Derivations of a family of quantum second Weyl algebras

In view of a well-known theorem of Dixmier, its is natural to consider primitive quotients of $U_q^+(\mathfrak{g})$ as quantum analogues of Weyl algebras. In this work, we study these primitive quotients in the $G_2$ case and compute their Lie algebra of derivations.


Introduction
Weyl algebras have been extensively studied in the last 60 years due to their link to Lie theory, differential operators, quantum mechanics, etc. One of the main questions remaining is the famous Dixmier Conjecture that asserts that every endomorphism of a complex Weyl algebra is an automorphism.
Let K be a field and q be a non-zero element of K that is not a root of unity. The aim of this article is to produce quantum analogues of the second Weyl algebra and to compare their properties to those of the second Weyl algebra. There exist in the literature various families of "quantum Weyl algebras", e.g. the so-called quantum Weyl algebras and generalised Weyl algebras (GWA for short). Most of the time, they are obtained by generators and relations through a deformation of the classical defining relation of the first Weyl algebra: xy − yx = 1.
To produce potential quantisations, we take a different approach in this article. Our inspiration comes from a Theorem of Dixmier (see, for instance, [7,Théorème 4.7.9]) that asserts that primitive quotients of enveloping algebras of complex nilpotent Lie algebras are isomorphic to Weyl algebras.
We have at hand a quantum analogue of at least some enveloping algebras of complex nilpotent Lie algebras, namely the positive part U + q (g) of a quantised enveloping algebra U + q (g) of a complex simple Lie algebra g. As a consequence, it is natural to consider primitive quotients of U + q (g) as quantum analogues of Weyl algebras. In the A 2 and B 2 cases, primitive ideals of U + q (g) have been classified and it turns out that in the B 2 case, some of the resulting primitive quotients provide 'nice' quantum analogues of the first Weyl algebra. For instance, they are simple-this is not the case of quantum Weyl algebras-and do not possess non-trivial units-this is not the case of a quantum GWA over a Laurent polynomial ring. (See [15] for details.) The present article is concerned with the G 2 case. More precisely, we identify a family of primitive ideals of U + q (G 2 ) and then proceed in proving that the corresponding primitive quotients have (at least for some choice of the parameters) properties similar to those of the second Weyl algebra. More precisely, the center of U + q (G 2 ) is a polynomial algebra in two variables K[Ω 1 , Ω 2 ] and we prove that the quotient algebra A α,β := U + q (G 2 )/ Ω 1 − α, Ω 2 − β is simple for all (α, β) = (0, 0). We then proceed and study these quotient algebras. In particular, we show that A α,β has the same (Gelfand-Kirillov) dimension as the second Weyl algebra A 2 (K). We also establish that for certain choice of the parameters α and β, the algebra A α,β is a deformation of a quadratic extension of A 2 (K) at q = 1.
In the final section, we compute the derivations of A α,β . Our results show that when α and β are both non-zero, all derivations of A α,β are inner, a property that is well known to hold in A 2 (K).
In view of the celebrated Dixmier Conjecture, it would be interesting to describe automorphisms and endomorphisms of A α,β when α and β are both non-zero. We intend to come back to these questions in the future. This article is organized as follows. In Section 2, we recall the presentation of U + q (G 2 ) as a so-called quantum nilpotent algebra (QNA for short). This allows the use of two different tools to study the prime and primitive spectra of U + q (G 2 ): the H-stratification theory of Goodearl and Letzter, and the Deleting Derivation Theory of Cauchon. We recall both theories in the context of U + q (G 2 ) in Section 2. In Section 3, we use these two theories to establish that Ω 1 − α, Ω 2 − β is a maximal ideal of U + q (G 2 ) when (α, β) = (0, 0).
In Section 4, we focus on comparing A α,β with the second Weyl algebra A 2 (K). In particular, we show that both have Gelfand-Kirillov dimension equal to 4. Through a direct computation, we also establish that A 1, 1 9(q 6 −1) is a quadratic extension of A 2 (K) at q = 1. In this section we also compute a linear basis for A α,β .
