Some isomorphism results for graded twistings of function algebras on finite groups

We provide isomorphism results for Hopf algebras that are obtained as graded twistings of function algebras on finite groups by cocentral actions of cyclic groups. More generally , we also consider the isomorphism problem for finite-dimensional Hopf algebras fitting into abelian cocentral extensions. We apply our classification results to a number of concrete examples involving special linear groups over finite fields, alternating groups and dihedral groups.


Introduction
Hopf algebras are useful and far-reaching generalizations of groups. In the semisimple (hence finite-dimensional) setting, the framework that is the closest from the one of finite groups, all the known examples arise from groups via a number of sophisticated constructions, and a general fundamental question [2,Problem 3.9] is whether any semisimple Hopf algebra is "group-theoretical" in an appropriate sense. An answer to the above question, positive or not, still would leave open the hard problem of the isomorphic classification of the "group-theoretical" Hopf algebras. This paper proposes contributions to this classification problem, mainly for the class of Hopf algebras that are obtained as graded twisting of function algebras of finite groups.
The graded twisting of Hopf algebras, which differs in general from the familiar Hopf 2-cocycle twisting construction [9], was introduced in [4], and is the formalization of a construction in [26] that solved the quantum group realization problem of the Kazhdan-Wenzl categories [18]. The initial data is that of a graded Hopf algebra A, acted on by a group Γ. The resulting twisted Hopf algebra then has a number of pleasant features related to those of initial one. Among those features, the following one [4,5] is of particular interest: if A = O(G) is the coordinate algebra on a linear algebraic group G and Γ has prime order, then all the noncommutative quotients of the graded twisted Hopf algebra again are graded twist of O(H), for a wellchosen "admissible" closed subgroup H ⊂ G. This applies in particular to O −1 (SL 2 (C)), whose noncommutative quotients have been discussed and classified in [3,27]. The results in [4,5], however, leave open the question of the isomorphic classification of the Hopf algebras that are obtained by graded twisting, and this is precisely the problem that we discuss in this paper.
We prove 3 isomorphism results for graded twisting of Hopf algebras of functions on finite groups. These results all have in common strong cohomological assumptions on the underlying group, which we believe to be difficult to overcome to obtain general results, but yet are broad enough to cover a number of interesting cases. Namely, we obtain classification results for Hopf algebras that are graded twists of (1) O(SL n (F q )) by Z m , where q is a power of a prime number, m = gcd(n, q − 1) is prime and (n, q) ∈ { (2,9), (3,4)} (see Theorem 5.2); (2) O( A n ) by Z 2 , where A n is the unique Schur cover of the alternating group A n , with n = 6 (see Theorem 5.4); (3) O( S n ) by Z 2 , where S n is any of the two Schur covers of the symmetric group S n , with n = 6 (see Theorem 5.5).
While the first two isomorphism theorems (Theorem 3.1 and Theorem 3.3) are obtained rather directly and early in the paper (in Section 3), the third one (Theorem 4.27) is obtained by considering the more general problem of the classification of the Hopf algebras fitting into an abelian cocentral extension. This is a classical topic in the field, which has been quite studied and very successful to obtain several classification results [15,20,25].
-502 -Most of our analysis in Section 4 is thus well-know to specialists, but we feel that certain formulations and our focus on extensions that are universal bring some novelty, and we get as applications some results in this framework that seem to be new. Indeed we obtain classification results (i.e. parameterizations by concrete and explicitly known group-theoretical data) for noncommutative Hopf algebras A fitting into an abelian cocentral extension k → O(H) → A → kZ m in the following cases: (1) H = PSL 2 (F p ), with p odd prime and m = 2; (2) H = A n , with n = 5 or n 8 and m = 2; (3) H = A 5 , for any m 1; (4) H = S n , with n = 6 and m = 2; (5) H = D n , the dihedral group of order 2n with n odd and m 1; (6) H = D n with n even, with the above extension universal and m = 2; (7) H = Z p × Z p with p an odd prime and m a power of a prime such that m|(p − 1).
Among those examples, it is interesting to note that the one with D n and n even is certainly the most intricate one, and does not follow from a general result, although the structure of this group is certainly not the richest one.
The paper is organized as follows. Section 2 consists of reminders and preliminaries. In Section 3 we provide our first two isomorphism results for graded twistings of function algebras on finite groups. Section 4 deals with general abelian cocentral extensions and provides our third isomorphism result for graded twistings. The final Section 5 discusses applications of the previous results to the concrete examples mentioned above.

