The Prime Spectrum and Simple Modules Over the Quantum Spatial Ageing Algebra

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The quantum plane and the quantized enveloping algebra U q (sl 2 ) are important examples of generalized Weyl algebras and ambiskew polynomial rings, see e.g., [6] and [31]. Let U 0 q (sl 2 ) be the 'positive part' of U q (sl 2 ). It is the subalgebra of U q (sl 2 ) generated by K ±1 and E. There is a Hopf algebra structure on U 0 q (sl 2 ) defined by The notion of smash product has proved to be very useful in studying Hopf algebra actions [40]. For example, the enveloping algebra of a semi-direct product of Lie algebras can naturally be seen as a smash product algebra. The smash product is constructed from a module algebra, see [40, 4.1] for details and examples. We can make the quantum plane a U 0 q (sl 2 )-module algebra by defining and introduce the smash product algebra A := K q [X, Y ] U 0 q (sl 2 ). We call this algebra the quantum spatial ageing algebra. The defining relations for the algebra A are given explicitly in the following definition.

Definition
The quantum spatial ageing algebra A = K q [X, Y ] U 0 q (sl 2 ) is an algebra generated over K by the elements E, K, K −1 , X and Y (where K −1 is the inverse of K) subject to the defining relations: The algebra A can be seen as the quantum analogue of the enveloping algebra U(a) of the 4-dimensional (non-semisimple) Lie algebra a with basis {h, e, x, y} and Lie brackets: The, so-called, 1-spatial ageing algebra, which is studied in [38] where all the simple weight modules are classified, is a central extension of the Lie algebra a. Our aim is to study the structure of the algebra A and its representation theory. For an infinite dimensional non-commutative algebra, to classify its simple modules is a very difficult problem, in general. The known examples of algebras for which the simple modules were classified are mainly generalized Weyl algebras, see e.g. [4, 6-9, 11, 12, 14] and Ore extensions with Dedekind ring as coefficient ring, [11]. In the paper, we classify all simple unfaithful Amodules and the simple weight modules. Weight modules for certain (quantum) algebras of small Gelfand-Kirillov dimension are treated in [17-20, 45, 46].
The paper is organized as follows. In Section 2, we describe the partially ordered sets of the prime, maximal and primitive ideals of the algebra A. Using this description the prime factor algebras of A are given explicitly via generators and relations (Theorem 2.9). There are nine types of prime factor algebras of A. For two of them, 'additional' nonobvious units appear under factorization at prime ideals. It is proved that every prime ideal of A is completely prime (Corollary 2.13). In Section 3, the automorphism group of A is determined, which turns out to be a 'small' non-commutative group that contains an infinite discrete subgroup (Theorem 3.1). The orbits of the prime spectrum under the action of the automorphism group are described. In Section 4, we classify all simple unfaithful Amodules. This is the first instance where such a large class of simple modules is classified for a quantum algebra of Gelfand-Kirillov dimension larger than 2. In Section 5, we find the centralizer C A (K) of the element K in the algebra A and give a classification of all simple weight A-modules.

Prime Spectrum of the Algebra A
The aim of this section is to describe the prime, maximal and primitive spectrum of the algebra A (Theorem 2.9, Corollary 2.10 and Proposition 2.12). Every prime ideal of A is completely prime (Corollary 2.13). For all prime ideals P of A, the factor algebras A/P are given by generators and defining relations (Theorem 2.9).
Generalized Weyl Algebra Definition, [3,4,6]. Let D be a ring, σ be an automorphism of D and a is an element of the centre of D. The generalized Weyl algebra A := D(σ, a) := D[X, Y ; σ, a] is a ring generated by D, X and Y subject to the defining relations: The algebra A = ⊕ n∈Z A n is Z-graded where A n = Dv n , v n = X n for n > 0, v n = Y −n for n < 0 and v 0 = 1. It follows from the above relations that v n v m = (n, m)v n+m = v n+m n, m for some (n, m) and n, m ∈ D. If n > 0 and m > 0 then in other cases (n, m) = 1. Clearly, n, m = σ −n−m ((n, m)). Definition, [5]. Let D be an ring and σ be its automorphism. Suppose that elements b and ρ belong to the centre of the ring D, ρ is invertible and σ (ρ) = ρ. Then E := D σ ; b, ρ := D[X, Y ; σ, b, ρ] is a ring generated by D, X and Y subject to the defining relations: If D is commutative domain, ρ = 1 and b = u − σ (u) for some u ∈ D (resp., if D is a commutative finitely generated domain over a field K and ρ ∈ K * ) the algebras E were considered in [29] (resp., [30]).
The ring E is the iterated skew polynomial ring An element d of a ring D is normal if dD = Dd. The next proposition shows that the rings E are GWAs and under a certain (mild) conditions they have a 'canonical' normal element.