In the final section, we compute the derivations of A α,β . Our strategy here is to make use of the following tower of algebras arising from the Deleting Derivation Algorithm: The later algebra R 3 is a simple quantum torus whose derivations have been described by Osborn and Passman in [20]. We pull back their description to obtain a description of the derivations of A α,β through a step-by-step process consisting in "reverting" the Deleting Derivation Algorithm. Our results show that when α or β is equal to zero, then the first Hochschild cohomology group of A α,β is a 1-dimension vector space, whereas when both α and β are non-zero, all derivations are inner. and the monomials E k 1 1 . . . E k 6 6 (k 1 , . . . , k 6 ∈ N) form a basis of U + q (G 2 ) over K. Even better, one may write U + q (G 2 ) as a Quantum Nilpotent Algebra (QNA for short) or Cauchon-Goodearl-Letzter extension in the sense of [16,Definition 3.1], by adjoining the generators E i in lexicographic order. This means in particular that U + q (G 2 ) can be presented as an iterated Ore extension: where the σ i are automorphisms and the δ i are left σ i -derivations of the appropriate subalgebras. We would not need the precise definition of a QNA for what follows, but it is worth reminding the reader of the algebraic torus action involved in writing U + q (G 2 ) as a QNA. The algebraic torus H = (K × ) 2 acts by automorphisms on U + q (G 2 ) as follows: h · E i = h i E i for all i ∈ {1, 6} and h = (h 1 , h 6 ) ∈ H.
Note that the action of the automorphism h on the generators E 2 , . . . , E 5 follows from the above defining relations.
By [2,Theorem II.2.7], the action of H on U + q (G 2 ) is rational in the sense of [2, Definition II.2.6]. A consequence of the QNA condition is that important tools such as Cauchon's deleting derivations procedure and the Goodearl-Letzter stratification theory (this is the origin of the CGL terminology, see [16]) are available to study prime and primitive ideals. These ideas will be introduced in following sections. At the moment, we merely note that it is immediate that U + q (G 2 ) is a noetherian domain and that all prime ideals are completely prime (in the case of U + q (G 2 ), it was proved in [21,Section 5]). We denote by F q its skew-field of fractions, i.e. F q := Frac(U + q (G 2 )).

Prime ideals in U + q (G 2 ) and H-stratification
such that if J contains the product of two H-invariant ideals of U + q (G 2 ) then J contains at least one of them. We denote by H-Spec(U + q (G 2 )) the set of all H-prime ideals of U + q (G 2 ). Observe that if P is a prime ideal of U + q (G 2 ) then is an H-prime ideal of U + q (G 2 ). Indeed, let J be an H-prime ideal of U + q (G 2 ). We denote by Spec J (U + q (G 2 )) the H-stratum associated to J; that is, Spec J (U + q (G 2 )) = {P ∈ Spec(U + q (G 2 )) | (P : H) = J}.
Then the H-strata of Spec(U + q (G 2 )) form a partition of Spec(U + q (G 2 )) [2, Chapter II.2]; that is, This partition is the so-called H-stratification of Spec(U + q (G 2 )). It follows from the work of Goodearl and Letzter [9] that every H-prime ideal of U + q (G 2 ) is completely prime, so H-Spec(U + q (G 2 )) coincides with the set of H-invariant completely prime ideals of U + q (G 2 ). Moreover there are precisely |W | H-prime ideals in U + q (G 2 ), where W denotes the Weyl group of type G 2 (see [18,Remark 6.2.2]). As a consequence, the H-stratification of Spec(U + q (G 2 )) is finite and so the full strength of the H-stratification theory of Goodearl and Letzter is available to study Spec(U + q (G 2 )). For each H-prime ideal J of U + q (G 2 ), the space Spec J (U + q (G 2 )) is homeomorphic to the prime spectrum Spec(K[z ±1 1 , . . . , z ±1 d ]) of a commutative Laurent polynomial ring whose dimension depends on J, [2, Theorems II.2.13 and II.6.4]. These dimensions were computed in [1,23]. Finally, let us mention that the primitive ideals of A are precisely the prime ideals that are maximal in their H-strata [2,Theorem II.8.4].
