degree 2. In [4]it is shown that the closure U in A × A has four irreducible components P i , 1 ≤ i ≤ P i . Below we give the details of examples of four correspondingtypes of Poisson bracket on B for which B is Poisson simple. Full details, presented fromthe point of view of derivations of C [ x, y ], can be found in [4]. Type 1 , P : let a, b, c ∈ Q [ i ] \
0, with a = 1 be such that the quadratic polynomial y + bx + cxy is irreducible over Q [ i ]. Then, by [4, Proposition 4.1] and Corollary 3.4, C [ x, y, z ] is Poisson simple for the Poisson bracket such that { x, y } = 0 , { y, z } = c ( xy + a ) + bx and { z, x } = xy + a. Type 2 , P : let β ∈ Q [ i ][ x, y ], be homogeneous of degree 2 and irreducible over Q [ i ].Then, by [4, Proposition 5.4] and Corollary 3.4, C [ x, y, z ] is Poisson simple for the Poissonbracket such that { x, y } = 0 , { y, z } = − β and { z, x } = 1 . Type 3 , P : by [4, Proposition 6.1] and Corollary 3.4, C [ x, y, z ] is Poisson simple for thePoisson bracket such that { x, y } = 0 , { y, z } = − x and { z, x } = xy + 1 . Type 4 , P : in Examples 5.6(i), take p ( x ) = ρx where ρ ∈ C \{ } .For discussion of some classes of simple derivations δ = g∂ x − f ∂ y of A where the degrees of f and g may be greater than 2, see [3, Corollary 4.3, Theorems 4.4 and 5.5 and Proposition6.2]. Example 5.10.
Let f = − g = x , so that δ ( x ) = x and δ ( y ) = 1 and the Poissonbracket on A is such that { y, z } = − , { z, x } = x and { x, y } = 0. The Poisson triplehere is the cm-exact triple x grad( y − log x ). It is clear that xB is a Poisson prime ideal and that x / ∈ PZ( B ). Applying Proposition 5.5 with C = C [ x ± ], a = 0, b = 1 and δ | C = xd/dx , it is easy to see that xA is the only non-zero δ -prime ideal of A . As δ ( A ) * xA it follows from Theorem 3.2 that P . Spec( A ) = { , xA } . By Theorem 3.6, if R = A [ z ; δ ] thenSpec R = { , xR } . Example 5.11.
Let M = xB + yB and N = xA + yA and suppose that f, g ∈ A are suchthat if δ = g∂ x − f ∂ y then N is the unique non-zero δ -prime ideal of A , in other words,there are no height one prime ideals invariant under δ and N is the only maximal ideal of A invariant under δ . Then f = − δ ( y ) ∈ N , g = δ ( x ) ∈ N and δ ( A ) ⊆ N . By Theorems 3.2and 3.6, P . Spec B = { , xB + yB } ∪ { xB + yB + ( z − α ) B : α ∈ C } and, if R = C [ z ; δ ],Spec R = { , xR + yR } ∪ { xR + yR + ( z − α ) R : α ∈ C } . Note that P . Spec B = { } and Spec R = { } . In other words, there is no proper Poissonprime homomorphic image of B with a non-zero Poisson bracket and every proper primehomomorphic image of R is commutative. However if j is such that f / ∈ N j or g / ∈ N j then B/ ( N j B ) is a proper Poisson homomorphic image of B with a non-zero Poisson bracket and R/ ( N j R ) is a noncommutative proper homomorphic image of R . Such a j must exist as f and g must be non-zero and ∩ j ≥ N j = 0.Goodearl and Warfield [7, p. 61] specify such an example with f = − ( x + y ) and g = x + y . Although condition on the base field in [7] is satisfied by R but not by C , theconclusion is also valid for C . The details of this example were omitted from [7] as the proofwas ‘exceedingly tedious’. Interest was expressed in any similar example with a short proof.Here, subject to the reader’s interpretation of the word ‘short’, we present such an example.Let f = − x (1 + xy ) and g = y so that, δ ( y ) = x (1 + xy ) and δ ( x ) = y . Let ′ denotedifferentiation with respect to x . Clearly N is the unique maximal ideal of A invariantunder δ . Let Q = N be a non-zero δ -prime ideal of A . Then Q has height one and isprincipal, Q = qA , say, with 0 = q = P ni =0 q i ( x ) y i , each q i ( x ) ∈ C [ x ], q n ( x ) = 0 and, as δ ( q ( x )) = yq ′ ( x ), n >