Notation and conventions
We work over a fixed base field k, that we assume to be algebraically closed and of characteristic zero. We assume familiarity with the theory of Hopf algebras, for which [24] is a convenient reference, and we adopt the usual conventions: for example ∆, ε and S always respectively stand for the comultiplication, counit and antipode of a Hopf algebra, and we will use Sweedler's notation in the standard manner. A slightly less usual convention is that we will assume that Hopf algebras have bijective antipode. We also assume some familiarity with basic homological algebra, for which [12,14] are convenient references, and in particular we will use [14] as a reference for Schur multiplier computations. Other specific notations will be introduced throughout the text.

Preliminaries
This section consists of reminders about cocentral Hopf algebra maps, cocentral gradings, and the graded twisting construction. It also provides a number of simple but useful preliminary results.

Cocentral Hopf algebra maps, cocentral gradings
The concept of cocentral Hopf algebra map is dual to the familiar one of central algebra map. The precise definition is as follows [1], see [6,7] for extensive discussions on these notions. (1) A Hopf algebra map p : A → B is said to be cocentral if for any a ∈ A, we have p(a (1) ) ⊗ a (2) = p(a (2) ) ⊗ a (1) . (1) If p : A → B is a cocentral surjective Hopf algebra map, then B is necessarily cocommutative. (2) Given a Hopf algebra A, the existence of a universal cocentral Hopf algebra map A → B is easily shown as follows. Consider X, the linear subspace of A spanned by the elements ϕ(a (1) )a (2) − ϕ(a (2) )a (1) , ϕ ∈ A * , a ∈ A.
It is easy to see that X is a co-ideal in A, and then the ideal I generated by X is a Hopf ideal in A. The quotient Hopf algebra map p : A → A/I is then universal cocentral. Uniqueness of the universal cocentral Hopf algebra map is obvious from the definition. -504 -(4) If a Hopf algebra A is cosemisimple, it is not difficult to see, using the Peter-Weyl decomposition of A (decomposition of A into direct sum of matrix subcoalgebras), that A has a universal grading group.
The following lemma will be used several times in the text.
be Hopf algebras having the same universal finite cyclic grading group Γ 0 and suppose given two surjective cocentral Hopf algebra maps p : A → kΓ and q : B → kΓ for some finite cyclic group Γ, and a Hopf algebra isomorphism f : Proof. -Let p 0 : A → kΓ 0 and q 0 : B → kΓ 0 be the universal cocentral Hopf algebra maps. The Hopf algebra map q 0 • f : A → kΓ 0 being cocentral, there exists a unique group morphism v : Since q 0 • f is surjective, so is v and hence v is an automorphism since Γ 0 is finite. The Hopf algebra maps p : A → kΓ and q : B → kΓ being cocentral and surjective, the universality of p 0 and q 0 yields surjective group morphisms w, w : Γ 0 → Γ such that w • p 0 = p and w • q 0 = q. Let N = Ker(w) and N = Ker(w ). We have |N | = |Γ0| |Γ| = |N |, hence the uniqueness of a subgroup of given order in a finite cyclic group yields N = N = v(N ), and there exists a unique group morphism u : Γ → Γ such that u•w = w •v: Notice that the conditions 1 ∈ A e and S(A g ) ⊂ A g −1 follow from the other ones. Cocentral gradings by Γ correspond to cocentral Hopf algebra maps p : A → kΓ. Indeed, given a cocentral Hopf algebra map p : A → kΓ, the corresponding grading is defined by (1) )}.
-505 -We occasionally denote the set A g by A g,p to specify the dependence on p, in case there is a risk of confusion. Conversely, given a cocentral grading by Γ, the cocentral Hopf algebra map p : A → kΓ is defined by p |Ag = ε( · )g, and is surjective if and only if A g = {0} for any g ∈ Γ. We will freely circulate from cocentral Hopf algebra maps to cocentral gradings.
An important property of the cocentral gradings, provided that the corresponding cocentral Hopf algebra map is surjective, is that they are strong: for any g, h ∈ Γ, we have A g A h = A gh (see e.g. [5,Proposition 2.2] for this well-known fact). Here is a useful application, used in the proof of the forthcoming Lemma 2.13.
Lemma 2.5. -Let p : A → kΓ be a cocentral surjective Hopf algebra map. Let g ∈ Γ and let y, z ∈ A be such that xy = xz for any x ∈ A g . Then y = z.

