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Locally Finite Representations Over Noetherian Hopf algebras

We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products and affine Hopf algebras of low Gelfand-Kirillov dimension.


Introduction
This paper is concerned with the question under which conditions the category A -Mod loc of locally finite representations over a Noetherian K-algebra A is closed under taking injective hulls in A -Mod 1 , in which case we will say that A -Mod loc is essentially closed. An example of a locally finite left Amodule is its finite dual A • , which is known to be a coalgebra. Furthermore, any coalgebra is an injective object in the category of its comodules (see [9, 3.21]). Hence the coalgebra A • is an injective object in Comod-A • = A -Mod loc and so a necessary condition for A -Mod loc to be essentially closed is that the finite dual A • is an injective A-module. We will see, that the injectivity of A • as left A-module is also a sufficient condition, in case A is Noetherian.
Our study was motivated by various results in the literature: Let G be a polycyclic-by-finite group and A = K[G] its group ring over a field K. Then A -Mod loc is essentially closed (see [18,39]; S. Donkin attributes this result to K. Brown in [18]). Actually the interest in this question dates back to works by P. Hall and J. E. Roseblade from the 1960s and 1970s on finitely generated soluble groups (see [27,42]). Earlier, in [33,Proposition 3], E. Matlis had already shown, by using the Artin-Rees property, that over a commutative Noetherian ring, the injective hull of a simple module is Artinian. For a finite dimensional Lie algebra g over field K with enveloping algebra A = U (g), A -Mod loc is essentially closed if and only if g is solvable or char(K) > 0 (see [21]). The sufficiency for solvable Lie algebras in characteristic zero had been proven by S. Donkin and R. Dahlberg [14,19], while the necessity has been shown by J. Feldvoss. An old result of N. Jacobson [30,Proposition 2] states that U (g) is always a finite module over its center for a finite dimensional Lie algebra g in positive characteristic. Hence U (g) is a Noetherian PI-algebra for which locally finite representations are essentially closed (see Theorem 2.7 below).
There are instances when A -Mod loc is essentially closed due to "trivial" reasons. One extreme is when all representations over A are locally finite, which is precisely the case when A is a finite dimensional algebra. The other extreme is when there are no non-zero finite dimensional representations. This is the case, for example, for any infinite dimensional simple K-algebra A, like the Weyl algebras. In case A is a (not necessarily commutative) Noetherian K-algebra such that all irreducible representations over A are finite dimensional, then A -Mod loc is essentially closed if and only if every finitely generated essential extension of an irreducible representation is Artinian. The latter condition has been termed condition (⋄) in a series of papers [4, 10-12, 28, 40]. One of the main results by K. Brown et al. in [4] is that for an affine commutative Noetherian K-algebra R and automorphism α, the skew polynomial ring A = R[θ; α] satisfies (⋄) if and only if all irreducible left A-modules are finite dimensional. Hence in this case (⋄) implies A -Mod loc to be essentially closed. However, in general, these two conditions are independent.
For any Weyl algebra A, A -Mod loc is essentially closed but the second Weyl algebra over the complex numbers does not satisfy (⋄) by an old result of T. Stafford. On the other hand, the enveloping algebra A = U (sl 2 ) satisfies (⋄) by [15] but A -Mod loc is not essentially closed as we show below.
The structure of this paper is as follows: in Section 1, we will identify the category of locally finite representations of an algebra A with the category of right comodules over the finite dual A • . The first main result of Section 2 is Theorem 2.1, which characterizes Noetherian algebras A, whose locally finite representations are closed under essential extensions by reducing our problem to finite dimensional irreducible representations and by showing that this is equivalent to the injectivity of A • : Theorem 0.1. The following statements are equivalent for a Noetherian K-algebra A.
(a) A -Mod loc is essentially closed.
(b) The injective hull of any finite dimensional irreducible representation over A is locally finite. Section 2 contains also sufficient conditions to test whether A -Mod loc is essentially closed, with Theorem 2.7 being the main result of the section.
Theorem 0.2. The category of locally finite representations of a Noetherian K-algebra A is essentially closed if every maximal ideal of finite codimension contains an ideal Q that satisfies one of the following conditions: (1) Q has the Artin-Rees property and A/Q is an affine PI-algebra.
(2) Q is polynormal and A/Q -Mod loc is essentially closed.