[5, Corollary 1.4] If one of the equivalent conditions of statement 2 holds then the ring
The Algebra E is a GWA Let E be the subalgebra of A generated by the elements X, E and Y . The generators of the algebra E satisfy the defining relations is a GWA where σ (X) = qX and σ (ϕ) = q −1 ϕ. So, the algebra is a skew Laurent polynomial algebra where By the very definition, the algebra A is a Noetherian domain of Gelfand-Kirillov dimension GK (A) = 4. The algebra E is isomorphic to the algebra U + q (sl 3 ) and the algebra A is a subalgebra of U 0 q (sl 3 ) (generated by E 1 , E 2 and K ±1 1 , see [15, Section I.6] for the notation).
Let x 1 , x 2 , x 3 be a permutation of the elements X, Y, E.
The standard filtration associated with the canonical generators of the algebra A is equal to {F n } n 0 where F n = |i|+|α| n KK i x α where |α| = α 1 + α 2 + α 3 . Therefore, the Gelfand-Kirillov dimension of the algebra A is equal to GK (A) = 4.

Lemma 2.2
The following identities hold in the algebra A.
Proof Both equalities can be proved by induction on i and using the relation For a left denominator set S of a ring R, we denote by S −1 R = {s −1 r | s ∈ S, r ∈ R} the left localization of the ring R at S. If the left denominator set S is generated by elements X 1 , . . . , X n , we also use the notation R X 1 ,...,X n to denote the ring S −1 R. If M is a left R-module then the localization S −1 M is also denoted by M X 1 ,...,X n . By Lemma 2.2, the set S Y := {Y i | i 0} is a left and right Ore set in the algebra A. Note that the algebra U 0 q (sl 2 ) is the localization of its subalgebra, the quantum plane K K, E |KE = q 2 EK , at the powers of the element K (the quantum plane is generated by elements K and E subject to the defining relation KE = q 2 EK). Let A Y be the localization of A at the powers of Y . Recall that ϕ = EY − qY E = X + (q −1 − q)Y E, we have So, the element ϕ is a normal element of the algebra A. Recall that an element x of an algebra A is called a normal element if xA = Ax. Then of A (Theorem 2.9). In fact, explicit sets of generators and defining relations are found for all prime factor algebras of A (Theorem 2.9). Furthermore, all these algebras are domains, i.e., all prime ideals of A are completely prime (Corollary 2.13). For an element a of an algebra A, we denote by (a) the (two-sided) ideal it generates.

The Algebra A/(X)
The element X is a normal element in the algebras E and A. By Eq.
3, the factor algebra is a GWA.
is isomorphic to the quantum plane. It is a Noetherian domain of Gelfand-Kirillov dimension 2. Now, the factor algebra is a skew Laurent polynomial algebra where τ (E) = q 2 E and τ (Y ) = q −1 Y. It is a Noetherian domain of Gelfand-Kirillov dimension 3. The element of the algebra A/(X), Z := ϕY K −1 = (1 − q 2 )EY 2 K −1 , belongs to the centre of the algebra A/(X). By Eq. 8, the localization of the algebra A/(X) at the powers of the central element Z, is the tensor product of algebras where the algebra

The Algebra A/(ϕ)
The element ϕ is a normal element in the algebras E and A. By Eq.

