A class of finite-by-cocommutative Hopf algebras

We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a Tannakian perspective. We determine when the new Hopf algebras are co-Frobenius, or cosemisimple, or Noetherian, or regular, or have finite Gelfand-Kirillov dimension.


Introduction
A Hopf algebra H is commutative-by-finite if it has a normal Hopf subalgebra A such that A is commutative and H is a finitely generated A-module.In other words there is an exact sequence of Hopf algebras A / / H / / / / u where A is commutative and u is finite-dimensional.There are various remarkable families of commutative-by-finite Hopf algebras arising from the theory of quantum groups.A systematic study of affine commutative-by-finite Hopf algebras was started in [14]; here "affine" means that H is a finitely generated algebra.
A Hopf algebra K is finite-by-cocommutative if it fits into an exact sequence of Hopf algebras a / / K / / / / U where U is cocommutative and a is finite-dimensional.Lusztig's quantum groups at roots of 1 are finite-by-cocommutative.An example of a finite-bycocommutative Hopf algebra appeared in [6] to disprove a conjecture on co-Frobenius Hopf algebras.A family of examples containing that one and characterized by suitable Nicolás Andruskiewitsch et al.
The goal of this paper is to present and study a family of finite-by-cocommutative Hopf algebras.We assume that the base field k is algebraically closed and has characteristic 0. To start with, consider a Noetherian Hopf algebra H with a central Hopf subalgebra A; we set H ε = H/HA + , where A + = ker ε |A .Thus we have an extension of Hopf algebras In Section 3, for any subgroup Γ of the pro-affine algebraic group G = Alg(A, k), we define a suitable subgroup Γ fd of Γ and a Hopf subalgebra H(Γ ) of the finite dual H • , which is an extension of Hopf algebras From a Tannakian perspective, the category of finite-dimensional comodules over H(Γ ) is equivalent to the full subcategory C Γ of the category rep H of finite-dimensional Hmodules such that the action of A is semisimple and by characters in Γ ; thus the objects of C Γ bear a Γ -grading.
In Section 4 we assume further that the extension (E) is cleft and that dim H ε < ∞.Then H is a finitely generated A-module, A is Noetherian, G is an algebraic group, Γ fd = Γ , (F Γ ) becomes H * ε → H(Γ ) ↠ kΓ and so H(Γ ) is finite-by-cocommutative.We establish several properties of H(Γ ):  16, 4.17, 4.20, 4.23 and 4.26.The keys to these results are that H(Γ ) is strongly Γ -graded and that Γ , being a subgroup of an algebraic group, is linear, i.e., embedable into GL(n, k) for some n.
The contents of the paper are organized in the following way.Section 2 contains expositions of known facts needed along the article.In Section 3 we present the Hopf algebras H(Γ ) and the extensions (F Γ ) in a general context and establish some basic properties.In Section 4 we study the Hopf algebras H(Γ ) under the restrictions above.Section 5 is devoted to examples.
Notations.The natural numbers are denoted by N, and N 0 = N∪{0}.Given m < n ∈ N 0 , we set I m,n = {i ∈ N 0 : m ≤ i ≤ m} and I n = I 1,n ."Algebra" means associative unital algebra.The space of algebra homomorphism from a k-algebra A to a k-algebra B is denoted by Alg(A, B).The category of finite-dimensional left R-modules, where R is an algebra, is denoted by rep R. All Hopf algebras are supposed to have bijective antipode.We write M ≤ N to express that M is a subobject of N in a given category.The notation for Hopf algebras is standard: ∆ is the comultiplication, ε is the counit, S is the antipode.For the comultiplication and the coactions we use the Heynemann-Sweedler notation.

Preliminaries
We refer the reader to [42,45] for the basic facts about Hopf algebras used throughout the paper.Given a coalgebra K and an algebra H, the group of invertible elements in Hom(K, H) with respect to the convolution is denoted by Reg(K, H).
2.1.Cleft comodule algebras.These were studied in [13,25]; we recall the relevant facts.Let H be a Hopf algebra, let R be a right comodule algebra with coaction ρ : R → R ⊗ H and let R co H = {x ∈ R : ρ(x) = x ⊗ 1}.One defines similarly co H T for a left comodule algebra T .
For instance, if π : C → B is a Hopf algebra map, then C is a right, respectively left, comodule algebra via ρ = (id ⊗π)∆, resp.λ = (π ⊗ id)∆.The algebras of right and left coinvariants of π are We consider three properties of the extension of algebras R co H ⊂ R: Theorem 2.1 ([25]).The extension R co H ⊂ R is cleft if and only if it is H-Galois and has the normal basis property.

