Poisson derivations of a semiclassical limit of a family of quantum second Weyl algebras

In previous paper, we studied deformations $A_{\alpha,\beta}$ of the second Weyl algebra and computed their derivations. In the present paper, we identify the semiclassical limits $\mathcal{A}_{\alpha,\beta}$ of these deformations and compute their Poisson derivations. Our results show that the first Hochschild cohomology group HH$^1(A_{\alpha,\beta})$ is isomorphic to the first Poisson cohomology group HP$^1( \mathcal{A}_{\alpha,\beta}).$

future after we have successfully studied the automorphism group of A α,β . In the present case and as already mentioned, we only focus on studying the Lie algebra of Poisson derivations of A α,β , and comparing them to their non-commutative counterparts in [15].
In the noncommutative world, the knowledge of the derivations of twisted group algebras, studied by Osborn and Passman [19], has helped in studying the derivations of other non-commutative algebras such as the quantum second Weyl algebra (see [15]), quantum matrices (see [13]), generalized Weyl algebras (see [11]) and some specific examples of quantum enveloping algebras (see [14], [20], and [21]). In view of this, we also study the Poisson derivations of the Poisson analogue of the twisted group algebras-called Poisson group algebras-and apply the results to study the Poisson derivations of a semiclassical limit A α,β of A α,β . The rest of the paper is organised as follows.
In Section 2, we recall some basics on Poisson algebras and semiclassical limit. We then proceed to study the Poisson derivations of the Poisson group algebras. Similarly to their non-commutative counterparts in [19], every Poisson derivation of a Poisson group algebra is the sum of an inner Poisson derivation and a central/scalar Poisson derivation.
In Section 3, we study a semiclassical limit A of the quantum algebra U + q (G 2 ) and establish that A is a Poisson polynomial K-algebra generated by six indeterminates X 1 , . . . , X 6 . Since A = K[X 1 , . . . , X 6 ] satisfies the conditions in [12,Hypothesis 1.7], we can apply the Poisson deleting derivations algorithm [12] to study its Poisson centre A: a polynomial ring K[Ω 1 , Ω 2 ] in two variables. In Section 4, we study some Poisson H-prime ideals of A using Goodearl's H-stratification theory [8], and proceed to study some family ( Ω 1 − α, Ω 2 − α ) (α,β)∈K 2 \{(0,0)}) of maximal and primitive Poisson prime ideals of A. Consequently, we study their corresponding Poisson simple quotients A α,β := K[X 1 , . . . , X 6 ]/ Ω 1 − α, Ω 2 − β , and conclude that the Poisson algebra A α,β is a semiclassical limit of the quantum second Weyl algebra A α,β . Having a complete description of the semiclassical limit A α,β of A α,β , we proceed to study its Poisson derivations in the final section of this paper, by following procedures similar to its non-commutative counterpart A α,β (see [15, §5]). That is, we successively embed A α,β into a suitable Poisson torus R 3 via localization as follows: These embeddings and localization allow us to extend every Poisson derivation of A α,β successively and uniquely to a Poisson derivation of each of the Poisson algebras R i through to the Poisson torus R 3 . Since a Poisson torus is an example of a Poisson group algebra, we have that every Poisson derivation of R 3 is the sum of an inner Poisson derivation and a central/scalar Poisson derivation. Given the Poisson derivations of R 3 , we backwardly and successively pull the Poisson derivations of R 3 to A α,β . This gives us a complete description of the Poisson derivations of A α,β . Similarly to their non-commutative counterparts in [15], every Poisson derivation of A α,β is an inner Poisson derivations provided αβ = 0, and the sum of inner Poisson and scalar Poisson derivations whenever α or β is zero. More precisely, the first Poisson cohomology group HP 1 (A α,β ) is a one-dimensional vector space in the case where α or β is zero.

Poisson derivations of Poisson group algebras
This section begins with a reminder about Poisson algebras and semiclassical limit. We will then proceed to introduce Poisson group algebras, and consequently, study their derivations.