In this article, we will mainly focus on one specific H-stratum. Since U + q (G 2 ) is a domain, 0 (technically, 0 ) is clearly an H-invariant completely prime ideal of U + q (G 2 ), and so an H-prime, and we will focus on computing its stratum, the so-called 0-stratum. The motivation here is twofold: first, in the B 2 case, we obtain "new" quantum deformation of the first Weyl algebra as U + q (B 2 )/P , where P is a primitive ideal from the 0-stratum of Spec(U + q (B 2 )) [15]. Next, in the present case, we would like to construct algebras of GK dimension 4 as explained in the introduction. Since Tauvel's height formula holds in U + q (G 2 ) [8], we need to quotient U + q (G 2 ) by a primitive ideals of height 2. Given that the H-spectrum of U + q (G 2 ) is homeomorphic to the Weyl group of type G 2 , such primitive ideals can only be found in the 0-stratum and the strata associated to one of the two height 1 H-primes. In this article, we mainly present results for the 0-stratum, but we will also indicate results obtained for the primitive quotients coming from the height 1 H-prime strata.

Deleting derivations algorithms in
is a QNA, we can apply Cauchon's Deleting Derivation Algorithm to study its prime spectrum.
Recall first that U + q (G 2 ) is an itereated Ore extension of the form: where, σ 2 denotes the automorphism of K[E 1 ] defined by: and δ 6 denotes the σ 6 -derivation of K[E 1 ] · · · [E 5 ; σ 5 , δ 5 ] defined by: The deleting derivations algorithm (DDA for short) constructs by a decreasing induction a family {E 1,j , . . . , E 6,j } of elements of the division ring of fractions F q = Fract(U + reason that Cauchon used the expression "effacement des dérivations". More precisely, let M = (µ i,j ) ∈ M 6 (K * ) be the multiplicatively antisymmetric matrix defined as follows: Then we have where

Canonical embedding
is a QNA, one can use Cauchon's DDA in order to relate the prime spectrum of A to the prime spectrum of the associated quantum affine space A. More precisely, the DDA allows the construction of embeddings Recall from [4,Section 4.3] that these embeddings are defined as follows. Let P ∈ Spec(A (j+1) ). Then where g j denotes the surjective homomorphism whose inverse is also an increasing homeomorphism. Also, ψ j induces an increasing homeomorphism from {P ∈ Spec(A (j+1) ) | E j,j+1 ∈ P } onto its image by ψ j whose inverse similarly is an increasing homeomorphism. Note however that, in general, ψ j is not an homeomorphism from Spec(A (j+1) ) onto its image. Composing these embeddings, we get an embedding which is called the canonical embedding from Spec(A) into Spec(A). The canonical embedding ψ is H-equivariant so that ϕ(H −Spec(A)) ⊆ H −Spec(A). Interestingly, the set H−Spec(A) has been described by Cauchon as follows. For any subset C of {1, . . . , 6}, let K C denote the H-prime ideal of A generated by the T i with i ∈ C, that is It follows from [4,Proposition 5.5.1] that The aim of this section is to give explicit generating sets for the primitive ideals of U + q (G 2 ) that belong to the 0-stratum. They are intimately related to the centre of U + q (G 2 ) and so we start this section by making explicit the centre of U + q (G 2 ) and related algebras.
where the parameters a, b, a ′ , b ′ , c ′ , d ′ can be found in Appendix B. Note, Ω 1 and Ω 2 are central elements of A (j) for each 2 ≤ j ≤ 7 since Fract(A (j) ) = Fract(A). We now want to show that the centre of A and other related algebras is a polynomial ring generated by Ω 1 and Ω 2 over K. The following discussions will lead us to the proof.