Example 2.2 ([55]
).Let G be a group with unit e and let R be an algebra.A kGcomodule algebra structure on R is the same as a G-grading of algebras R = ⊕ g∈G R g ; here R co H = R e .Now "R is strongly G-graded" means that Then R e ⊂ R is kG-Galois if and only if R is strongly G-graded.

Extensions of Hopf algebras.
This notion was considered in many papers.see e.g.[33,8,49,31,50,52,38,16,17].Following [1,8] together with [49] we say that the sequence of morphisms of Hopf algebras In this case we also say that C is an extension of B by A.
The left and right adjoint actions of C, denoted by ad ℓ , ad r : C → End C, are given by Notice that ad r (x)(y) = S −1 (ad ℓ (S(x))S(y)) , x, y ∈ C.

Nicolás Andruskiewitsch et al.
A Hopf subalgebra A of a Hopf algebra C is normal if it stable under one of, hence both, the adjoint actions.Proof.(i), (ii) are easy; see [42, 3.4.3]

Lemma 2.3. Let A be Hopf subalgebra of a Hopf algebra Cand
Let C be a Hopf algebra.The left and right coadjoint actions of C are the (left and right) comodule structures ϱ ℓ , ϱ r : C → C ⊗ C given by Let τ be the usual flip.Notice that A surjective Hopf algebra map π : C → B is conormal, or simply B is a conormal quotient of C, if ker π is a subcomodule for one of, hence both, ϱ ℓ and ϱ r (notice a change of terminology with respect to [8]).
There exists a minimal cocentral Hopf algebra map q : H → HC(H); by abuse of notation, HC(H) is called the Hopf cocenter of H.Here minimal means that any cocentral map q : H → K factorizes through q.
(i) If H is not commutative, then HZ(H) ≃ k. (ii) If H is not cocommutative, then HC(H) ≃ k. (iii) (N. A. and H.-J.Schneider, Appendix to [1]).The small quantum groups associated to simple Lie algebras and their parabolic subalgebras are simple Hopf algebras, hence their Hopf centers and cocenters are trivial.

Cocycles and twists.
Let H be a Hopf algebra.A Hopf 2-cocycle [24] or simply a cocycle is a convolution invertible inear map σ : for all x, y, z ∈ H. Then we have a new Hopf algebra H σ , the coalgebra H with multiplication conjugated by σ, i. e.
Dually, a twist for H is an element F = F (1) ⊗ F (2) ∈ H ⊗ H that satisfies and has an inverse 2) .Then H F , the algebra H with the comultiplication ∆ F := F ∆F −1 is a Hopf algebra.These definitions are compatible with duals, i.e. the transpose of a twist is a cocycle etc.
Example 2.10.Let now Γ be a group with unit e and A a Hopf algebra.Assume that Then A e is a Hopf subalgebra of A. Let F = F (1) ⊗F (2) ∈ A e ⊗A e be a twist for A e .Then F is a twist for A, which remains a Γ -graded algebra: κ is again a sucoalgebra, namely A κ with comultiplication conjugated by F .

3.
Hopf algebras with a central Hopf subalgebra

Hopf systems.
Let A be a central Hopf subalgebra of a Noetherian Hopf algebra H and let G = Alg(A, k) be the pro-affine algebraic group defined by A; its unit is the counit ε.Given κ ∈ G, let Since A is central, H κ is an algebra (with multiplication m κ and unit u κ ) and the natural projection Hence for any κ, γ ∈ G there are well-defined algebra morphisms that satisfy ( By the coassociativity and antipode axioms, for any κ, γ, ν ∈ G we have ) In particular H ε is a quotient Hopf algebra of H.In other words, (H κ ) κ ∈ G is a Hopf system in the sense of [1].By Lemma 2.3 (iii) and Remark 2.4 (ii), we have an exact sequence of Hopf algebras