Poisson algebras
A Poisson algebra A is a commutative algebra over K endowed with a skew-symmetric K-bilinear map Given a Poisson ideal I of A, it is well known that the quotient algebra A/I is a Poisson algebra with an induced Poisson bracket defined as {x,ȳ} = {x, y}, wherex := x + I andȳ := y + I, note that x, y ∈ A. Finally, the subalgebra Z P (A) := {a ∈ A | {a, x} = 0, ∀x ∈ A} is called the Poisson centre of A.
Remark 2.1. If A is a Poisson algebra and {x 1 , . . . , x n } is a generating set for A (as an algebra), then (1) it is always enough to define a Poisson bracket {−, −} on A by defining it on only the generating set.

Semiclassical limit
Given a non-commutative algebra, one can move from the 'Non-commutative World' to the 'Poisson World' through a process called semiclassical limit, and reverse this process through quantization. This transformation (semiclassical limit) and its reverse transformation (quantization) have been widely studied (for example, see [5, § §1.1.3], [2,Chapter III.5], and [9, §2]). In line with the presentation in [5, § §1.1.3], we present the following overview of semiclassical limit. Let R be a commutative principal ideal domain containing the field K and hR be a maximal ideal of R for a fix h ∈ R. Let A be an algebra which is not necessarily commutative torsion-free R-algebra such that the quotient A := A/hA is a commutative algebra. For u, v ∈ A; we have thatū := u + hA andv := v + hA are their respective canonical images in A. . We say that A is a quantization of A, and A is a semiclassical limit of A. Fix λ ∈ K. The algebra A λ := A/(h − λ)A is a deformation of the Poisson algebra A = A 0 if the central element h − λ is not invertible in A. We refer the interested reader to [9, §2] for some known examples of semiclassical limit of some families of quantum algebras.