j+1 · · · T i 6 6 | i j , · · · , i 6 ∈ N and λ ∈ K * } for each 2 ≤ j ≤ 6. One can observe that S j is a multiplicative system of non-zero divisors of A (j) = K E i,j | for all i = 1, · · · , 6 . Furthermore, the elements T j , · · · , T 6 are all normal in A (j) . Hence, S j is an Ore set in A (j) . We can therefore localize A (j) at S j as follows: Recall that Σ j := {T n j | n ∈ N} is an Ore set in both A (j) and A (j+1) for each 2 ≤ j ≤ 6, and that For all 2 ≤ j ≤ 6, we have that: Note, R 7 := A. Again, one can also observe that T 1 is normal in R 2 . As a result, we can form the localization The algebra R 1 is the quantum torus associated to the quantum affine space A. As a result, R 1 = K q M [T ±1 1 , · · · , T ±1 6 ], where T i T j = q µ ij T j T i for all 1 ≤ i, j ≤ 6 and µ ij ∈ M. Similar to [17, §31], we construct the following tower of algebras: The aim of this paragraph is to show that Ω 1 and Ω 2 are (completely) prime. We will make use of DDA to establish these facts. Note that we could also have used the results of [11] to obtain these results. However, we will need some of the intermediate steps obtained here to compute the derivations of certain primitive quotients of U + q (G 2 ) in the final section. From Section 2.4 we know that there is a bijection between {P ∈ Spec(A (j+1) ) | P ∩ Σ j = ∅} and {Q ∈ Spec(A (j) ) | Q ∩ Σ j = ∅} via P = QΣ −1 j ∩ A (j+1) . Note, T 1 and T 2 are prime ideals of the quantum affine space A, since each of the factor algebras A/ T 1 and A/ T 2 is isomorphic to a quantum affine space of rank 5 which is well known to be a domain.
The following result and its proof show that T 1 belongs to the image Im(ψ) of the canonical embedding ψ and that Ω 1 is the completely prime ideal of A such that ψ( Ω 1 ) = T 1 .
We now proceed to prove the above claims. 1. One can easily verify that A (3) / T 1 is isomorphic to a quantum affine space of rank 5, which is a domain, hence T 1 is a prime ideal in A (3) .
Using similar techniques, one can prove that T 2 ∈ Im(ψ) and that Ω 2 is the completely prime ideal of A such that ψ( Ω 2 ) = T 2 . Again, we refer the interested reader to [19] for details. We record these facts in the following lemma.
Since H-Spec(U + q (G 2 )) is homeomorphic to the Weyl group W of type G 2 by [22], there are only 2 H-primes in U + q (G 2 ) of height 1. Since Ω 1 and Ω 2 are central, the prime ideals that they generate have height less than or equal to 1, and so equal to 1. As an immediate consequence, we get the following result.

Description of the 0-stratum and beyond
In this section, we will often assume that our base field K is algebraically closed. This assumption is actually not necessary for the main result of this section, Theorem 3.12, but makes the description of the 0-stratum easier to present. This section focuses on finding the height two maximal ideals of A = U + q (G 2 ). Note first that such ideals can only belong to the H-stratum of an H-prime of height less than or equal to 1 (since H-Spec(A) is isomorphic to W ). It follows from the previous sections that we need to compute the H-strata of 3 H-primes: 0, Ω 1 and Ω 2 . We start with the 0-stratum.
The strategy is similar to [15,Propositions 2.3 and 2.4]. Note, in this subsection, all ideals in A will simply be written as Θ , where Θ ∈ A. However, if we want to refer to an ideal in any other algebra, say R, then that ideal will be written as Θ R , where in this case, Θ ∈ R.
Proposition 3.8. Assume K is algebraically closed. Let P be the set of those unitary irreducible polynomials P (Ω 1 , Ω 2 ) ∈ K[Ω 1 , Ω 2 ] with P (Ω 1 , Ω 2 ) = Ω 1 and P (Ω 1 , Proof. We claim that Spec 0 (A) = {Q ∈ Spec(A) | Ω 1 , Ω 2 ∈ Q}. To establish this claim, let us assume that this is not the case. That is, suppose there exists Q ∈ Spec 0 (A) such that Ω 1 , Ω 2 ∈ Q; then the product Ω 1 Ω 2 which is an H−eigenvector belongs to Q. Consequently, . This confirms our claim. Since is an increasing bijection from Spec 0 (A) onto Spec(R). Since Ω 1 and Ω 2 are H−eigenvectors, and H acts on A, we have that H also acts on R. Since every H-prime ideal of A contains Ω 1 or Ω 2 , one can easily check that R is H−simple (in the sense that the only H-invariant ideal of R is the 0 ideal).