Tensor categories.
Let C be the full subcategory of rep H whose objects are those where A acts in a semisimple way.Thus, if V ∈ C , then Because of (3.3), C is closed under tensor products and duality.Clearly it is a full subcategory closed under taking subquotients, hence it is a tensor subcategory of rep H [26, 4.11.1].Notice however that C is not closed under extensions, see the discussion on rep H in Subsection 3.4.By (3.3), we also see that (3.7) is a grading of tensor categories, that is This statement is also true when dim H ε is finite, as we show next.
Proof.Let A be the full subcategory of rep A whose objects W are semisimple, i.e., where We have a grading A = κ ∈ G A κ , where A κ is the full subcategory of rep A whose objects are those W with W = W κ .This grading is faithful since the one-dimensional representation supported by κ belongs to A κ .
The restriction functor rep H → rep A induces a functor F : C → A. We claim that F is dominant, that is, for every object W ∈ A there exists an object V ∈ C such that W is a subobject of F (V ).To see this, recall that H is faithfully flat over A, cf.Remark 2.

(ii). Then the inclusion
Given a morphism of algebras ϕ : R → T , the transpose t ϕ : If in addition R and T are Hopf algebras and ϕ is a morphism of Hopf algebras, then so is ϕ We consider the subcoalgebras of In other words p • κ : Let Γ be any subgroup of G.We introduce Then H(Γ ) is a Hopf subalgebra of H • by (3.3).Notice that the subcategory C κ in Subsection 3.2 is equivalent to the category of finite-dimensional right C(κ)-comodules.Also C Γ is tensor-equivalent to the tensor category of finite-dimensional H(Γ )-comodules. Let Clearly Γ fd is a subgroup of Γ but it could be strictly smaller.
Example 3.5.Let h n be the n th Heisenberg Lie algebra, with basis where R is the polynomial ring in 2n variables.Now G = Alg(k[z], k) is the algebraic group (k, +); for any κ ∈ G \ 0, H κ = U (h n )/⟨z − κ⟩ is isomorphic to the Weyl algebra in 2n variables which has no non-zero finite-dimensional representation, being simple and infinite-dimensional. Thus for any Γ ≤ G, Γ fd is trivial.
The following is the main result of this Section.Recall that ι :

graded and the following is an exact sequence of Hopf algebras:
Proof.Let (κ j ) be a finite family of different elements in G. Then Here (⋆) is a standard fact in commutative algebra and (#) is evident.Let us prove ( * ): We next claim that Im ϖ ≃ kΓ fd , so that the map ϖ in (3.11) makes sense.Indeed, if V ∈ rep H κ , then A acts on V via κ and thus C Res A V ⊆ kκ and the equality holds iff V ̸ = 0.This implies that Im ϖ ⊆ kΓ fd and the equality follows by definition of Γ fd .
Our next goal is to show that We start with the following observation.By standard arguments on matrix coefficients, if W ∈ rep H κ and V = W pκ , then where the counit of H • is denoted by ε.Now (3.12) follows because each Then by (3.13), For the exactness of (3.11), it remains to see that ker ϖ = H(Γ )(H • ε ) + .Now H(Γ ) is faithfully coflat over kΓ fd by Lemma 2.6.By Lemma 2.5 (iii) we are reduced to show that ϖ is conormal, but this follows from the centrality of A: Thus, if f ∈ ker ϖ, then f (1) S(f (3) ) ⊗ ϖ(f (2) ) = 0, hence ker ϖ is a subcomodule with respect to ϱ ℓ , i.e. ϖ is conormal.
) is a Hopf subalgebra of H(Γ ), Γ ′ fd ≤ Γ fd and there is a morphism of exact sequences  Proof.Let V, W ∈ rep H.We first observe that V ∈ C (κ) iff there exists a filtration Indeed, given r and s, take v ∈ V r and w ∈ W s .By hypothesis, for any z ∈ A we have Iterating we see that (z Observe that the previous arguments imply that the category rep A also bears a grading rep A = κ ∈ G A (κ) , where A (κ) is the full subcategory of rep A whose objects are those V with V = V (κ) .Now this grading is faithful since the one-dimensional representation supported by κ belongs to A (κ) .
Assume now that dim H ε < ∞.As in Proposition 3.3, the restriction functor F : rep H → rep A is dominant, implying the faithfulness of the grading (3.14).□ Definition 3.12.Let Γ ≤ G. Then C (Γ ) denotes the full subcategory of rep H generated by C (κ) , κ ∈ Γ , in other words this is a Hopf subalgebra of H • and C (Γ ) is tensor-equivalent to the tensor category of finite-dimensional H((Γ ))-comodules.
Proof.First recall that for any Hopf algebra K and right K-comodule V , one has an isomorphism of right K-comodules where dim C(κ) < ∞ by (4.1).Recall that C κ is the category of finite-dimensional C(κ)comodules.By the previous discussion, for any γ ∈ Γ , By a similar argument, using the isomorphism C(κ)⊗H(Γ ) ≃ H(Γ ) dim C(κ) , we also obtain that κ) , for all κ, γ ∈ Γ.In the rest of the paper, we shall assume that Assumption 4.2 holds and that the extension (E) is cleft, hence Assumption 4.1 also holds.