Introduction to Poisson group algebras
In [19, §1&2], Osborn and Passman studied the derivations of twisted group algebras. In line with their results, we also study the Poisson derivations of Poisson group algebras. The results in this section will be crucial in the final section of this paper where we study the Poisson derivations of a semiclassical limit of the quantum second Weyl algebra A α,β . Let G represent a finitely generated abelian group and λ : G × G −→ K be a map such that λ(y, x) = −λ(x, y) and λ(x, yz) = λ(x, y) + λ(x, z). We define a Poisson group algebra K λ P [G] as a commutative K-algebra which has a copy G := {ḡ | g ∈ G} of G as a basis and satisfies the Poisson bracket via {x,ȳ} = λ(x, y)xȳ = λ(x, y)xy for all x, y ∈ G (note thatxȳ = xy). Observe that λ(x, y) = 0 if and only if {x,ȳ} = 0.
As an example, take the group algebra K[Z 2 ] generated by x ±1 , y ±1 over K, with basis Z 2 := {x i y j | (i, j) ∈ Z 2 }. A Poisson structure can be defined on K[Z 2 ] via {x i y j , x k y l } = λ((i, j), (k, l))x i+k y j+l , where λ((i, j), (k, l)) := il − jk, to obtain a rank 2 Poisson torus K λ P [Z 2 ]. In general, K λ P [Z n ] is a Poisson torus of rank n over the field K for some λ : Z n × Z n → K, where Z n is the usual additive group.
Let γ ∈ K λ P [G]. One can write γ as γ = g c gḡ with g ∈ G and c g ∈ K. Note that c g = 0 for almost all g. The set supp(γ) := {g ∈ G | c g = 0 in γ} is called the suppor t of γ. Furthermore, the set C := {g ∈ G | {ḡ,x} = 0 for all x ∈ G} and ∆(x) := {g ∈ G | {ḡ,x} = 0} are both subgroups of G. The following remark establishes a relationship between these two subgroups.
for all x ∈ G.
Proof. We need to show that D(xȳ) = D(x)ȳ +xD(ȳ) and Similarly to [19], we will call D in Lemma 2.5 as a central Poisson derivation when Z P (K λ P [G]) strictly contains K, and a scalar Poisson derivation when for all x ∈ G.
We can now state our main result in this section in the theorem below.
where g := hx −1 and a g (x) := b gx (x). Note that a g : G −→ K and a g (x) = 0 for almost all x ∈ G.
Since D is a Poisson derivation, we have that D(xȳ) = D(x)ȳ +xD(ȳ) for all x, y ∈ G. As a result, Identifying the coefficients in the above equality reveals that a g (xy) = a g (x) + a g (y).
Suppose that g ∈ C. It follows that λ(g, y) = λ(g, x) = 0 for all x, y ∈ G. Since a g (xy) = a g (x) + a g (y), the map θ : Now, let g ∈ C. There exists y ∈ G such that λ(g, y) = 0. Fix y and define c g := a g (y) λ(g, y) .
Take any arbitrary element x ∈ G. It follows that c g λ(g, x) = a g (y)λ(g, x) λ(g, y) .
From (3), we have that From (4)  3 Poisson prime spectrum and Poisson deleting derivations algorithm of a semiclassical limit of U + q (G 2 ) This section aims to study a semiclassical limit of the positive part of the quantized enveloping algebra of type G 2 , U + q (G 2 ). Given the semiclassical limit of U + q (G 2 ), we will study its Poisson prime spectrum using Goodearl's H-stratification theory [8], and its Poisson deleting derivations algorithm-introduced by Launois and Lecoutre [12]. Given the data of the Poisson deleting derivations algorithm, we will study the Poisson centre of the semiclassical limit.
where f (z) = z 4 + z 2 + 1. Fix λ ∈ K * . Observe that the element z − λ is central and not invertible in is a Poisson algebra with the Poisson bracket defined as follows: Therefore, A 1 is a semiclassical limit of the non-commutative algebra U + q (G 2 ), and A q is a deformation of the Poisson algebra A 1 . For simplicity, we set in the rest of this paper. One can write the Poisson algebra A as an iterated Poisson-Ore extension over K (see [17,Theorem 1.1] for the definition of iterated Poisson-Ore extension) as follows: where, σ i and δ i are respectively the Poisson derivations and Poisson σ i -derivations of (2 ≤ i ≤ 6 and δ 2 = 0) defined as follows: 3.2 Poisson prime spectrum of the semiclassical limit of the algebra U + q (G 2 ) We study the Poisson prime spectrum of the Poisson algebra A = K[X 1 , . . . , X 6 ] in this subsection. Let P be a proper Poisson ideal of a Poisson algebra A and I 1 , I 2 be Poisson ideals of A such that P ⊇ I 1 I 2 .
The ideal P is called a Poisson prime ideal provided P ⊇ I 1 or P ⊇ I 2 . Since the Poisson algebra A is a noetherian domain, every Poisson ideal which is also a prime ideal is a Poisson prime ideal and vice versa (see [ [8, §2] for the definition of a rational torus action). A Poisson prime ideal P is H-invariant if h · P = P for all h ∈ H. Moreover, J := h∈H h · P is the largest Poisson H-invariant prime ideal contained in P.
(H2) The derivations δ i are all locally nilpotent and δ i α − αδ i = η i δ i for some non-zero scalar η i , for all 2 ≤ i ≤ 6.
For each 2 ≤ j ≤ 7, the algebra A (j) represents the subalgebra of Fract(A) generated by all the X i,j . That is, by an isomorphism that maps X i,j to X i , and τ j , . . . , τ 7 denote the Poisson derivations defined by τ l (X i ) = µ li X i for all 1 ≤ i < l ≤ 7. With a slight abuse of notation, one can identify τ j , . . . , τ 7 with σ j , . . . , σ 7 respectively.
One can easily check that A is a Poisson affine space associated to the skew-symmetric matrix That is, A satisfies the relation {T i , T j } = µ ij T j T i for all 1 ≤ i, j ≤ 6, where µ ij are the entries of M.