This allows us to conclude in Section 3 that locally finite representations are essentially closed over any (unmixed) Ore extensions R[x; σ] and R[x; δ] of a commutative Noetherian algebra R (see 3.2). One of the additional tools here is the Rees ring of an ideal which we examine for certain maximal ideals of an Ore extension. In the final section, we turn to Noetherian Hopf algebras and iterated Hopf Ore extensions and find in Theorem 4.2 conditions under which our problem can be decided by just looking at essential extensions of the trivial representation. Throughout the paper K denotes an algebraically closed field of characteristic 0 and algebras are considered to be K-algebras. The paper finishes with a look at Hopf crossed products and affine Hopf algebras of finite Gelfand-Kirillov dimension, using the classification result of Brown, Goodearl, Zhang and others (see [5,8,26,45,46]). In particular, we prove that locally finite representations of an algebra A are closed under essential extensions in any of the following cases with R commutative affine, σ an automorphism and δ a derivation.
(3) A = R# σ H is an affine Noetherian Hopf algebra with bijective antipode and a crossed product of a Hopf subalgebra R and commutative Hopf algebra H, such that R + satisfies the strong Artin-Rees property. (4) A is an affine Hopf algebra domain of Gelfand-Kirillov dimension 2 with Ext 1 H (K, K) = 0.

Locally finite representations of an algebra
Let A be a K-algebra, and A * = Hom K (A, K) be the space of linear forms on A. Denote the category of left A-modules by A -Mod. By a locally finite (left) representation of A we mean a left A-module M such that Am is a finite dimensional subspace of M for any m ∈ M . We will denote the subcategory of locally finite representations of A by A -Mod loc . A particular example of a locally finite representation is the finite dual of A, which is the coalgebra . The functions f i exist since f factors through the finite dimensional algebra A/I and hence the composition f • µ of the multiplication µ of A/I with f is a linear form from A/I ⊗ A/I to K and hence belongs to We will denote the category of right A • -comodules by Comod-A • . Any right A • -comodule (M, ρ M ) becomes also a left A-module with the action given by Km i is finite dimensional and hence any right A • -comodule is a locally finite representation of A. Conversely, for any locally finite representation M ∈ A -Mod loc and element m ∈ M , one can choose a K-basis (m 1 , . . . , m n ) of Am and elements f 1 , . . . , f n ∈ A * such that a · m = n i=1 f i (a)m i for all a ∈ A. Since Ann A (Am) is a cofinite ideal of A and contained in the kernel of each f i , these linear forms are actually elements of the [6,9,17,38]). In general, the category of right comodules over a coalgebra C can be identified with the subcategory σ[ C * C] of C * -Mod, which consists of those C * -modules that are isomorphic to submodules of factor modules of direct sums of copies of C (see [9, 4.3]). Here C * = Hom K (C, K) is a K-algebra with the convolution product and acts on C by f · c = (c) c 1 f (c 2 ), for all c ∈ C and f ∈ C * . Therefore, the category of locally finite representations over A is identified as where the last equality results from the inclusion of A as a subalgebra of (A • ) • . Note that categories of type σ[ R M ] (termed Wisbauer categories), for a left R-module M over a ring R, are known to be Grothendieck categories and hence have enough injective modules. However, in general the injective hulls in σ[ R M ] are properly contained in the injective hulls in R -Mod.
Note that essential extensions of locally finite modules may not be locally finite over an Ore extension of a not necessarily commutative Noetherian ring. Neither need locally finite representations be closed under essential extensions over Ore extension of a Noetherian K-algebra R where R -Mod loc is essentially closed. The following applies in particular to the enveloping algebra of semisimple Lie algebras. Lemma 1.1. Let A be an augmented K-algebra that is a domain, such that all finite dimensional representations of A are completely reducible. If A -Mod loc is essentially closed, then A = K.
Proof. Let α : A → K be any algebra homomorphism and consider α K, a simple representation of A via α. If A -Mod loc is essentially closed, then the injective hull E(K) of K is locally finite. Since by hypothesis any finitely generated submodule of E(K) is completely reducible, we have E(K) = K and hence K is an injective left A-module. As an injective A-module, K is divisible, i.e. for any non-zero a ∈ A, one has a · K = K. Since for any nonzero element a ∈ Ker(α), a · K = 0 holds, we must have Ker(α) = 0, i.e. A = K.
By Weyl's Theorem, finite dimensional left U (g)-modules are completely reducible in case char(K) = 0. Moreover U (g) is a Noetherian domain with algebra homomorphism ǫ : U (g) → K given by ǫ(h) = ǫ(e) = ǫ(f ) = 0. Although finitely generated essential extensions of irreducible U (g)-modules are Artinian, they do not need to be finite dimensional since there are non-split extensions of finite dimensional and infinite dimensional irreducible representations: let I = U (g)e 2 +U (g)f e+U (g)(h+2) and J = U (g)e+U (g)(h+2) be left ideals of U (g). We leave it to the reader to verify that is a non-split short exact sequence, with J/I ≃ K being the trivial representation and U (g)/J being an infinite dimensional representation of U (g).
The last example shows that for a Noetherian algebra A, such that A -Mod loc is essentially closed then Ext 1 A (V, W ) = 0 for all finite dimensional representations W and infinite dimensional representations V of A.