Assumptions on H. Let H be a Hopf algebra with a central Hopf subalgebra A such that H is a finitely generated
In particular, G = Alg(A, k) is an algebraic group.These equivalences follow from Lemma 4.9 below whose proof requires two results.Theorem 4.7 ([27]).If R is a right Notherian ring which is finitely generated as a right module over a commutative subring S, then S is Noetherian.

Theorem 4.8 ([41]). Let A be a commutative Hopf algebra. Then
That is, I ∩ A is the ideal of functions on G that vanish in the Zariski closure Γ of Γ .Hence, if Γ ̸ = G, then I ∩ A ̸ = 0 and so I ̸ = 0. We have proved: Lemma 4.12.If the pairing ⟨ , ⟩ : is natural to ask whether the Hopf algebras H(Γ ) and H(Γ ′ ) are isomorphic (here is an abuse of notation, as the initial data are different).We offer a partial answer, applicable in many examples.Recall that ϖ : H(Γ ) → kΓ is cocentral, cf.Remark 3.8.

Coradical filtration and co-Frobenius property.
Recall that a Hopf algebra K is co-Frobenius if it admits a non-zero (right) integral, i.e., a linear functional : K → k, ̸ = 0, invariant under the dual of the left regular representation.See e.g.[5,6] for details and a list of equivalent characterizations.
Let Γ ≤ G. Corollary 3.10 implies that H(Γ ) is co-Frobenius.The next Theorem gives a refinement of this fact.

Theorem 4.16. The coradical filtration of H(Γ ) is
Proof.The proof of (4.4) is by induction on n ≥ 0. For n = 0 this is a consequence of [45, 3.4

4.3.
Gelfand-Kirillov dimension.We refer to [34] for the definition and basic properties of this notion.
The main result of this subsection characterizes the algebras H(Γ ) with finite Gelfand-Kirillov dimension.We start by a general remark.Let Γ be a group and let R = κ ∈ Γ R κ be a strongly Γ -graded algebra.
for all κ ∈ Γ. ( In particular, if GKdim R < ∞, then Γ has polynomial growth. where in the second inequality we assume (4.5).Now ♠ implies that lim sup Assume that (4.5) holds.Let V be a finite-dimensional subspace of R; clearly there exists a finite and the equality GKdim R = GKdim kΓ follows.□ Recall that a finitely generated group is nilpotent-by-finite if it has a normal nilpotent subgroup of finite index.Here is a celebrated result by Gromov: Theorem 4.19 ([28]).If Γ is a finitely generated group, then GKdim kΓ < ∞ if and only if Γ is nilpotent-by-finite.
Assume next that Γ is a not necessarily finitely generated group.Then kΓ has finite GKdim if and only if there exists N ∈ N such that GKdim kΥ < N for any finitely generated Υ ≤ Γ.
In particular any finitely generated subgroup of Γ should be nilpotent-by-finite.We then conclude: Thus GKdim H(Γ ) < ∞ iff there exists N ∈ N such that any finitely generated Υ ≤ Γ is nilpotent-by-finite and GKdim kΥ < N .
In general any finitely generated subalgebra of H(Γ ) is contained in H(Υ ) for some finitely generated Υ ≤ Γ , cf.Remark 3. (a) GKdim kQ = 1 because any finitely generated subgroup of Q is cyclic (and torsionfree).Since Q is a subgroup of the additive group k, it embeds in any algebraic group which is not a torus.(b) Let G ∞ ≤ k × be the group of all roots of 1. Then GKdim kG ∞ = 0 because any finitely generated subgroup is cyclic (and torsion).Now G ∞ embeds in any algebraic group that contains a torus.