Canonical embedding
The set Σ := {X n j,j+1 | n ∈ N} = {X n j,j | n ∈ N} is a multiplicative system of regular elements of A (j) and A (j+1) . Moreover, One can use the PDDA to relate P.Spec(A) to P.Spec(A) by constructing an embedding ψ j : P.Spec(A (j+1) ) ֒→ P.Spec(A (j) ) defined by for each 2 ≤ j ≤ 6 (see [12,Lemma 2.3]). The map g j is a surjective homomorphism defined by g j (X j,j ) := X j,j+1 + X j,j+1 (further details can be found in [12, §2]). From [12, §2.1], there exists an increasing homeomorphism from the topological space {P ∈ P.Spec(A (j+1) ) | X j,j+1 ∈ P } onto the topological space {Q ∈ P.Spec(A (j) ) | X j,j ∈ P } whose inverse is also an increasing homeomorphism (the topology being the Zariski topology). The map ψ j is injective but not necessarily bijective. However, ψ j induces a bijection between {P ∈ P.Spec( is obtained by composing all the ψ j . This canonical embedding ψ helps to construct a partition of P.Spec(A) into a disjoint union of strata known as the canonical partition via the Cauchon diagrams. The torus invariant Poisson prime spectrum has generally been described in [12, §2.2] via Cauchon's diagrams. Based on that, we can describe the set H-P.Spec(A) as follows. For any subset C of {1, . . . , 6}, . . , i 6 ∈ N} is a multiplicative system of non-zero divisors of A (j) for each 2 ≤ j ≤ 6. One can therefore localize A (j) at S j as follows: Note that the set Σ j := {T n j | n ∈ N} is also a multiplicative set in both A (j) and A (j+1) for each 2 ≤ j ≤ 6. It follows from [12, Prop. 1

.11] that
One can easily verify that Moreover, the localization ] also holds in R 2 , since T 1 generates a multiplicative system in R 2 . In fact, R 1 is the Poisson torus associated to the Poisson affine space A. As a result, The PDDA helps to construct the following embeddings: Note that the family ( . For the reverse inclusion, let y ∈ Z P (R 1 ). Then, y can be written in terms of the basis of R 1 as y = (i,...,n)∈Z 6 a (i,...,n) T i 1 T j 2 T k 3 T l 4 T m 5 T n 6 . One can verify that {y, T 1 } = (−3j − k + m + 3n)yT 1 . Since y ∈ Z P (R 1 ), it follows that −3j − k + m + 3n = 0. Following the same pattern for T 2 , T 3 , T 4 , T 5 and T 6 , one can confirm that 3i − 3k − 3l + 3n = 0, i + 3j − 3l − m = 0, 3j + 3k − 3m − 3n = 0, −i + k + 3l − 3n = 0, and −3i − 3j + 3l + 3m = 0. Solving this system of six equations will reveal that i = k = m and j = l = n. One can therefore write 2. Similar argument as in (1) will prove the result.

Height one Poisson H-invariant prime ideals of A
In this subsection, we study the Poisson H-invariant prime ideals of A = K[X 1 , . . . , X 6 ] with height one. We will begin by showing that Ω 1 and Ω 2 are Poisson prime ideals. Note that Θ R will denote an ideal generated by the element Θ in any the ring R. Where no doubt arises, we will simply write Θ .
The following result shows that T 1 ∈ Im(ψ), and that Ω 1 is the Poisson prime ideal of A such that ψ( Ω 1 ) = T 1 .
Proof. We will prove this result in several steps by showing that: We proceed to prove the above claims.