When locally finite representations are essentially closed
As in the previous section, let us denote by A • the finite dual of a K-algebra A. Theorem 2.1. The following statements are equivalent for a Noetherian K-algebra A.
(a) A -Mod loc is essentially closed.
Let V be a locally finite left A-module and let E be an essential extension of V . We first show that V can be assumed to be finite dimensional. Note that E is locally finite if every cyclic submodule of it is finite dimensional. If x is a nonzero element of E, then Ax is an essential extension of V ∩ Ax. Since A is Noetherian, V ∩ Ax is a finitely generated submodule of V and hence it is finite dimensional. Thus every cyclic submodule of E is an essential extension of a finite dimensional submodule of V , and hence it suffices to consider the case in which V is finite dimensional. Now assume that V is finite dimensional and let S be its (finite dimensional) socle. Since V is Artinian, it has an essential socle and any essential extension E of V is also an essential extension of S. Thus E can be embedded into the injective hull of S, which is by hypothesis locally finite.
(a) ⇔ (d) Following the explanation in the introduction, the category of locally finite representations A -Mod loc is equal to the category of right A • -comodules. By [9, 3.21 and therefore also locally finite.
(a) ⇔ (c). Let E be any injective left A-module and E(Loc(E)) the injective hull of Loc(E) in A -Mod. By hypothesis E(Loc(E)) is also locally finite and belongs to A -Mod loc . Since F (E) = Loc(E) is injective in A -Mod loc , as seen above, and essential in E(Loc(E)), these two modules must coincide. Hence Loc(E) = E(Loc(E)) is injective in A -Mod and splits off in E as a direct summand. Conversely, let M be a locally finite left A-module with injective hull E in A -Mod. Then M ⊆ Loc(E) which is injective by (c) and hence equal to E. Theorem 2.1 shows that the question of determining whether A -Mod loc is essentially closed for a Noetherian K-algebra can be reduced to the study of essential extensions of finite dimensional irreducible representations. Over an algebraically closed field K and a Noetherian K-algebra A with all maximal ideals being completely prime, we can go a step further and reduce the problem to the study of essential extensions of one-dimensional representations. First we note that if V is a finite dimensional irreducible representation over a K-algebra A with ring of endomorphisms D = End( A V ), then D is a finite dimensional division algebra over K and End(V D ) is a matrix ring over D. Let P = Ann A (V ) be the annihilator of V . Then A/P ∼ = End(V D ) by the Jacobson Density Theorem and P is in particular a maximal ideal of A.
Corollary 2.2. Let A be a Noetherian K-algebra over an algebraically closed field K such that maximal ideals are completely prime. Then A -Mod loc is essentially closed if and only if the injective hull of onedimensional representations are locally finite.
Proof. Let P = Ann A (V ) the annihilator of a finite dimensional irreducible representation over A, which as mentioned before is a maximal ideal and by hypothesis completely prime. Thus A/P is a finite dimensional division algebra over K and since K is algebraically closed, V = A/P = K is one-dimensional. Hence the corollary follows from Theorem 2.1(b).
Among the examples of algebras in which maximal ideals are completely prime are certain quantum affine spaces: is the K-algebra generated by x 1 , . . . , x n and relations ij (see [25, p.14]). Theorem 2.4 (Goodearl-Letzter, [24, Theorem 2.1]). Let O q (K n ) be a quantum affine space for some data q = (q ij ). Assume that the subgroup Λ of K × generated by the q ij is torsionfree. Then all of the prime ideals of O q (K n ) are completely prime.
2.2. Sufficient conditions. The second part of this section will gather some sufficient conditions for A -Mod loc to be essentially closed. This is for example the case if all irreducible representations are finite dimensional and A satisfies the second layer condition 2 (see [4,Theorem 3.3]).