3, the factor algebra
is isomorphic to the quantum plane. It is a Noetherian domain of Gelfand-Kirillov dimension 2. Now, the factor algebra is a skew Laurent polynomial algebra where τ (E) = q 2 E and τ (Y ) = q −1 Y . The algebra A/(ϕ) is a Noetherian domain of Gelfand-Kirillov dimension 3. The element C := XY K ∈ A/(ϕ) belongs to the centre of the algebra A/(ϕ).
The localization E X,Y of the algebra E at the Ore set is a skew Laurent polynomial algebra. Then the localization A X,Y of the algebra A at the Ore set S is equal to A X,Y = E X,Y [K ±1 ; τ ]. By Eqs. 14 and 15, the localization of the algebra A/(ϕ) at the powers of the element C, is a tensor product of algebras where Y is the central simple algebra as in Eq. 12. Hence, the centre of the algebra A/(ϕ) Let R be a ring. Then each element r ∈ R determines two maps from R to R, r· : x → rx and ·r : x → xr where x ∈ R. Recall that for an element r ∈ R, we denote by (r) the ideal of R generated by the element r. Proposition 2.7 Let R be a Noetherian ring and s be an element of R such that S s := {s i | i ∈ N} is a left denominator set of the ring R and (s i ) = (s) i for all i 1 (e.g., s is a normal element such that ker (·s R ) ⊆ ker (s R ·)). Then Spec Proof Clearly, Spec (R) = A 1 A 0 is a disjoint union where A 1 and A 0 are the subsets of Spec (R) that consist of prime ideals p of R such that p∩S s = ∅ and p∩S s = ∅, respectively. If p ∈ A 1 then s i ∈ p for some i 1, and so p ⊇ (s i ) = (s) i , by the assumption. Therefore, p ⊇ (s) (since p is a prime ideal), i.e., p s. This means that A 1 = Spec (R, s). We have shown that s ∈ p iff s i ∈ p for some i 1. By the very definition, s p, is a bijection with the inverse q → σ −1 (q) and the statement (b) follows. The statement (c) is obvious.
In the proof of Theorem 2.9 the following very useful lemma is used repeatedly.

Lemma 2.8 Let A be a ring, S be a left denominator set of
The Prime Spectrum of the Algebra A The key idea in finding the prime spectrum of the algebra A is to use Proposition 2.7 repeatedly and the following diagram of algebra homomorphisms (18) that explains the choice of elements at which we localize. Using Eq. 18 and Proposition 2.7, we represent the spectrum Spec (A) as the disjoint union of the following subsets where we identify the sets of prime ideals via the bijections given in the statements (a) and (b) of Proposition 2.7: A prime ideal p of an algebra A is called a completely prime ideal if A/p is a domain. We denote by Spec c (A) the set of completely prime ideals of A, it is called the completely prime spectrum of A. The theorem below gives an explicit description of the prime ideals of the algebra A together with inclusions of prime ideals. (19). In the diagram (20), all the inclusions of prime ideals are given (lines represent inclusions of primes). More precisely,