Noetherianity.
A general reference for this Subsection is [40].Recall that a solvable group is polycyclic if every subgroup is finitely generated; equivalently, if it admits a subnormal series with cyclic factors.Also, a group is polycyclic-by-finite if it has a normal polycyclic subgroup of finite index.The Hirsch number of a polycyclic group is the number of infinite factors in any subnormal series; the Hirsch number of a polycyclic-by-finite group is that of a polycyclic normal subgroup with finite index.
It is a classical result that the group algebra of a polycyclic-by-finite group is Noetherian [29]; a well-known open question is whether the converse holds.
Recall that a group is Noetherian if it satisfies the maximal condition on subgroups; if the group algebra of a given group is Noetherian, then so is the group but the converse is not true, see [32].However for linear groups there is a remarkable result of Tits: Theorem 4.21 ([54]).A linear Noetherian group is polycyclic-by-finite.
In consequence, if Γ is a linear group, then the following are equivalent: that is, K is not regular.Here ♡ is e.g. by [37, 2.4] As is known [8,31,38], such an extension can be described by a collection (⇀, σ, ρ, τ ) made up of a weak action, a cocycle, a weak coaction and a cycle that satisfies a long set of axioms; centrality slightly simplifies the requirements (e.g. the weak action is trivial) but otherwise this situation seems to be difficult to handle.
Then we also need information on the following classical problem.
Question 5.3.Given an algebraic group G (that admits a Noetherian Hopf algebra H as in Question 5.1), describe its subgroups, in particular those that are finitely generated nilpotent-by-finite, or polycyclic-by-finite.
Towards this question, it is worth recalling the following celebrated result.

Quantum algebras of functions.
Let G be a semisimple simply connected algebraic group with Lie algebra g and algebra of functions O(G).Let ℓ be an odd integer (prime to 3 if G has a component of type G 2 ), and ϵ a primitive ℓ th root of unity.Recall the quantized algebra of functions O ϵ (G), see e.g.[21], and the small quantum group u ϵ (g).
There is an exact sequence of Hopf algebras which is cleft by [46, 3.4.3].By Theorem 5.4 any polycyclic-by-finite group Γ is a subgroup of a suitable G and so gives rise to a Hopf algebra H(Γ ).More examples are given by the Hopf algebra quotients of the quantized algebras of functions classified in [9].