Recall that an ideal I of a ring A is said to have the left Artin-Rees property if for every finitely generated left A-module M and every submodule N ≤ M , there is a positive integer n such that N ∩ I n M ≤ IN . An ideal generated by normal elements has the Artin-Rees property by [36, 4.2.6]. A sequence x 1 , . . . , x n of elements of a ring R is called a normalizing sequence if for each j ∈ {0, · · · , n − 1} the image of x j+1 in R/ j i=1 Rx i is normal. An ideal generated by a normalizing sequence is called polynormal (see [36, 4.1.13]). The following is an adaptation of an argument by Jategaonkar: Proof. We proceed by induction on the number of elements of a normalizing sequence of generators of Q. Suppose Q is generated by one normal element being the center of A, and it maps A-submodules of E to A-submodules of E. Since Q has the Artin-Rees property by [36, 4.2.6], there exists a natural number n ≥ 1 such that Q n E = x n 1 V = 0. This induces a finite filtration whose subfactors are A/Q-modules and f induces a submodule preserving chain of embeddings Assume that the assertion holds for all Noetherian algebras and essential extensions V ≤ E of simple left A-modules such that Ann A (V ) contains a polynormal ideal Q with a normalizing sequence of generators of less than n elements. Let V ≤ E be an essential extension of a simple left A-module and assume that there exists a polynormal ideal Q ⊆ Ann A (V ) with normalizing sequence of generators {x 1 , . . . , x n }. By hypothesis, the submodule E ′ = Ann E (x 1 ) is finite dimensional if and only if E is finite dimensional. Let A ′ = A/Ax 1 and Q ′ = Q/Ax 1 . Then Q ′ ⊆ Ann A ′ (V ) is polynormal and generated by the normalizing sequence {x 2 , . . . ,x n }, wherex i denotes the image of the element contains an ideal Q satisfying one of the following conditions: (1) Q has the Artin-Rees property, such that A/Q is an affine PI-algebra (2) Q is polynormal and A/Q -Mod loc is essentially closed. Proof.
(1) Since Q has the Artin-Rees property, Q n E = 0 for some n ≥ 1 and E is a finitely generated essential extension of V over the affine PI-algebra A := A/Q n . By [31,Theorem 2], the injective hull of any irreducible representation over A has finite length and so does E. Since any irreducible representation over an affine PI-algebra is finite dimensional (see [36, 13.10.3]), E is finite dimensional.
(2) Since A/Q -Mod loc is essentially closed and Ann E (Q) is a (finitely generated) essential extension of V , Ann E (Q) is finite dimensional. By Lemma 2.5, E is finite dimensional.
From Proposition 2.6 it follows obviously that A -Mod loc is essentially closed for any Noetherian affine PI-algebra, which in particular applies to any Noetherian affine K-algebra that is finitely generated over a commutative affine PI-algebra. Moreover, a direct application of the proposition is the following theorem.
Theorem 2.7. The category of locally finite representations of a Noetherian K-algebra A is essentially closed if every maximal ideal of finite codimension contains an ideal Q that satisfies one of the following conditions: (1) Q has the Artin-Rees property and A/Q is an affine PI-algebra.
(2) Q is polynormal and A/Q -Mod loc is essentially closed.
Before we apply Theorem 2.7 to quantum affine spaces, we first turn our attention to finite normalizing ring extensions. Recall that a ring extension R ⊆ S is said to be finite normalizing, if there exists a finite set of elements {x 1 , . . . , x n } of S that generates S as R-module, such that Rx i = x i R for all i.
Moreover, ϕ is a monomorphism because if f m = 0 for some m, then f m (1) = m = 0.
Lemma 2.8. Let R ⊆ S be a finite normalizing extension of K-algebras. If R -Mod loc is essentially closed, then so is S -Mod loc .
Proof. Let M be a finite dimensional left S-module. We need to show that any finitely generated essential extension of M is finite dimensional. By assumption, the R-injective hull E of M is a locally finite Rmodule and we will show that Hom R (S, E) is a locally finite R-module. Since R ⊆ S is a finite normalizing extension, there exist elements x 1 , . . . , x n ∈ S such that S = n i=1 Rx i and Rx i = x i R, for all i. Define the K-linear map ϕ : Hom R (S, E) → E n given by Note that ϕ is injective, because if f (x i ) = 0, for all i, then as S = n i=1 Rx i also f (S) = n i=1 Rf (x i ) = 0, i.e. f = 0. Moreover, for all f ∈ Hom R (S, E) and r ∈ R, there exist r 1 , . . . , r n ∈ R with r i x i = x i r, for all i, and ϕ(r · f ) = (f (x 1 r), . . . , f (x n r)) = (r 1 f (x 1 ), . . . , r n f (x n )) ∈ Rf (x 1 ) ⊕ · · · ⊕ Rf (x n ).
Therefore ϕ(R · f ) ⊆ Rf (x 1 ) ⊕ · · · ⊕ Rf (x n ) is finite dimensional, as all cyclic R-submodules Rf (x i ) of E are finite dimensional. Since ϕ is an injective K-linear map, also R · f is finite dimensional. Thus, Hom R (S, E) is locally finite as an R-module.
It is a standard fact that Hom R (S, E) is an injective left S-module (see for example [32, (3 Since U is also finitely generated as an R-module and Hom R (S, E) is locally finite, U must be finite dimensional. This completes the proof.
Denote by Ξ(A) := Alg K (A, K) the set of K-algebra homomorphisms α : A → K. For any α ∈ Ξ(A), we define α K to be the one-dimensional representation with the action given by a · 1 := α(a) for all a ∈ A. Theorem 2.9. Let K be algebraically closed. Let O q (K n ) be a quantum affine space for some data q = (q ij ). Then locally finite representations of O q (K n ) are closed under essential extensions.