Theorem 2.9 The prime spectrum Spec (A) of the algebra A is the disjoint union of sets
Proof As it was already mentioned above, we identify the sets of prime ideals via the bijection given in the statements (a) and (b) of Proposition 2.7. Recall that the set S X = {X i | i ∈ N} is a left and right denominator set of A and A X : The element X is a normal element of A. By Proposition 2.7, and none of the ideals of the set Spec (A, X) is contained in an ideal of the set Spec (A X ). Similarly, the element ϕ is a normal element of A X and, by Proposition 2.7 By Lemma 2.4, the algebra A X,ϕ is a simple domain. Hence, Spec (A X,ϕ ) = {0}, and statement 6 is proved.
The second isomorphism holds, by Eq. 17. Using the we see that the elements Y and E are invertible in the algebra A X (ϕ) X , and so the first isomorphism holds.
, the non-zero element C = XY K ∈ L r is invertible in the field L r . Hence, the elements X and Y are invertible in the algebra A/(ϕ, r). Hence, Now, the statement (ii) follows from Eq. 23 and the statement (i).
(iii) Statement 5 holds: Recall that the algebra Y is a central simple algebra. By the statement (i), the set Spec Since Y is a central simple algebra, the statement (b) of statement 5 follows. The statement (a) of statement 5 is obvious (see Eq. 15). Hence, (ϕ) = A ∩ (ϕ) X , by Lemma 2.8. So, statement 5 holds.
By Proposition 2.7, and none of the ideals of the set Spec (A/(X, Z)) is contained in an ideal of the set Spec ((A/(X)) Z ).
, the non-zero element Z = ϕY K −1 ∈ L q is invertible in the field L q . Hence, the elements ϕ and Y are invertible in the algebra A/(X, q). Therefore, Now, by Eq. 12, the statement (iv) holds.
(v) Statement 4 holds: The algebra Y is a central simple algebra. By Eq. 12, the set The last equality follows from the statement (iv) (the algebra A/(X, q) is simple and . Now, the statement (b) of statement 4 holds. The statement (a) is obvious, see Eq. 11. Hence, (X) = A ∩ (X) Y,ϕ , by Lemma 2.8. So, the proof of the statement (v) is complete.
In the algebra A, using the equality ϕ = X + (q −1 − q)Y E we see that The first equality follows from the relation X = EY − q −1 Y E. Then the second equality follows from Eq. 26.
The first equality follows from the relation X = EY − q −1 Y E. Then the second equality follows from the definition of the element ϕ = X + (q −1 − q)Y E.
(viii) The elements Y and E are normal in A/(X): The statement follows from Eqs. 10 and 11.

11,
By the statement (viii) and Proposition 2.7, By the statement (vi), ; τ ] is a domain, the result follows from Lemma 2.8. So, statement 3 holds.
The element E is a normal element of the algebra U . By Proposition 2.7, Since (vi)) is a domain, the set Spec (U E ), as a subset of Spec (A), consists of a single ideal (Y ), and statement 2 follows.
We proved that Eq. 19 holds. Clearly, we have the inclusions as on the diagram (20). It remains to show that there is no other inclusions. The ideals (Y, E, p), (X, q) and (ϕ, r) are the maximal ideals of the algebra A (see statement 1, 4, and 5). By Eq. 25 and the relations given in the diagram (20), there are no additional lines leading to the maximal ideals (X, q). Similarly, by Eq. 23 and the relations given in the diagram (20), there are no additional lines leading to the maximal ideals (ϕ, r). The elements X and ϕ are normal elements of the algebra A such that (X) ⊆ (ϕ) and (X) ⊇ (ϕ), by Eq. 4. The proof of the theorem is complete.
For an algebra A, Max(A) is the set of its maximal ideals. The next corollary is an explicit description of the set Max(A).

Corollary 2.10 Max
Proof The corollary follows from the diagram (20). ( Proof See Theorem 2.9.

The Dixmier-Moeglin Equivalence
The Dixmier-Moeglin equivalence states that for prime ideals the following properties are equivalent: locally closed ⇐⇒ primitive ⇐⇒ rational.
The reader is referred to the book of Brown and Goodearl [15] for the details.
Lemma 2.14 The algebra A satisfies the Nullstellensatz.
Proof Using similar arguments as in [42, 4.5-4.9] we can prove that the algebra A satisfies the Nullstellensatz.
By the diagram (20) and Lemma 2.15, all ideals but (X), (ϕ) and (Y, E) are locally closed, hence, rational (by Eq. 29). The ideals (X), (ϕ) and (Y, E) are not rational as the centre of each factor algebra A/(X), A/(ϕ) and A/(Y, E) contains a transcendental element (Z, C and K, respectively). Therefore, every rational prime ideal is locally closed, i.e., the algebra A satisfies the Dixmier-Moeglin equivalence.