Cocentral actions and graded twisting
The following notion is introduced in [4] under the name "invariant cocentral action". In the present paper, to simplify terminology, we will simply say "cocentral action". Definition 2.6. -A cocentral action of a group Γ on a Hopf algebra A consists of a pair (p, α) where p : A → kΓ is a surjective cocentral Hopf algebra map and α : Γ → Aut Hopf (A) is a group morphism, together with the compatibility condition p • α g = p for any g ∈ Γ.
In the graded picture, the compatibility condition is α g (A h ) = A h for any g, h ∈ Γ.
Definition 2.7. -Given a cocentral action (p, α) of a group Γ on a Hopf algebra A, the graded twisting A p,α is the Hopf algebra having A as underlying coalgebra, and whose product and antipode are defined by The present definition of a graded twisting differs from the original one in [4], but is equivalent to it: see [5,Remark 2.4], the underlying algebra structure is that of a twist in the sense of [29].
-506 -Lemma 2.8. -Let q : A → B be a universal cocentral Hopf algebra map and let (p, α) be a cocentral action of a group Γ on A. Then q : A p,α → B still is a universal cocentral Hopf algebra map.
Proof. -Recall from Remark 2.2 that we can assume that q is the quotient map A → A/I where I is the ideal of A generated by X, the linear subspace of A spanned by the elements ϕ(a (1) )a (2) − ϕ(a (2) )a (1) , ϕ ∈ A * , a ∈ A. The space X is as well the linear subspace of A spanned by the elements Let I be the ideal of A p,α generated by X. The computation, for shows that I ⊂ I. In a symmetric manner, since ab = a · α g −1 (b) for a ∈ A g and b ∈ A, we have I ⊂ I and hence I = I . Therefore the quotient map q : A p,α → A p,α /I , which is universal cocentral, equals q, and we have our result.
Since our main goal is to compare the different Hopf algebras obtained via graded twisting, an obvious thing to do first is to compare the various cocentral actions, and for this the following notion is quite natural. Definition 2.9. -Two cocentral actions (p, α) and (q, β) of a group Γ on a Hopf algebra A are said to be equivalent if there exist u ∈ Aut(Γ) and f ∈ Aut Hopf (A) such that Lemma 2.10. -Let (p, α) and (q, β) be cocentral actions of a group Γ on a Hopf algebra A. If (p, α) and (q, β) are equivalent, then the Hopf algebras A p,α and A q,β are isomorphic.
Proof. -Let u ∈ Aut(Γ) and f ∈ Aut Hopf (A) as in the above definition. The condition u • p = q • f ensures that f (A g ) = A u(g) for any g ∈ Γ. Hence for a ∈ A g and b ∈ A, we have which shows that f is as well a Hopf algebra isomorphism from A p,α to A q,β .
There is also another weaker notion of equivalence for cocentral actions, as follows.
-507 -Definition 2.11. -Two cocentral actions (p, α) and (q, β) of a group Γ on a Hopf algebra A are said to be weakly equivalent if there exist u ∈ Aut(Γ) and a Hopf algebra isomorphism f : A e,p → A e,q such that Not surprisingly, equivalent cocentral actions are weakly equivalent. Proof. -Let u ∈ Aut(Γ) and f ∈ Aut Hopf (A) be such that u = A e,q , and the conclusion follows.
It is unclear to us whether the existence of a Hopf algebra isomorphism between A p,α and A q,β forces the cocentral actions (p, α) and (q, β) to be weakly equivalent. However this holds true in the following special situation.
Lemma 2.13. -Let A be a commutative Hopf algebra having a finite cyclic universal grading group, and let (p, α), (q, β) be cocentral actions of a cyclic group Γ on A. If the Hopf algebras A p,α and A q,β are isomorphic, then the cocentral actions (p, α) and (q, β) are weakly equivalent.
Proof. -Let f : A p,α → A q,β be a Hopf algebra isomorphism. By Lemma 2.8, we can apply Lemma 2.3 to get u ∈ Aut(Γ) such that u•p = q•f , so that f (A g,p ) = A u(g),q for any g ∈ Γ, and in particular f (A e,p ) = A e,q . For a ∈ A g and b ∈ A e , we have By the commutativity of A, we have as well We conclude from Lemma 2.5 that β u(g) (f (b)) = f (α g (b)), so our cocentral actions are indeed weakly equivalent.