Proof. Assume first, that the group Λ = q ij ⊆ K × is torsionfree. By Goodearl and Letzter's Theorem 2.4, every prime ideal of O q (K n ) is completely prime. By Corollary 2.2, O q (K n ) -Mod loc is essentially closed if and only if essential extension of the one-dimensional modules χ K over O q (K n ) are locally finite, for all χ ∈ Ξ(O q (K n )). Let χ ∈ Ξ(O q (K n )) and M = Ker(χ). The defining relations of the quantum affine space, Let I = {i : χ(x i ) = 0}. Then for any i, j ∈ I, equation (1) yields q ij = 1. Let P = {x i : i ∈ I} ⊆ M be the ideal generated by those x i with χ(x i ) = 0. Then P is generated by normal elements and O q (K n )/P ≃ K[{x i : i ∈ I}] is a commutative Noetherian domain and has the property that locally finite representations are closed under essential extensions. Hence O q (K n ) satisfies condition (2) of Theorem 2.7 and has the property that locally finite representations are closed under essential extensions. Now suppose Λ = q ij : i, j is possibly not torsionfree, then as a finitely generated Abelian group, its torsion submodule T (Λ) is finite. Let N = |T (Λ)| be the order of its torsion group, then λ ∈ T (Λ) if and only if λ N = 1. In particular {λ N : λ ∈ Λ} is a torsionfree subgroup of Λ. For each i, j set p ij := q N 2 ij and y i := x N i . Then y i y j = p ij y j y i , for all i, j, and the subring of O q (K n ), generated by y 1 , . . . , y n can be identified with O q ′ (K n ) for q ′ = (p ij ). Since Λ ′ = p ij : i, j is torsionfree, the first part of this proof applies and shows that O q ′ (K n ) is essentially closed. Moreover, O q (K n ) is generated over O q ′ (K n ) by the set of monomials {x i 1 1 · · · x in n | 0 ≤ i 1 , . . . , i n < N 2 }, such that all these monomials commute with the generators y i of O q ′ (K n ) up to a scalar. Thus, O q (K n ) is a finite normalizing extension of O q ′ (K n ) and Lemma 2.8 shows that O q (K n ) is essentially closed. Example 2.10. A particular example is the quantum plane O q (K 2 ). In this case, O q (K 2 ) -Mod loc is always essentially closed, while by [12], O q (K 2 ) satisfies (⋄) if and only if q is a root of unity.

Locally finite representations over Ore extensions
In order to study locally finite representations over Ore extensions, we will first look at conditions to deduce that certain ideals have the Artin-Rees property. A stronger condition is the property that these ideals have a Noetherian Rees ring. We will revise the necessary background first. Let A be a ring and I be an ideal in A. The Rees ring of I is the subring R A (I) of the polynomial ring A[t] generated by A + It. In other words, and hence a typical element f of R A (I) has the form f = a k t k where a k ∈ I k for each k ≥ 1. The reason why we are interested in Rees rings is that an ideal with a Noetherian Rees ring satisfies the Artin-Rees property (see [25,Lemma 13.2]). We will say that an ideal I of A has the strong Artin-Rees property if R A (I) is Noetherian. The following result is basically [22,Lemma 5] (see also [2,Lemma 7.1]). Recall that an ideal I of a ring R with automorphism σ and σ-derivation δ is called (σ, δ)-invariant if σ(I) ⊆ I and δ(I) ⊆ I. Lemma 3.1. Let R be a Noetherian ring, σ be an automorphism of R, δ be a σ-derivation and A = R[x; σ, δ] be the associated Ore extension. If I is an σ-and δ-invariant ideal of R having the strong Artin-Rees property, then IA is an ideal of A with the strong Artin-Rees property.
Proof. Let t be an indeterminate and identify A[t] with R[t][x, σ, δ] where we extend (σ, δ) to R[t] by σ(t) = t and δ(t) = 0. By hypothesis, all powers I n are σ-and δ-invariant ideals, whence the Rees ring is closed under σ and δ. In particular, σ restricts to an automorphism of R R (I). Moreover, IA = AI and so (IA) j = I j A for all j ≥ 0. It follows that R R (I)[x; σ, δ] = R A (IA). By assumption, R R (I) is Noetherian and, as an Ore extension, so is R A (IA). Lemma 3.1 and Theorem 2.7 allow us to conclude that locally finite representations over unmixed Ore extensions of commutative Noetherian K-algebras are essentially closed as we will show in the next theorem.
Theorem 3.2. Let R be a commutative Noetherian K-algebra with a K-algebra automorphism σ and a Klinear derivation δ. Then locally finite representations over R[x; σ] as well as locally finite representations over R[x; δ] are essentially closed.