Similarly, we show that Ω 1 −α, , where m, n ∈ A. Choose (i, j) ∈ N 2 minimal (in the lexicographic order on N 2 ) such that the equality holds. Without loss of generality, suppose that i > 0 and let f : This contradicts the minimality of (i, j). Hence, (i, j) = (0, 0) and so y = m( Using similar techniques, we obtain the following description for the H-strata of Ω 1 and Ω 2 . Proposition 3.9. Assume K is algebraically closed.
Since maximal ideals in their stratum are primitive for a QNA, we obtain the following result.
Remark 3.11. The statement of the above corollary is still valid without the assumption that K is algebraically closed. The proof is actually similar as we only use this assumption to get a full description of the strata we were interested in.
We can actually prove a stronger result.
. We claim that J cannot be 0 , Ω 1 or Ω 2 . For instance, if α, β = 0, then J cannot be equal to 0 since in this case Ω 1 − α, Ω 2 − β is maximal in the 0-stratum. Moreover, J = Ω 1 as otherwise I would contain α = Ω 1 − (Ω 1 − α), a contradiction. The other cases are similar and left to the reader. This means that J is an H-prime of height at least equal to 2. As the poset of H-primes is isomorphic to W , this forces J to contain both Ω 1 and Ω 2 . Moreover, since J ⊆ I, it implies that Ω 1 , Ω 2 ∈ I. Given that Ω 1 − α, Ω 2 − β ⊂ I, we have that Ω 1 − α, Ω 2 − β ∈ I. It follows that α, β ∈ I, hence I = A, a contradiction! This confirms that Ω 1 − α, Ω 2 − β is a maximal ideal in A.
4 Simple quotients of U + q (G 2 ) and their relation to the second Weyl algebra Now that we have found primitive ideals of A = U + q (G 2 ), we are going to study the corresponding simple quotient algebras. In view of Dixmier's theorem, we consider these simple quotients as deformations of a Weyl algebra (of appropriate dimension), and so we compare their properties with some known properties of the Weyl algebras. In this section, we prove that the Gelfand-Kirillov dimension of A α,β is 4 and consequently prove that the height of the maximal ideal Ω 1 − α, Ω 2 − β is 2 as expected. Then we focus on describing a linear basis of A α,β ; we use this basis in the following section to study the derivations of A α,β . Finally, we show that with appropriate choices of α and β, the algebra A α,β is a quadratic extension of the second Weyl algebra A 2 (K) at q = 1.
Recall from Theorem 3.12 that Ω 1 − α and Ω 2 − β, where (α, β) ∈ K 2 \ {(0, 0)}, generate a maximal ideal of A. As a result, the corresponding quotient The algebra A α,β satisfies the following relations: e 2 e 4 e 6 + be 2 e 3 5 + be 3 The following additional relations of A α,β in the lemma below will be helpful when computing linear basis for A α,β . Note, we put constant coefficients of monomials in a square bracket [ ] in order to distinguish them from monomials easily. These constants are defined in Appendix B.     Proof. This is proved by brute-force computation, left to the reader.

Gelfand-Kirillov dimension of A α,β
We refer the reader to [13] for background on the Gelfand-Kirillov dimension (GKdim for short). Assume first that α, β = 0. Recall from Section 3.1 that is the quantum torus associated to the quantum affine space A = A (2) . Also, Ω 1 = T 1 T 3 T 5 and Ω 2 = T 2 T 4 T 6 in A. It follows from [4,Theorem 5.4.1] that there exists an Ore set S α,β in A α,β such that A α,β S −1 The algebra A α,β is generated by t ±1 1 , · · · , t ±1 6 subject to the following relations: , where the skew-symmetric matrix N can easily be deduced from M (by deleting the first two rows and columns) as follows: Secondly, suppose that α = 0 and β = 0.
for all 1 ≤ i, j ≤ 6 and µ ij ∈ M. We also have that ]. Finally, when α = 0 and β = 0, then one can also verify that A α, . As a result of the above discussion, in all cases, we have that A α,β S −1

Linear basis for
From Propositions 3.8 and 3.9, one can conclude that Ω 2 − β is a completely prime ideal (since it is a prime ideal) of A for all β ∈ K. Hence, the algebra A β is a noetherian domain.