One can easily verify that
. . , 6. Therefore, I is a Poisson ideal in A (4) . In addition, A (4) /I is isomorphic to a polynomial ring in five variables which is a domain, hence I is a prime ideal. Since I is both Poisson and prime ideal, it is a Poisson prime ideal in A (4) .
3. Similar to (2). (6) . We establish the reverse inclusion. Let is not a factor of Ω 1 , hence, it must be a factor of v ′ . Now A (6) can be viewed as a free K[X 1,6 , X 2,6 , X 3,6 , T 4 , as expected. 5. The proof is similar to (4).
Following similar procedures, one can also prove that T 2 ∈ Im(ψ), and that Ω 2 is the Poisson prime ideal of A such that ψ( Ω 2 ) = T 2 (the interested reader can check out the details of the proof in [18]). We state only the result in the following lemma.
Observe that T 1 , T 2 and T 2 , T 3 are Poisson prime ideals of A. In the next lemma, we will show that T 1 , T 2 , T 2 , T 3 ∈ ψ(P.Spec(A)).
Proof. Let J (j) ∈ P.Spec(A (j) ) for all 2 ≤ j ≤ 6. We already know that We begin by showing that T 1 , T 2 ∈ ψ(P.Spec(A)). Set J 1,2 , a contradiction! Therefore, 1,2 . Similarly, one can show that T 6 ∈ J 1,2 . Hence, J Recall that Ω 1 = T 1 T 3 T 5 and Ω 2 = T 2 T 4 T 6 in A. Observe that Ω 1 , Ω 2 are both elements of T 1 , T 2 and T 2 , T 3 . From Lemma 4.3, we know that J 1,2 and J 2,3 are the elements of P.Spec(A) such that ψ(J 1,2 ) = T 1 , T 2 and ψ(J 2,3 ) = T 2 , T 3 . In the next lemma, we show that J 1,2 and J 2,3 contain Ω 1 and Ω 2 . Proof. Recall that Ω 1 and Ω 2 are central elements of A (j) for all 2 ≤ j ≤ 7. Given the set-up in the proof of Lemma 4.3, we know that Ω 1 , We now want to find the height one Poisson H-invariant prime ideals of A, and show that the height 2 Poisson H-invariant prime ideals of A contain those of height one.
Denote the kernel of g j by ker(g j ) and the image of ψ by Im(ψ). We have the following lemma.
Note that the map ψ induces a canonical embedding from H-P.Spec(A) to H-P.Spec(A). Observe that { T i | i = 1, . . . , 6} is the set of only height one Poisson H-invariant prime ideals we have in A. Since ψ preserves the height of a prime ideal, if ψ −1 ( T i ) ∈ P.Spec(A), then it is a height one Poisson H-invariant prime ideal in A for all 1 ≤ i ≤ 6. For example, we already know that ψ −1 ( T 1 ) = Ω 1 and ψ −1 ( T 2 ) = Ω 2 . Therefore, Ω 1 and Ω 2 are height one Poisson H-invariant prime ideals in A. We will show in the next lemma that Ω 1 and Ω 2 are the only height one Poisson H-invariant prime ideals in A.
The corollary below is deduced from the proof of Lemma 4.6.
Recall from Lemma 4.3 that there exist J 1,2 and J 2,3 of P.Spec(A) such that ψ(J 1,2 ) = T 1 , T 2 and ψ(J 2,3 ) = T 2 , T 3 . As a result of Corollary 4.7, the Poisson ideals T 1 , T 2 and T 2 , T 3 are the only height two Poisson H-invariant prime ideals of ψ(P.Spec(A)). Since ψ preserves Poisson H-invariant prime ideals and the height of a Poisson prime ideal, this implies that J 1,2 and J 2,3 are the only height two Poisson H-invariant prime ideals of A. It follows from Lemma 4.4 that the height two Poisson H-invariant prime ideals of A contain Ω 1 and Ω 2 .
Remark 4.8. Since the height two Poisson H-invariant prime ideals of A contain Ω 1 and Ω 2 , every non-zero Poisson H-invariant prime ideal of A will contain either Ω 1 or Ω 2 . Note that those Poisson H-invariant primes of at least height 2 will contain both Ω 1 and Ω 2 .
In the remainder of this section, we study a Poisson torus arising from a localization of A α,β , and a linear basis of A α,β , both of which will be useful in computing the Poisson derivations of A α,β in the next section.