Proof. Let S = R[x; σ] and V be a finite dimensional simple left S-module, P = Ann S (V ) be its annihilator and E any finitely generated essential extension of V . If x ∈ P , then Q = Sx is an ideal contained in P that is generated by the normal element x. Since S/Q ≃ R is commutative Noetherian, S/Q -Mod loc is essentially closed and by Lemma 2.6(2), E is finite dimensional.
If x ∈ P , then P ∩ R is an σ-prime ideal of R [36, 10.6.4(ii)]. Moreover, P ∩ R has a Noetherian Rees ring, since it is an ideal of the commutative Noetherian ring R. By Lemma 3.1, Q = (P ∩ R)S ⊆ P is an ideal contained in P with the Artin-Rees property. If P = Q, then P has the Artin-Rees property and by Lemma 2.6(2), E is finite dimensional. If P = Q, then by [29,Theorem 4.3] σ induces a finite order automorphism σ on R/(P ∩ R) and by [16,Corollary 10] it follows that S/Q ∼ = (R/(P ∩ R))[x; σ] is a PI ring. In this case it follows from Theorem 2.7 that E is finite dimensional. Now let S = R[x; δ]. By Sigurdsson's Theorem [25,Theorem 10.23], every prime ideal of S = R[x; δ] is completely prime. Therefore, by Corollary 2.2 it is enough to consider one-dimensional simple modules V = χ K, for some character χ ∈ Ξ(S). Let P = Ker(χ) = Ann S (V ). Then P ∩ R is δ-stable, since for any a ∈ P ∩ R, we have δ(a) = xa − ax ∈ P ∩ R. Now P ∩ R is a δ-invariant character ideal with the strong Artin-Rees property, hence by Lemma 3.1 Q := (P ∩ R)S has the strong Artin-Rees property too. Moreover, by [7,Theorem 4.2], Q is an ideal of S such that S/Q ∼ = K[x], i.e. a PI-algebra. It follows from Theorem 2.7 that finitely generated essential extensions of V are finite dimensional.
For the simplest mixed case, a general Ore extension over K[x], we can also show that locally finite representations are essential closed. Let A = A 1 (k, q) = k[x][y; σ, δ] be the quantized Weyl algebra, q = 1 and q is not a root of unity. Then the element u = yx − xy is a normal element in A, every nonzero prime ideal of A contains u by [22,Theorem 8.4(a)] and A/uA ∼ = K[x, x −1 ] (see [22,Proposition 8.2]). Since A/uA is commutative Noetherian, A/uA -Mod loc is essentially closed and again by Theorem 2.7, A has the same property. When q is a root of unity, A becomes a finitely generated module over a central subring, hence a PI-algebra and Theorem 2.7 applies again.
For an arbitrary commutative coefficient ring R, we do not have a definite answer when the Ore extension S = R[x; σ, δ] is essentially closed. However, in the case where the extension R ⊆ S is a Hopf-Ore extension, we can get a positive answer. Recall from [7, Definition 2.1] that an Ore extension S = R[x; σ, δ] of a Hopf algebra R over K is a Hopf-Ore extension of R if S is a Hopf algebra over K with Hopf subalgebra R such that there are a, b ∈ R and v, w ∈ R⊗R such that ∆(x) = a⊗x+x⊗b+v(x⊗x)+w. Proof. By Molnar's Theorem [37], R is an affine K-algebra. According to [7,Theorems 5.1 & 5.2], there is always a change of variables so that the Ore extension is either a differential type or an automorphism type. Since the coefficient ring is commutative, it follows from Theorem 3.2 that A is essentially closed.

Locally finite representations over some Hopf algebras
In this last section, we restrict ourselves mostly to Hopf algebras H. We have seen in Section 2 that it is sometimes possible to reduce the study of locally finite representations to the study of onedimensional representations. In particular we will prove conditions under which H -Mod loc is essentially closed provided the injective hull of the trivial representation is locally finite. This will be always the case when the antipode S of H is invertible. This is the case for Noetherian semiprime Hopf algebras as shown by Skryabin in [43, Corollary 1].

4.1.
Reduction to the trivial representation. The category of left H-modules is a tensor category. In particular, for any left H-modules U and V , the tensor product U ⊗ V is a left H-module with the left H-action defined as h · (u ⊗ v) = (h 1 · u) ⊗ (h 2 · v), for all h ∈ H, u ∈ U and v ∈ V . Also, Hom K (U, V ) is a left H-module with left H-action defined as (h · f )(u) = h 1 · f (S(h 2 ) · u), for all h ∈ H, u ∈ U and f ∈ Hom K (U, V ). In case the antipode S is invertible, we define another left H-module structure on the dual space V * of V by (h · ϕ)(v) = ϕ(S −1 (h) · v), for all h ∈ H, v ∈ V and ϕ ∈ V * Lemma 4.1. If H is a Hopf algebra with invertible antiopode, U and E are left H-modules and V is a finite dimensional representation, then Hom H (U, E ⊗ V ) ≃ Hom H (U ⊗ V * , E) as left H-modules.