The Automorphism Group of A
In this section, the group G := Aut K (A) of automorphisms of the algebra A is found (Theorem 3.1). Corollary 3.2 describes the orbits of the action of the group G on Spec (A) and the set of fixed points. We introduce a degree filtration on the algebra A by setting deg(K) = deg( (where A[−1] := 0) be the associated graded algebra of A with respect to the filtration {A[n]} n 0 . For an element a ∈ A, we denote by gr a ∈ gr A the image of a in gr A. It is clear that gr A is an iterated Ore extension, gr In particular, gr A is a Noetherian domain of Gelfand-Kirillov dimension GK (gr A) = 4 and the elements X, Y and E are normal in gr A.
The group of units A * of the algebra A is equal to The next theorem is an explicit description of the group G.
Proof Using the defining relations of the algebra A, one can verify that σ λ,μ,γ ,i ∈ G for all λ, μ, γ ∈ K * and i ∈ Z. The subgroup G generated by these automorphisms is isomorphic to the semi-direct product (K * ) 3 Z. It remains to show that G = G . Recall that the elements X and ϕ are normal in the algebra A. Let σ ∈ G, we have to show that σ ∈ G . By the diagram (20), there are two options either the ideals (X) and (ϕ) are σ -invariant or, otherwise, they are interchanged. In more details, either, for some elements λ, λ ∈ K * and i, j ∈ Z, (a) σ (X) = λK i X and σ (ϕ) = λ K j ϕ, or, otherwise, (b) σ (X) = λK i ϕ and σ (ϕ) = λ K j X.
We will show that the case (b) is not possible (see the statement (v) below) and then the group G will be found. Arguments are given in steps (i)-(viii).
(i) σ (K) = γ K for some γ ∈ K * : The group of units A * of the algebra A is equal to {γ K s | γ ∈ K * , s ∈ Z}. So, either σ (K) = γ K or, otherwise, σ (K) = γ K −1 for some γ ∈ K * . Let us show that the second case is not possible. Notice that KX = qXK and Kϕ = qϕK, i.e., the elements X and ϕ have the same commutation relation with the element K. Because of that it suffices to consider one of the cases (a) or (b) since then the other case can be treated similarly. Suppose that the case (a) holds and that σ (K) = γ K −1 . Then the equality σ (K)σ (X) = qσ (X)σ (K) yields the equality γ K −1 · λK i X = qλK i X · γ K −1 = qλγ qK i−1 X. Hence, q 2 = 1, a contradiction.

By applying σ to this equality we obtain the equality σ (Y )σ (E)
(iv) i = j (see the cases (a) and (b)): The elements X and ϕ commute, hence σ (X)σ (ϕ) = σ (ϕ)σ (X). Substituting the values of σ (X) and σ (ϕ) into this equality yields q −i = q −j in both cases (a) and (b), i.e., i = j (since q is not a root of unity).

Classification of Simple Unfaithful A-Modules
The aim of this section is to classify all simple A-modules with non-zero annihilators. The set of simple A-modules are partitioned according to their annihilators which were described in Proposition 2.12. For each primitive ideal p of A, the factor algebra A/p is described by Theorem 2.9. They are of the type  Given elements α, β ∈ D \ {0}, we write α < β if there are no maximal ideas p and q that belong to the same infinite orbit and such that α ∈ p, β ∈ q and p q. In particular, if α ∈ D * is a unit of D then α < β for all β ∈ D \ {0}. The relation < is not a partial order on D \ {0} as 1 < 1. Clearly, α < β iff σ j (α) < σ j (β) for some/all j ∈ Z.

Classification of Simple R-modules
Definition An element b = t m β m + t m+1 β m+1 + · · · + t n β n ∈ R where β i ∈ D, m < n and β m , β n = 0 is called an l-normal element if β n < β m .
Definition We say that the automorphism σ of D is of (I)-type automorphism, if all Gorbits in Max(D) are infinite. We say that the automorphism σ of D is of (F)-type if there is a maximal ideal p ∈ Max(D) which is σ -invariant (i.e., σ (p) = p), {p} is the only finite orbit and the automorphismσ : The next theorem is about classification of simple, D-torsionfree R-modules.

Theorem 4.2 1. [4, Theorem 5] Suppose that σ is of (I)-type. Then
The simple R-modules R/R ∩ Bb and R/R ∩ Bb are isomorphic iff the B-modules B/Bb and B/Bb are isomorphic.

[9, Theorem A] Suppose that σ is of (F)-type. Then
The simple R-modules R/R ∩ Bb and R/R ∩ Bb are isomorphic iff the B-modules B/Bb and B/Bb are isomorphic.
where τ (Y ) = q −1 Y and so the automorphism τ is of (I)-type.
LetÂ (fin. dim.) be the set of isomorphism classes of simple finite dimensional A-modules.