0. Let h ∈ A be such that δ ( q ) = hq . Note that, for 0 ≤ i ≤ n , δ ( q i ( x ) y i ) = q ′ i ( x ) y i +1 + ix q i ( x ) y i + ixq i ( x ) y i − . Note also that q ′ n ( x ) ∈ q n ( x ) C [ x ] whence q ′ n ( x ) = 0 and q n ( x ) ∈ C ∗ . Therefore deg y ( δ ( q )) ≤ n so h = h ( x ) ∈ C [ x ]. Comparing coefficients of y i , 0 ≤ i ≤ n , in the equation δ ( q ( x )) = h ( x ) q ( x ), we obtain(5.2) ( i + 1) xq i +1 ( x ) = ( h ( x ) − ix ) q i ( x ) − q ′ i − ( x ) , where q − ( x ) = 0 = q n +1 ( x ). Note that q ( x ) = 0, otherwise q i ( x ) = 0 for all i . For i ≥
0, let d i = deg( q i ( x )), let d = d and let e i = deg( h ( x ) − ix ). By (5.2) with i = 0, d = e + d − d i + e i > d i − − d i +1 = d i + e i − . RE EXTENSIONS AND POISSON ALGEBRAS 13
In the following five situations, (5.3) can be used to show, inductively, that the sequence { d i } is eventually strictly increasing. Hence these cases can be excluded.(i) If h ( x ) = 0 then e i = 2, when i > q ( x ) = 0 and d i = d − i whenever i > h ( x ) has degree r = 0 or 1 then e = r , e i = 2 when i > d = d + r − d i = d − r − i when i > h ( x ) has degree r ≥ h ( x ) = ax + bx + c has degree r = 2 and a / ∈ N then d i = d + i ( r −
1) for i > h ( x ) = ax + bx + c has degree 2, a ∈ N and b = 0 then d i = d + i for 0 ≤ i ≤ a , d a +1 = d + a and d a + j = d + a + j − j ≥ h ( x ) = ax + c has degree 2, a ∈ N and c = 0 then d i = d + i for 0 ≤ i ≤ a , d a +1 = d + a − d a + j = d + a + j − j ≥ h ( x ) = ax , a ∈ N , in which we need to keep track of leadingcoefficients as well as degrees. Let α be the leading coefficient of q ( x ). By repeated useof (5.2), the leading coefficient of q i ( x ) is (cid:0) ai (cid:1) α for 0 ≤ i ≤ a . In particular, the leadingcoefficients of q a − ( x ) and q a ( x ) are aα and α respectively. By (5.3), d i = d + i for 0 ≤ i ≤ a .By (5.2), with i = a , ( a + 1) xq a +1 ( x ) = − q ′ a − ( x ) so d a +1 = d + a − q a +1 ( x ) is − ( d + a − aα/ ( a + 1).From (5.2), with i = a + 1, we see that d a +2 ≤ d + a − x d + a − in q a +2 ( x ) is − ( d + 2 a ) α/ (( a + 1)( a + 2)) = 0. Therefore d a +2 = d + a − > d a +1 − e a +2 − d a + j = d + a + j − j ≥
3, which is impossible. Thiscompletes the proof that P . Spec and Spec R are as stated above. Example 5.12.
Here we consider the Poisson bracket on B arising from [7, Example 2.15],where δ = 2 y∂ x + ( y + x ) ∂ y so that { y, z } = − ( y + x ), { z, x } = 2 y and { x, y } = 0. In [7],it is shown that the only non-zero δ -prime ideals of A are the maximal ideal M := xA + yA and the height one prime Q := ( y + x + 1) A . Note that Q * M and that δ ( A ) ⊆ M but δ ( A ) * Q . By Theorem 3.2,P . Spec B = { , ( y + x + 1) B, xB + yB } ∪ { xB + yB + ( z − α ) B : α ∈ C } . If R = A [ z ; δ ] thenSpec R = { , ( y + x + 1) R, xR + yR } ∪ { xR + yR + ( z − α ) R : α ∈ C } . Note that P . Spec B = { , ( y + x + 1) B } and Spec R = { , ( y + x + 1) R } . In contrast toExample 5.11, there is a unique non-zero Poisson prime ideal that is not residually null. Remark . In both Examples 5.11 and 5.12, the Poisson algebra B has a Poisson primeideal P = xB + yB which has height two as a prime ideal but is minimal as a non-zero Poissonprime ideal. In both cases P is residually null. To obtain examples of this phenomenon inwhich P is not residually null, pass to B ′ = B [ u, v ] = C [ x, y, z, u, v ] with the Poisson bracketsuch that { u, b } = { v, b } = 0 for all b ∈ B and { u, v } = 1. This is the tensor product,as Poisson algebras, of B and a copy of the coordinate ring of the symplectic plane. Then xB ′ + yB ′ again has height two as a prime ideal and is minimal as a non-zero Poisson prime ideal but it is not residually null, having factor isomorphic to C [ z, u, v ] with { u, v } = 1 and { u, z } = { v, z } = 0. References [1] K. A. Brown and K. R. Goodearl,
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