Proof. Define the map Φ : for all f ∈ Hom K (U, E ⊗ V ), ϕ ∈ V * and u ∈ U . It is clear that Φ is an isomorphism of K-vector spaces with inverse given by Φ −1 : where {b 1 , . . . , b n } is a basis of V , {p 1 , . . . , p n } ⊆ V * a dual basis satisfying p i (b j ) = δ ij , for all i, j.
We will prove that Φ is H-linear. For all h ∈ H, we denote the action of h on U by λ U h , i.e. λ U h (u) = h·u. Note that for any ϕ ∈ V * , h ∈ H and e ⊗ v ∈ E ⊗ V we have Note also that ϕλ V h (v) = ϕ(h · v) = (S(h) · ϕ)(v), for all v ∈ V and h ∈ H. Hence, Therefore:  The inverse of σ is given by In particular, for a character χ ∈ Ξ(H) and the winding automorphism σ = τ r χ we have that the H-action on σ −1 ( χ K) is equal to the trivial action, because Thus, if χ K ⊆ M is an essential extension of left H-modules, then ǫ K = σ −1 ( χ K) ⊆ σ −1 M is an essential extension of left H-modules and if furthermore any finitely generated essential extension M of ǫ K is finite dimensional, then σ −1 M must be finite dimensional and so must be M . ; σ 2 , δ 2 ] · · · [x n ; σ n , δ n ] be an iterated Ore extension over K.
and assume that the following conditions hold: Either q i is not a root of unity or q i = 1 and the char(K) = 0.
Further assume that the subgroup Λ of K × generated by the λ ij is torsionfree. Then all prime ideals of A are completely prime.
In particular the only finite dimensional representations are the 1-dimensional ones, under our running assumptions of K being algebraically closed and having characteristic zero. Examples of algebras that can be presented as in Goodearl and Letzter's Theorem are certain quantum algebras over K, like the multiparameter coordinate ring of quantum n × n matrices, O λ,q (M n (K)), the multiparameter quantized Weyl algebras of degree n, A Q,Γ n (K), the coordinate ring of quantum symplectic 2n-space, O q (spK 2n ) and the coordinate ring of quantum Euclidean n-space, O q (oK n ) (see the survey [23] for definitions).   If I is a polynormal ideal of R with a normalizing sequence (x 1 , . . . , x n ) such that δ(x i ) ∈ Rx 1 + · · · + Rx i holds, for all 1 ≤ i ≤ n. Then AI is polynormal.
Proof. We prove this lemma by induction on the number of generators of I. For n = 1, I = Rx = xR is generated by a normal element, which satisfies δ(x) = f x = xg for some f, g ∈ R. Then yx = xy + δ(x) = x(y + g) and xy = yx − δ(x) = (y − f )x shows Ax = xA, i.e. AI = Ax is polynormal.
Let n ≥ 1 and suppose we have proven the statement of the lemma for all rings R with derivations δ and polynormal ideals I generated by a normalizing sequence (x 1 , . . . , x n ) that satisfies δ(x i ) ∈ i j=1 Rx j , for all 1 ≤ i ≤ n. Suppose that I is a polynormal ideal generated by a normalizing sequence (x 1 , . . . , x n , x n+1 ) that satisfies δ(x i ) ∈ i j=1 Rx j , for all 1 ≤ i ≤ n + 1. Let A = R[y; δ]. Since x 1 is normal in R, we know that x 1 is normal in A and hence Ax 1 is an ideal. Let R = R/Rx 1 and A = A/Ax 1 . Then δ extends to a derivation δ of R such that R[y; δ] ≃ A. The ideal I = I/Rx 1 is polynormal, generated by the normalizing sequence (x 2 , . . . , x n+1 ) that satisfies δ(x i ) ∈ i j=1 R j x j , where x i = x i + Rx 1 . By induction AI = AI/Ax 1 is a polynormal ideal of A and therefore AI is a polynormal ideal of A.