Graded twisting of function algebras
In this subsection we translate in group theoretical terms the notions discussed in the previous subsections when A = O(G), the function algebra on a finite group G (this of course runs as well if we assume that G is a linear algebraic group, but for simplicity we restrict to the finite case). The translations are rather obvious, convenient, and induce a few new notations. As usual, if Γ is group, the dual group Hom(Γ, k × ) is denoted Γ. If G is a group and T ⊂ G is a subgroup, we denote by Aut T (G) the group of automorphisms of G that preserve T , and by Aut • T (G) the subgroup of automorphisms that fix each element of T .
(1) A cocentral action (p, α) of the finite group Γ on O(G) corresponds to a pair (i, α) where i : Γ → Z(G) is an injective group morphism and α : Γ → Aut • i( Γ) (G) a group morphism. We then consider cocentral actions of Γ on O(G) as such pairs (i, α), call them cocentral actions on G, and denote the corresponding graded twisting O(G) p,α by O(G) i,α .
Assuming that the finite group G has a cyclic center, there is a convenient way to describe the equivalence classes of cocentral actions of Z m on G, as follows.
For m a divisor of |Z(G)|, let T m be the unique subgroup of order m of Z(G), and let X m (G) be the set of elements α 0 ∈ Aut • Tm (G) such that α m 0 = id G modulo the equivalence relation For α 0 ∈ Aut • Tm (G), we denote byα 0 its equivalence class in X m (G). We will also denote by X • m (G) the set of equivalence classesα 0 such that α 0 does not induce the identity on G/T m . Proof. -Fix a generator g of Z m and an injective group morphism i : , and associate to α 0 ∈ Aut • Tm (G) the cocentral action (i, α) of Z m on G with α g = α 0 . It is clear that for α 0 , β 0 ∈ Aut • Tm (G), we haveα 0 =β 0 if and only if the cocentral actions (i, α) and (j, β) are equivalent, so we get the announced injective map.
Start now with a cocentral action (j, β) of Z m on G. Let u be the automorphism of Z m defined by u = i −1 • j: u = ( · ) l for l prime to m. For l such that ll ≡ 1 [n], we then see that the cocentral actions (j, β) and (i, β l ) are equivalent, and this proves that our map is surjective.