Pointed Hopf algebras.
Recall that a Hopf algebra is pointed if every simple comodule is one-dimensional.See [10] for this notion and [2] for the related notion of a Nichols algebra.We start by a result needed later.Theorem 5.5 ([39, 1.3]).Let π : U → u be a surjective map of Hopf algebras.If U is pointed, then the u-comodule algebra U , with coaction (id ⊗π)∆, is cleft.
Let (V, c) be a braided vector space.Then the tensor algebra T (V ) is a braided graded Hopf algebra.A pre-Nichols algebra of V is a factor of T (V ) by a graded Hopf ideal supported in degrees ⩾ 2. The Nichols algebra of V is B(V ) = T (V )/J (V ), where J (V ) is the maximal Hopf ideal among those.Thus any pre-Nichols algebra B of V lies between T (V ) and B(V ).
We refer to [3,4,11] for details on the following material.Let (V, c) be a braided vector space of diagonal type with braiding matrix q = (q ij ) ∈ k × I×I , where I = {1, . . ., θ}, θ ∈ N. Assume that the Nichols algebra B q := B(V ) has finite dimension, thus q belongs to the classification in [30].As shown in [11], (V, c), i.e., the matrix q, gives rise to the following data: • The distinguished pre-Nichols algebra B q .
• The Hopf algebra U q (the Drinfeld double of the bosonization of B q ).• The subalgebra Z q of U q as modified in [4, § 4.5].
The class of Examples of this Subsection arises from the following result.Theorem 5.6.If q satisfies the technical condition [4, (4.26)], then Z q is a central Hopf subalgebra of U q , M q := Alg(Z q , k) is a solvable algebraic group, u q = U q /U q Z + q is finitedimensional, and the exact sequence of Hopf algebras Z q / / U q / / / / u q is cleft.
Proof.This follows from [11,Theorem 33 & Remark 11], see also the discussion in [4, § 4.5].Theorem 5.5 implies the cleftness of the exact sequence.□ Let g be a semisimple Lie algebra and q a root of 1 of odd order, coprime with 3 if g has an ideal of type G 2 .Then there is a suitable q such that U q is isomorphic to the De Concini-Kac-Procesi quantized enveloping algebra U q (g), see [19,20,22].But this class of examples covers also quantum supergroups and more, see [3,4].
where W ≃ V * linearly.Hence for any Γ ≤ k × , we have an extension 5.4.5.Example: (finite) quantum linear spaces.The input for this example is a matrix q = (q ij ) i,j ∈ I θ whose entries are roots of 1 that satisfy Let M i be the least common multiple of {ord q ij : j ∈ I θ }, for every i ∈ I θ .Thus N i divides M i .To this input we attach: • The finite abelian group L = ⟨g 1 ⟩ ⊕ • • • ⊕ ⟨g θ ⟩ where ord g i = M i , i ∈ I θ .
• A vector space V with a basis x 1 , . . .x θ , realized in kL kL YD and kΛ kΛ YD by δ(x i ) = g i ⊗ x i , g Here δ : V ⊗ kL and δ : V ⊗ kΛ are the coactions.• The algebra B = k⟨x 1 , . . ., x θ |x i x j − q ij x j x i , i ̸ = j ∈ I θ ⟩.Then the following facts hold: • B(V Proof.The algebra H is generated by x 1 , . . ., x θ , g ±1 1 , . . ., g ±1 θ with relations x i x j = q ij x j x i , i ̸ = j ∈ I θ , g i x j = q ij x j g i , g i g j = g j g i , i, j ∈ I θ .
The subalgebra A is generated by x N 1 1 , . . ., x N θ θ , g ±N 1 1 , . . ., g ±N θ θ .Now for all i, j ∈ I θ .This implies the Lemma.□ Now the comultiplication of A is determined by Therefore G := Alg(A, k) is isomorphic to B θ , where B is the Borel subgroup of SL (2, k).
In conclusion, for any Γ ≤ B θ , we have an extension Proof.Since the subcoalgebras C(κ) of H(Γ ) are C(ε)-bimodules, then C(κ) F is a subcoalgebra of H(Γ ) F , for all κ ∈ Γ.Notice that (H σ ).σ M κ = HM κ for all κ ∈ Γ. Hence the subcoalgebra C σ (κ) := (H σ /(H σ ).σ M κ ) * of (H σ ) • is contained in (H • ) F and it coincides with the subcoalgebra C(κ) F .Therefore □ Proposition 5.8 has the following application.Let u be a finite-dimensional pointed Hopf algebra.In all known examples, the graded Hopf algebra gr u with respect to the coradical filtration is of the form B(V )#kL as in §5.4 and u ≃ (gr u) σ for a suitable cocycle σ.Then for any pair (H, A) with H Noetherian, A central in H, the extension (E) cleft and H ε ≃ gr u, the pair (H σ , A) has analogous properties and (H ε ) σ ≃ u.
then it is a Hopf ideal, B is a Hopf algebra and the quotient map π : C → B is a morphism of Hopf algebras.(iii) If A + C = CA + and C is a faithfully flat A-module (under left or right multiplication), then A B is exact, π is faithfully coflat and A is normal.The converse in (i) and whether Hopf algebras are faithfully flat over Hopf subalgebras are open questions.