We are now going to find a linear basis for A α,β , where (α, β) ∈ K 2 \ {(0, 0)}. Since A α,β is identified with A β / Ω 1 − α , we will first and foremost find a basis for A β , and then proceed to find a basis for A α,β . Note, the relations in Lemma A.1 are also valid in A β and A α,β , and are going to be very useful in this section. Proof. Since the family (Π 6 s=1 E is s ) is∈N is a PBW-basis of A over K, it follows that the family (Π 6 s=1 e s is ) is∈N is a spanning set of A β over K. We want to show that S spans A β . We do this by showing that Π 6 s=1 e s is can be written as a finite linear combination of the elements of S for all i 1 , · · · , i 6 ∈ N by an induction on i 4 . The result is obvious when i 4 = 0 or 1. For i 4 ≥ 1, assume that where v := (i, j, k, l, m) ∈ N 5 and a (ξ,v) are all scalars. Note, I is a finite subset of {0, 1} × N 5 . It follows from the commutation relations of A β (see Lemma A.1) that From the inductive hypothesis, Hence, we proceed to show that Π 6 s=1 e s is e 4 is also in the span of S. From the inductive hypothesis, we have Using the commutation relations in Lemma A.1, we have that All the terms in the above expression belong to the span of S except e 1 i e 2 j e 3 k e 4 2 e 5 l e 6 m . From (11), we have that where β 0 = −1/d ′ . Substituting (12)  Therefore, e 1 i 1 e 2 i 2 e 3 i 3 e 4 i 4 +1 e 5 i 5 e 6 i 6 can be written as a finite linear combination of the elements of S over K for all i 1 , · · · , i 6 ∈ N. By the principle of mathematical induction, S is a spanning set of A β over K.
In A β , we have that where Υ is defined as follows:  Before we continue the proof, the following point needs to be noted.
We note for future use the following immediate consequence of Proposition 4.4. Remark 4.6. Given the basis of A α,β , we have computed the group of units of A α,β , however, we do not include the details in this manuscript due to the voluminous computations involved. We only summarise our findings below. Set h 1 := e 3 e 5 + ae 4 and h 2 := (q −3 − q −9 )e 2 e 4 − (q 4 − 2q 2 + 1)/(q 4 + q 2 + 1)e 3 3 .

A α,β as a q-deformation of a quadratic extension of A 2 (K)
Recall that GKdimA α,β = 4 and so we should compare A α,β to the second Weyl algebra. In this section, we prove that, for a suitable choice of α and β, the simple algebra A α,β is a q-deformation of (a quadratic extension of) A 2 (K).
Since e 4 is invertible, one can also verify that 9e 2 e 4 , 3e 3 e 4 , e 4 , e 5 and e 6 generate A 1 . Let R be an algebra generated by f 2 , f 3 , f 4 , f 5 , f 6 subject to the following defining relations: Proof. One can easily check that we define a homomorphism φ : Recall, e 2 4 = 1/9. To check that φ is indeed a homomorphism, we just need to check its compatibility with the defining relations of R. We check this on the relation f 6 f 2 − f 2 f 6 = 1 and f 3 f 5 − f 5 f 3 = 1, and leave the remaining ones for the reader to verify. We do that as follows: Conversely, one can check that we define a homomorphism ϕ : We check this on the relation e 2 3 −3e 1 e 4 −3e 2 e 5 = 0, and leave the remaining ones for the reader to verify. We do that as follows: ϕ(e 3 ) 2 − 3ϕ(e 1 )ϕ(e 4 ) − 3ϕ(e 2 )ϕ(e 5 ) = (3f 3 To conclude we just observe that φ and ϕ are inverse of each other.
The corollary below can easily be deduced from the above proposition.
is the second Weyl algebra over the ring F.