holds for all j ≤ i, we can conclude that H + is a polynormal ideal with normalizing sequence {x 1 , . . . , x n }, which can be proven by using Lemma 4.4 at each step. More concretely, at the first step, the augmentation ideal of the Hopf algebra H (1) = K[x 1 ] is generated by x 1 and hence polynormal. Suppose we have already verified that the augmentation ideal H + (i) is polynormal with normalizing sequence (x 1 , . . . , x i ), for some 1 ≤ i ≤ n. Then assumption (4) implies δ i+1 (x j ) ∈ H (i) x 1 + · · · + H (i) x j , for all j ≤ i. Thus we can apply Lemma 4.4 and conclude that that a finite dimensional Lie algebra g is nilpotent if and only if every ideal of U (g) has the Artin-Rees property. Hence in particular the augmentation ideal P = Ker(ǫ) of U (g) has this property and since U (g)/P = K, Proposition 2.6 applies, showing that U (g) -Mod loc is essentially closed. Donkin proved in [18, Proposition 2.2.2] that essential extensions of the trivial representation of a solvable Lie algebra g are locally finite, by using the unique maximal nilpotent ideal h of g. By [41,Theorem 7], the augmentation ideal U (h) + := U (h) ∩ Ker(ǫ) of U (h) has the strong Artin-Rees property and Donkin goes on to prove that U (g)U (h) + also has the strong Artin-Rees property in U (g). Since g/h is abelian, U (g)/U (g)U (h) + is commutative and Proposition 2.6 applies again. Note that U (g) = U (h)# σ U (g/h) is a crossed product. A similar argument works in the case of the group ring of a polycyclic-by-finite group.
Let π : H → H be a surjective homomorphism of Hopf algebras, which splits as a coalgebra map, i.e. there exists a coalgebra map γ : H → H with γ • π = id and such that γ(1) = 1. Then is a normal Hopf subalgebra and H ≃ R# σ H is a crossed product, i.e. H = R ⊗ H as vector spaces, H acts weakly on R by h · a = γ(h 1 )aS(γ(h 2 )), for all a ∈ R and h ∈ H. The multiplication in R# σ H is given by a#h b#k = a(h 1 · b)σ(h 2 , k 1 )#h 3 k 2 and the cocycle σ : H × H → R is given by σ(h, k) = γ(h 1 )γ(k 1 )S(γ(h 2 k 2 ))(see [3,38] By hypothesis, R R (R + ) is Noetherian and H acts also on it. Since HR + = R + H we also have (HR + ) n t n = (R + ) n t n ⊗ H ⊆ R R (R + )#H, showing that the Rees ring R H (HR + ) of HR + can be identified with the crossed product R R (R + )# σ H.
As H is an affine commutative Hopf algebra and hence the coordinate ring O(V ) of an affine variety V , it is an image of the enveloping algebra U (a) of a finite dimensional abelian Lie algebra (e.g. a polynomial ring). Hence R H (HR + ) = R R (R + )# σ H is an image of a crossed product R R (R + )# σ U (a), which is Noetherian by [36].  [44] says that an affine prime K-algebra of Gelfand-Kirillov dimension one is finitely generated over its center. Hence locally finite representations over an affine, prime Hopf algebra of Gelfand-Kirillov dimension one are always closed under essential extensions. We go through the Hopf algebras appearing in the theorem one by one in relation to being essentially closed. Consider the quantum affine space O q (K 3 ) with x 1 x 2 = x 2 x 1 and x 1 x 3 = qx 3 x 1 while x 3 x 2 = q −1 x 2 x 3 . Then A ≃ O q (K 3 )/ x 1 x 2 −1, x 2 x 1 −1 . If q is not a root of unity, then 1, q, q −1 = q ⊆ K × is torsionfree and by Corollary 2.9, locally finite representations over O q (K 3 ) are essentially closed. The same is true for the quotient algebra A. If q is a root of unity, then O q (K 3 ) is a PI-algebra and so is A.

Hopf algebras of low
Example 4.8.
[The Hopf algebras B(n, p 0 , . . . , p s , q)] Let n, p 0 , . . . , p s be positive integers and let q ∈ K × be such that (a) s ≥ 2 and 1 < p 0 < p 1 < . . . < p s ; (b) p 0 | n and p 1 , . . . , p s are relatively prime; (c) q is a primitive lth root of unity, where l = (n/p 0 )p 1 . . . p s . Let m = p 1 . . . p s and m i = m/p i . The Hopf algebra B(n, p 0 , . . . , p s , q) is the skew Laurent polynomial ring A[x ± ; σ] where A is the subalgebra of the polynomial algebra K[y] generated by the powers y m i of the indeterminate y. The automorphism σ is the restriction of the K-algebra automorphism on K[y] which sends y to qy. Since q is a root of unity, K[y][x ± ; σ] is a PI-algebra, and so is B(n, p 0 , . . . , p s , q). Thus, by Theorem 2.7, locally finite representations over B(n, p 0 , . . . , p s , q) are essentially closed. . There is a unique Hopf algebra structure on C(n) such that y is grouplike and x is skew primitive, with ∆(x) = x ⊗ y n−1 + 1 ⊗ x. The counit ǫ : C(n) → K is such that ǫ(x) = 0 and ǫ(y) = 1. By Theorem 3.2, locally finite representations over the differential operator ring C(n) are essentially closed.