Definition 3 . 4 .
the latter being the restriction of the finite dimensional H-module H ⊗ A W , by assumption.Clearly, the image of W is contained in the A-socle V of H ⊗ A W , which is an object of C , because A is central in H.The faithfulness of the grading (3.7) follows from the fact that F (C κ ) ⊆ A κ , for all κ ∈ G.□ Let Γ ≤ G. Then C Γ denotes the full subcategory of C generated by C κ , κ ∈ Γ , that is the full subcategory of rep H with objects where A acts in a semisimple way by characters in Γ .In other words

3 . 10 )
and since all H κ 's are finite-dimensional.Let κ ∈ Γ , W ∈ rep H κ and let c ∈ C W pκ be a matrix coefficient.Then ⟨c, a⟩ = κ(a)c, for any a ∈ A. Hence

. 1 ) 5 . 7 .
and B are both Hopf algebras in kL kL YD and kΛ kΛ YD.The natural projection π from B to the Nichols algebra B(V ) induces an isomorphism B(V ) ≃ B/⟨x N 1 1 , . . ., x N θ θ ⟩. • The subalgebra of B generated by x N 11 , . . ., x N θ θ coincides with B co π and we have an exact sequence of Hopf algebrasA := B co π #kΛ 0 / / H := B#kΛ π#℘ / / / / K. (5Lemma If N i = M i , for all i ∈ I θ , then A is central in H. is cosemisimple if and only if H ε is semisimple.(iv) If Γ is finitely-generated, then GKdim H(Γ ) < ∞ if and only if Γ is nilpotent-byfinite.(v) H(Γ ) is Noetherian if and only if Γ is polycyclic-by-finite. (vi) If H(Γ ) is Noetherian, then it is regular iff H ε is semisimple.Theorem 1.1 is proved in Section 4, see Remark 4.11 and Theorems 4.
□Remark 2.4.A Hopf algebra C is faithfully flat over a Hopf subalgebra A provided that either of the following conditions hold:

Hopf center and Hopf cocenter. Let
is exact and ι is faithfully flat.
Finally we establish some properties of H(Γ ), see § 4.2.First we recall: B be an exact sequence of Hopf algebras with C faithfully coflat as a B-comodule.Then C is co-Frobenius (respectively cosemisimple) if and only if A and B are co-Frobenius (respectively cosemisimple).3.9 applied to the exact sequence(3.11),see Lemma 2.6, gives: A grading of rep H.Given V ∈ rep H, we set / kΓ fd .Remark 3.8.The projection H(Γ ) → kΓ fd is cocentral.Proof.This follows from the centrality of A in H. □ π − → → Corollary 3.10.H(Γ ) is co-Frobenius (respectively cosemisimple) if and only if H • ε is co-Frobenius (respectively cosemisimple).
As in the previous Section we fix a Noetherian Hopf algebra H with a central Hopf subalgebra A; thus we have an exact sequence of Hopf algebras (E): A 4. Finite-by-cocommutative Hopf algebras 4.1.Discussion of the assumptions.ι → H pε ↠ H ε .Let G = Alg(A, k).4.1.1.Assumptions on H ε .Consider the following assumptions: Assumption 4.1.H is a finitely generated A-module, say (h i ) i ∈ I N generate H over A. Question 4.4.Does Assumption 4.2 imply Assumption 4.1 always?Example 4.5.Assumption 4.2 does not imply cleftness of (E); see the example by Oberst and Schneider in [42, Section 3].If (4.1) holds, then the subcoalgebra C(κ) of the finite dual H • defined in (3.9) is identified with the finite-dimensional coalgebra H * κ , via the injective map of coalgebras p t κ : H A is Noetherian if and only if it is affine.Γ ) is cleft.The second claim follows from Theorem 2.1 and Example 2.2.
[40,ecall that Γ , being a subgroup of an affine algebraic group, is linear.Since Regularity.A reference for this Subsection is[40, Chapter 7].As in loc.cit.weuse the following abbreviations: pd stands for projective dimension; r. gldim, l. gldim, gldim stand for right global dimension, respectively left global dimension, and global dimension.Given an algebra R, if r. gldim R = l.gldimR then we set gldim R = l.gldimR; we say that R is regular if gldim R < ∞.B where we assume that dim A < ∞.First, we record Lemma 4.24.In the situation above, if A is semisimple and B is regular, then C is regular and gldim C ≤ gldim B.Nicolás Andruskiewitsch et al.Now any finite-dimensional Hopf algebra is Frobenius, thus either A is semisimple, or gldim A = ∞.The latter case is dealt with the following result, that relies on a theorem by Schneider and a well-known argument.