Linear basis for A α,β
Set A β := A/ Ω 2 − β , β ∈ K. Denote the canonical image of X i in A β by x i := X i + Ω 2 − β for each 1 ≤ i ≤ 6. It can be verified that A α,β ∼ = A β / Ω 1 − α . Note that A β satisfies the relation: Proof. Since (Π 6 s=1 X is s ) is∈N is a basis of A over K, we have that (Π 6 s=1 x s is ) is∈N is a spanning set of A β over K. We want to show that F is a spanning set of A β . It is sufficient to do that by showing that i 6 can be written as a finite linear combination of the elements of F over K for all i 1 , . . . , i 6 ∈ N. We do this by an induction on i 4 . The result is clear when i 4 = 0. For i 4 ≥ 0, suppose that where v := (i, j, k, l, m) ∈ N 5 , I is a finite subset of {0, 1} × N 5 , and the a (ξ,v) are scalars. It follows that We have to show that The result is obvious when ξ = 0. For ξ = 1, then using (10), one can verify that Before we continue the proof, the following ordering < 4 needs to be noted.
We proceed to show that F is a linearly independent set. Suppose that It follows that where ν ∈ A. Write ν = (i,...,n)∈J b (i,...,n) X i 1 X j 2 X k 3 X l 4 X m 5 X n 6 , where J is a finite subset of N 6 , and b (i,...,n) are scalars. From Subsection 3.5, we have that It follows that where LT < 4 contains lower order terms with respect to < 4 (see item ♣). Moreover, LT < 4 vanishes when b (i,...,n) = 0 for all (i, . . . , n) ∈ J. One can easily confirm this when the previous line of equality (right hand side) is fully expanded. Suppose that there exists (i, j, k, l, m, n) ∈ J such that b (i,j,k,l,m,n) = 0. Let (i ′ , j ′ , k ′ , l ′ , m ′ , n ′ ) be the greatest element of J with respect to < 4 such that b (i ′ ,j ′ ,k ′ ,l ′ ,m ′ ,n ′ ) = 0. Identifying the coefficients of ..,n∈N is a basis for A and LT < 4 contains lower order terms). Therefore, b (i ′ ,j ′ ,k ′ ,l ′ ,m ′ ,n ′ ) = 0, a contradiction! As a result, b (i,j,k,l,m,n) = 0 for all (i, j, k, l, m, n) ∈ J, and Consequently, a (ξ,i,j,k,l) = 0 for all (ξ, i, j, k, l) ∈ I.
We are now ready to find a basis for A α,β . 1}×N 5 is a spanning set of A α,β over K. We want to show that P spans A α,β by showing that e i 1 1 e i 2 2 e i 3 3 e ξ 4 e i 5 5 e i 6 6 can be written as a finite linear combination of the elements of P over K for all (i 1 , i 2 , i 3 , ξ, i 5 , i 6 ) ∈ {0, 1} × N 5 . By Proposition 4.15, it is sufficient to do this by an induction on i 3 . The result is obvious when i 3 = 0 or 1. For i 3 ≥ 1, suppose that where v := (i, j, k, l) ∈ N 4 , and the a (ǫ 1 ,ǫ 2 ,v) are all scalars. Moreover, I is a finite subset of {0, 1} 2 × N 4 . It follows from the inductive hypothesis that We need to show that The result is obvious when (ǫ 1 , ǫ 2 ) = (0, 0), (0, 1). Using Lemma 4.13(1),(3), one can also show that (1,1); and i, j, k, l ∈ N. Therefore, x ξ 4 x i 5 5 x i 6 6 ∈ Span(P). As a result, P spans A α,β .
We proceed to show that F is a linearly independent set. Suppose that Then, where ν ∈ A β . Set w := (i, j, k, l, m) ∈ N 5 , and let J 1 , J 2 be finite subsets of N 5 . One can write ν in terms of the basis F of A β as: where b w and c w are scalars. Note that Given this expression, and the relation (10), one can express the above equality as follows: Note that r 1 , . . . , r 11 are some non-zero rational numbers. Before we continue the proof, the following ordering < 3 needs to be noted.
Proof. The result is obvious when k, l ≥ 0 due to Proposition 4.16. When k (resp. l) is negative, then one can multiply the above equality enough times by t 5 (resp. t 6 ) to kill all the negative powers and then apply Proposition 4.16 to complete the proof.
5 Poisson derivations of the semiclassical limit of the quantum second Weyl algebra A α,β This section focuses on studying the Poisson derivations of the Poisson algebra A α,β .