Derivations of the simple quotients of
In this section, we compute the derivations of the algebra A α,β using DDA that allows to embed A α,β into a suitable quantum torus. Derivations of quantum tori are known, thanks to the work of Osborn and Passman [20]. In our cases, such derivations are always the sum of an inner derivation and a scalar derivation (of the quantum torus). Since A α,β can be embedded into a quantum torus, we first extend every derivation of A α,β to a derivation of such quantum torus, and then pull back their description as a derivation of the quantum torus to a description of their action on the generators of A α,β by "reverting" DDA process. We conclude that every derivation of A α,β is inner when α and β are both non-zero. However, when either α or β is zero, we conclude that every derivation of A α,β is the sum of an inner and a scalar derivation. In fact, the first Hochschild cohomology group of A α,β is of dimension 0 when α and β are both non-zero and 1 when either α or β is zero.

Preliminaries and strategy
where A (j) is defined in Section 3.1 and, Ω 1 and Ω 2 are the generators of the center of A (j) , see Remark 3.3. Note in particular that A α,β = A α,β . For each 2 ≤ j ≤ 7, denote the canonical images of the α,β by e i,j for all 1 ≤ i ≤ 6. As usual we denote by t i the canonical image of T i in A (2) α,β for each 1 ≤ i ≤ 6. For each 3 ≤ j ≤ 6, define S j := λt i j j t i j+1 j+1 · · · t i 6 6 | i j , · · · , i 6 ∈ N and λ ∈ K * . One can observe that S j is a multiplicative system of non-zero divisors (or regular elements) of A (j) α,β . Furthermore; t j , · · · , t 6 are all normal elements of A (j) α,β and so S j is an Ore set in A (j) α,β . One can localize A (j) α,β at S j as follows: As a consequence, similar to (7), we have that for all 2 ≤ j ≤ 6. By convention, R 7 := A α,β . We also construct the following tower of algebras in a manner similar to (8): ] studied in Section 4.1.
Our strategy to compute the derivations of R 7 is to extend these derivations to derivations of the quantum torus R 3 . Then we can use the description of the derivations of a quantum torus obtained by Osborn and Passman in [20]. Once this is done, we will have a "nice" description but involving elements of R 3 and we will then use the fact that these derivations fix (globally) all R i to obtain a description only involving elements of R 7 . This is a step by step process requiring knowing linear bases for R i . We find such bases in the next section.
Before doing so, we note from [4, Lemme 5.3.2] that DDA theory predicts the following relations between the elements e i,j : where, as usual, the necessary parameters can be found in Appendix B.
We also note that we have complete control over the centers of the algebras R i .
Proof. One can easily verify that Z(R 3 ) = K. Note,

Linear bases for R 3 , R 4 and R 5
Let (α, β) ∈ K 2 \ {(0, 0)}. We aim to find a basis of R j for each j = 3, 4, 5. Since R 3 = A α,β , the set {t i 3 t j 4 t k 5 t l 6 | i, j, k, l ∈ Z} is a K−basis of R 3 . For simplicity, we set Basis for R 4 . Observe that where Ω 1 = F 1 T 3 T 5 + aT 2 T 5 and Ω 2 = T 2 T 4 T 6 in A (4) . Recall from Section 4.2 that finding a basis for the algebra A β served as a good ground for finding a basis for A α,β . In a similar manner, to find a basis for R 4 , we will first and foremost find a basis for the algebra where β ∈ K * . We will denote the canonical images of E i,4 (resp. T i ) in A (4) β by e i,4 (resp. t i ) for all Note, when β = 0, then one can easily deduce that A Proof. We begin by showing that S 4 is a spanning set for A can be written as a finite linear combination of the elements of S 4 for all (k 1 , · · · , k 6 ) ∈ N 3 × Z 3 . This can easily be done through an induction on k 2 using the fact that We now prove that S 4 is a linearly independent set. Suppose that i∈I a i f 1 This implies that and b j is a family of scalars. Given that Ω 2 = T 2 T 4 T 6 , it follows from the above equality that i∈I We denote by < 2 the total order on Z 6 defined by (i 1 , Suppose that there exists (i 1 , · · · , i 6 ) ∈ J such that b (i 1 ,··· ,i 6 ) = 0. Let (w 1 , · · · , w 6 ) ∈ J be the greatest element of J with respect to < 2 such that b (w 1 ,··· ,w 6 ) = 0. Note, , we have that b (w 1 ,··· ,w 6 ) = 0. This is a contradiction to our assumption, hence b (i 1 ,··· ,i 6 ) = 0 for all (i 1 , · · · , i 6 ) ∈ J. This implies that Consequently, a i = 0 for all i ∈ I. Therefore, S 4 is a linearly independent set.