Theorem 4.7
Proof A finite dimensional module over an infinite dimensional algebra has necessarily nonzero annihilator. Among the simple modules with nonzero annihilators, the modules described in Eq. 31 are the only finite dimensional ones.
For a module M, GK (M) is its Gelfand-Kirillov dimension.

Corollary 4.8 The Gelfand-Kirillov dimension of all simple, unfaithful, infinite dimensional A-modules is 1.
Proof The statement follows from the above classification of simple, unfaithful, infinite dimensional A-modules. The algebra A admits the inner automorphism ω K : a → KaK −1 . Each of the canonical generators K ±1 , E, X and Y of the algebra A are eigenvectors of the automorphism ω K with eigenvalues, 1, q 2 , q and q −1 , respectively. So, the algebra A = i∈Z A i is a Zgraded algebra where A i := {a ∈ A | ω K (a) = q i a} = 0 and A 0 is the centralizer

Classification of Simple Weight/K[K ±1 ]-Torsion
where  Proof Notice that the quantum plane (a tensor product of vector spaces),

The setÂ (D-torsion, 0)
where σ (K) = qK, σ (t) = q −1 t and σ (u) = qu. Clearly, where A 0,t,u = (A 0 ) t,u is the localization of A 0 at the powers of the elements t and u. The inner automorphism ω Y : a → Y aY −1 of the algebra A Y,X,ϕ preserves the subalgebras A 0 and A 0,t,u since So, ω Y ∈ Aut K (A 0 ) and ω Y ∈ Aut K (A 0,t,u ). Then the factor algebra A 0 /(p) D/p ⊗ = D/p t, u | tu = q 2 ut is the quantum plane over the field D/p. Let D/p ⊗ (∞-dim.) be the set of isomorphism classes of simple infinite dimensional D/p ⊗ -modules. A classification of all simple -modules is given in [8], [9] and [10]. In particular, the map One can verify this directly using the defining relations of the algebra A or, alternatively, this will be proved in the proof of Theorem 5.5. This construction gives all the elements of the setÂ(D-torsion, 0) (Theorem 5.5).

Lemma 5.4 Let M be a simple weight
is a bijection with the inverse [N ] → [soc A (N )] where the condition soc A = 0 means that an A Y,t,u -module has non-zero socle as an A-module. Since q is not a root of 1, the centralizer of the element K in A Y,t,u = (D ⊗ t,u )[Y ±1 ; σ ] is D ⊗ t,u . Therefore, every module V ∈ A Y,t,u (D-torsion, soc A = 0) is of the type for some N ∈ D/p ⊗ t,u where p ∈ Max(D). In more detail, the D-module N is a direct sum of copies of D/p. For each i ∈ Z, the left D-module Y i ⊗ N is a direct sum of copies D/σ i (p) (since σ i (p)(Y i ⊗ N ) = Y i p ⊗ N = Y i ⊗ pN = 0). The ideals {σ i (p) | i ∈ Z} are distinct (as q is not a root of 1). Therefore, the D-modules Y i ⊗ N and Y j ⊗ N are not isomorphic for all i = j . Any nonzero submodule M of the Amodule A Y,t,u ⊗ D⊗ t,u N = i∈Z Y i ⊗ N is necessarily of the form i∈I L i for some I ⊆ Z where for each i ∈ I, L i is a nonzero A 0,t,u -submodule of the simple A 0,t,umodule Y i ⊗ N . Therefore, L i = Y i ⊗ N for all i ∈ I. Then necessarily I = Z, i.e., M = A Y,t,u ⊗ D⊗ t,u N . The action of the elements K, E, X and Y are given by Eq. 38. In more detail, notice that since EY = ϕ + qY E = ϕ + q 2 (EY − X) and the equality (40) follows. Then Notice that Supp(V ) = O(p) as V σ i (p) = Y i ⊗ N = 0 for all i ∈ Z. By Eq. 37, we may assume that N = N t,u where N ∈ D/p ⊗ (∞-dim.). Then by Eq. 38, the direct sum