Preliminaries and strategy
where A (j) is defined in Subsection 3.3, and Ω 1 and Ω 2 are the generators of the centre of A (j) (see Subsection 3.3). Recall that A (7) = A = K[X 1 , . . . , X 6 ]. It follows that A α,β = A α,β . For each 2 ≤ j ≤ 7, denote the canonical images of the generators X i,j of A (j) in A (j) α,β by x i,j for all 1 ≤ i ≤ 6. Furthermore, from [12], one can deduce the following data of A α,β from the PDDA of A (see Subsection 3.3) as follows: x 2,5 = x 2,6 − 3x 2 3,6 x −1 5,6 + For simplicity, we will refer to the above data as the PDDA of A α,β . Note that the t i are the canonical images of T i in A (2) α,β for all 1 ≤ i ≤ 6. For each 2 ≤ j < 7, set . . , i 6 ∈ N, λ ∈ K * }. One can observe that S j is a multiplicative system of non-zero divisors (or regular elements) of A (j) α,β . As a result, one can localize A (j) α,β at S j . Let us denote this localization by R j . That is, Again, the set Σ j := {t k j | k ∈ N} is a multiplicative set in both A (j) α,β and A (j+1) α,β for each 2 ≤ j ≤ 6. Therefore, A It follows that for all 2 ≤ j ≤ 6, with the convention that R 7 := A α,β . Similarly to (7), we construct the following embeddings: Observe that Strategy to compute the Poisson derivations of A α,β = R 7 . From Corollary 2.7, we know that the Poisson derivations of the Poisson torus R 3 . Therefore, we will extend the Poisson derivations of A α,β to Poisson derivations of R 3 . We will then 'pull back' the Poisson derivations of R 3 sequentially to the Poisson derivations of A α,β , and this will give us a complete description of the Poisson derivations of A α,β . This process will be carried out in steps, and the linear basis for R i will play crucial role. As a result, we will compute these bases in the subsequent subsection.
The embeddings in (12) present an opportunity to compute the centre of each of the algebras R i , which will be helpful in studying the Poisson derivations of A α,β .
Proof. It is easy to verify that Z P (

Basis for R 4
Observe that where where β ∈ K. We will denote the canonical images of X i,4 (resp. T i ) in A (4) β by x i,4 (resp. t i ) for all 1 ≤ i ≤ 6. Observe that t 2 = β t 6 β S −1 4 . As usual, one can identify R 4 with A Proof. One can easily verify that f 1 k 3 ,...,k 6 )∈N 2 ×Z 3 spans R 4 . We show that P 4 is a spanning set of R 4 by showing that f k 1 1 t k 3 3 t k 4 4 t k 5 5 t k 6 6 can be written as a finite linear combination of the elements of P 4 for all (k 1 , k 3 , . . . , k 6 ) ∈ N 2 × Z 3 . It is sufficient to do this by an induction on k 1 . The result is clear when k 1 = 0. Assume that the statement is true for k 1 ≥ 0. That is, where i = (i 1 , i 4 , i 5 , i 6 ) ∈ I 1 ⊂ N × Z 3 and j = (i 3 , i 4 , i 5 , i 6 ) ∈ I 2 ⊂ N × Z 3 . Note that the a i and b j are all scalars.
Clearly, the monomial f i 1 +1 1 t i 4 4 t i 5 5 t i 6 6 ∈ Span(P 4 ). We have to also show that f 1 t i 3 3 t i 4 4 t i 5 5 t i 6 6 ∈ Span(P 4 ) for all i 3 ∈ N and i 4 , i 5 , i 6 ∈ Z. This can easily be achieved by induction on i 3 , and the use of the relation . Therefore, by the principle of mathematical induction, P 4 is a spanning set of R 4 over K.