α,β S −1 4 , we have the following two relations: f 1 t 3 t 5 + at 2 t 5 = α and t 2 t 4 t 6 = β. This implies that f 1 t 3 = αt −1 5 − at 2 and t 2 = βt −1 6 t −1 4 . Putting these two relations together, we have that Note, we will usually identify R 4 with A (4) Proof. Since f 1 can be written as a finite linear combination of the elements of B 4 for all (k 1 , k 3 , · · · , k 6 ) ∈ N 2 ×Z 3 . By Proposition 5.2, it is sufficient to do this by induction on k 1 . The result is clear when k 1 = 0. Assume that the statement is true for k 1 ≥ 0. That is, Note, a i and b j are all scalars. It follows that Clearly, the monomial f i 1 +1 . 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(B 4 ) for all i 3 ∈ N and i 4 , i 5 , i 6 ∈ Z. This can easily be achieved by an induction on i 3 , and the use of the relation f 1 t 3 = αt −1 5 − aβt −1 6 t −1 4 . Therefore, by the principle of mathematical induction, B 4 is a spanning set of R 4 over K.
Basis for R 5 . We will identify R 5 with A (5) α will be denoted by e i,j (resp. t i ). We now find a basis for A (5) α S −1 5 . Recall that Ω 1 = Z 1 T 3 T 5 + aZ 2 T 5 and Ω 2 = Z 2 T 4 T 6 + bT 3 3 T 6 in A (5) (remember, Z 1 := E 1,5 and Z 2 := E 2,5 ). Since z 2 t 4 t 6 + bt 3 3 t 6 = β and z 1 t 3 t 5 + a z 2 t 5 = α in R 5 and A α S −1 5 and, in R 5 , we have the following two relations: Proposition 5.4. The set Proof. The proof is similar to that of Proposition 5.2 and so is left to the reader. Details can be found in [19].
Proof. The proof is similar to that of Proposition 5.3 and so is left to the reader. Details can be found in [19].
We note for future reference the following immediate corollary.
Remark 5.7. 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 (Proposition 4.4).

Derivations of A α,β
We are now going to study the derivations of A α,β . We will only treat the case when both α and β are non-zero, and mention results when either α or β is zero without details. Throughout this subsection, we assume that α and β are non-zero. Let Der(A α,β ) denote the K−derivations of A α,β and D ∈ Der(A α,β ). Via localization, D extends uniquely to a derivation of each of the series of algebras in (16). Therefore, D extends to a derivation of the quantum torus ]. It follows from [20,Corollary 2.3] that D can uniquely be written as: where x ∈ R 3 , and δ is a scalar derivation of R 3 defined as δ(t i ) = λ i t i for each i = 3, 4, 5, 6. Note, λ i ∈ Z(R 3 ) = K. Also, ad x is an inner derivation of R 3 defined as ad x (L) = xL − Lx for all L ∈ R 3 . We aim to describe D as a derivation of A α,β = R 7 . We do this in several steps. Before starting the process we note the following relations that will be used in this section. They all follow from [4,Lemme 5.3.2]. Then We first describe D as a derivation of R 4 .
We proceed to describe D as a derivation of R 5 . Lemma 5.10. 1.
3. if α = 0 and β = 0, then every derivation D of A 0,β can uniquely be written as D = ad x + λθ, where λ ∈ K and x ∈ A 0,β . The above theorem shows that A α,β when both α and β are nonzero shares a number of properties with the second Weyl algebra over K: it is simple, units are reduced to scalars, and all derivations are inner.
It would be interesting to compute the automorphism group of these algebras and verify if all endomorphisms are automorphisms, i.e. an analogue of the celebrated Dixmier Conjecture [6].
In general, the present work and [15] suggest that the primitive quotients of U + q (g) by primitive ideals from the 0-stratum provide algebras that could (should?) be regarded (and studied) as quantum analogue of Weyl algebras.