Basis for R 5 .
We will identify R 5 with A (5) Note that the canonical images of X i,5 (resp. T i ) in A (5) α will be denoted by x i,5 (resp. t i ) for all 1 ≤ i ≤ 6. We now find a basis for A (5) α S −1 5 . Recall from Subsection 3.3 that Ω 1 = Z 1 T 3 T 5 − 1 2 Z 2 T 5 and Ω 2 = Z 2 T 4 T 6 − 2 3 T 3 3 T 6 in A (5) (remember that Z 1 := X 1,5 and Z 2 := X 2,5 ). Since z 2 t 4 t 6 − 2 3 t 3 3 t 6 = β and z 1 t 3 t 5 − 1 2 z 2 t 5 = α in R 5 and A (5) α S −1 5 respectively, we have the relation α S −1 5 and, in R 5 , we have the following two relations: Proposition 5.3. The set Proof. One can easily show that the family z 1 can be written as a finite linear combination of the elements of P 5 for all (k 1 , k 3 , k 4 , k 5 , k 6 ) ∈ N 3 × Z 2 . It is sufficient to do this by an induction on k 3 . The result is obvious when k 3 = 0, 1, 2. For k 3 ≥ 2, suppose that where I is a finite subset of {0, 1, 2} × N 2 × Z 2 , and the a (ξ,i) are all scalars. It follows that Now, z i 1 1 t ξ+1 3 t i 4 4 t i 5 5 t i 6 6 ∈ Span(P 5 ) when ξ = 0, 1. For ξ = 2, one can easily verify that z i 1 1 t 3 3 t i 4 4 t i 5 5 t i 6 6 ∈ Span(P 5 ) by using the relation in (14). Therefore, by the principle of mathematical induction, P 5 spans R 5 .
Corollary 5.4. Let I be a finite subset of {0, 1, 2} × N × Z 3 and (a (ξ,i) ) i∈I be a family of scalars. If Proof. When i 4 ≥ 0, then the result is obvious as a result of Proposition 5.3. For i 4 < 0, multiply both sides of the equality enough times by t 4 to kill all the negative powers of t 4 , and then apply Proposition 5.3 to complete the proof.
Remark 5.5. We were not successful in finding a basis for R 6 . However, this has no effect on our main results in this section. Since R 7 = A α,β , we already have a basis for R 7 (Prop. 4.16). Then, These selected data of the PDDA of A α,β will be useful in Subsections 5.3.1.

Poisson derivations of A α,β
We are now ready to study the Poisson derivations of A α,β . We will begin with the case where both α and β are non-zero, and subsequently look at the case where either α or β is zero.

Poisson derivations of
Throughout this subsection, we assume that α and β are non-zero. Let Der P (A) be the collection of all the Poisson K-derivations of A α,β , and D ∈ Der P (A). Now, D extends uniquely to a Poisson derivation of each of the algebras in (12) via localization. Hence, D is a Poisson derivation of the Poisson torus ]. It follows from Corollary 2.7 that D can be written as where ρ is a scalar Poisson derivation of R 3 defined as ρ(t i ) = λ i t i , i = 3, 4, 5, 6; with λ i ∈ Z P (R 3 ) = K, and ham x = {x, −} : R 3 → R 3 with x ∈ R 3 (see Corollary 2.7). We aim to describe D as a Poisson derivation of A α,β . We do this in several steps. We first describe D as a Poisson derivation of R 4 . Lemma 5.7. 1.
One can easily verify that z := t 4 t −1 5 t 6 is a Poisson central element of Q. Since R 3 is a Poisson torus, it can be presented as a free Q-module with basis (t j 3 ) j∈Z . One can therefore write We have that ham x + (z) + (λ 4 − λ 5 + λ 6 )z ∈ R 4 , hence ham x − (z) ∈ R 4 . Note that {t 3 , z} = 2zt 3 , and {γ, z} = 0 for all γ ∈ Q since z is Poisson central in Q. One can therefore express ham x − (z) as follows: We claim that for all n ∈ N >0 . Observe that hence the result is true for n = 1. Suppose that the result is true for n ≥ 1. Then, as expected. By the principle of mathematical induction, the claim is proved. Given that ℓ (n) = −m j=−1 (2j) n z n b j t j 3 , it follows that The above equality can be written as a matrix equation:  One can observe that the coefficient matrix  is similar to a Vandermonde matrix (since the terms in each column form a geometric sequence) which is well known to be invertible. This therefore implies that each b j t j 3 is a linear combination of the µ n ∈ R 4 . As a result, b j t j 3 ∈ R 4 for all j ∈ {−1, . . . , −m}. Consequently, x − = −m j=−1 b j t j 3 ∈ R 4 as desired.