Group-theoretical preliminaries
This last subsection consists of group theoretical preliminaries. As usual, if G is a group and M is a G-module, the second cohomology group of G with coefficients in M is denoted H 2 (G, M ). We mainly consider trivial Gmodules (the only exception is in the proof of Lemma 4.3). If τ ∈ Z 2 (G, M ) is a (normalized) 2-cocycle, its cohomology class in Our first lemma is certainly well-known. We provide the details of the proof for future use. where the map on the right is surjective when |H 2 (G/T, T )| 2 (or more generally when the natural actions of Aut(G/T ) and Aut(T ) on H 2 (G/T, T ) are trivial).
Proof. -Since any element in Aut T (G) simultaneously restricts to an automorphism of T and induces an automorphism of G/T , we get the group morphism on the right. Given χ ∈ Hom(G/T, T ), define an automorphism χ of G by χ(x) = xχ(π(x)), where π : G → G/T is the canonical surjection. This defines the group morphism Hom(G/T, T ) → Aut T (G) on the left, which is clearly injective and whose image is easily seen to be the kernel of the map on the right.
Put H = G/T . By the standard description of central extensions of groups, we can freely assume that G = H × τ T where τ ∈ Z 2 (H, T ) and the product of G is definedby ∀ x, y ∈ H, ∀ r, s ∈ T, (x, r) · (y, s) = (xy, τ (x, y)rs).
( ) Under this identification, the composition law in Aut T (G) is given by The map Aut T (G) → Aut(H) × Aut(T ) in the statement of the lemma is then the projection on the first and third factor, and elements of the kernel are exactly those of the form (id H , µ, id T ), where µ : H → T is a group morphism.
Assume now the natural actions of Aut(H) and Aut(T ) on H 2 (H, T ) by group automorphisms are trivial (which obviously holds when |H 2 (H, T )| 2). Let (θ, u) ∈ Aut(H) × Aut(T ). The cocycles u • τ and τ • (θ × θ) are then cohomologous to τ , and hence there exists µ : , which is exactly the condition ( ) that allows (θ, µ, u) to define an element of Aut T (G), and thus the map on the right in our exact sequence is surjective.
Our second lemma will be used at the end of Section 4.
Then there exists f ∈ Aut T (G) such that Proof. -Recall from the proof of the previous lemma that the elements of Aut T (G) are represented by triples (θ, µ, u) with θ ∈ Aut(H), u ∈ Aut(T ) and µ : and β l have the same image under the group morphism on the right in the previous lemma, and the assumption We have moreover f |T = u = ( · ) l , and this finishes the proof.
To finish this section, we record a last lemma, again to be used in Section 4. It is well known that inner automorphisms act trivially on the second cohomology of a group. Our next lemma is an explicit writing of this fact. The proof is a straightforward verification, but can also be obtained easily from the considerations in the proof of Lemma 2.15.

First results
We are now ready to state and prove our first two isomorphism results for graded twistings of function algebras on finite groups.  Denote by f → f the group morphism Aut i( Γ) (G) → Aut(H) of Lemma 2.15. Fix a generator g ∈ Γ and assume the existence of f ∈ Aut(H) and We . The condition Hom(H, Γ) = {1} and Lemma 2.15 then ensure that f −1 , and we conclude that the cocentral actions (i, α) and (j, β) are equivalent.

Example 3.2. -Let p
3 be a prime number. There are exactly two non-isomorphic non-trivial graded twistings of O(SL 2 (F p )). The details will be given in Section 5.
The previous theorem has the following very convenient consequence when Γ = Z 2 .  Proof. -The universal coefficient theorem provides the following exact sequence The assumption Hom(H, Z 2 ) = {1} implies that H H 1 (H) has odd order, so the group on the left vanishes. Moreover H 2 (H) H 2 (H, k × ) (again by the universal coefficient theorem), so the cyclicity of H 2 (H, k × ) yields that |H 2 (H, Z 2 )| 2, and we can apply Theorem 3.1.

Abelian cocentral extensions of Hopf algebras
To go beyond Theorem 3.1, it will be convenient to work in the more general framework of abelian cocentral extensions. As already said in the introduction, this is a very well studied and understood framework [1,15,19,20,22,25] (even in more general situations, dropping the cocentrality assumption), but we propose a detailed exposition of the structure of Hopf algebras fitting into abelian cocentral extensions, both for the sake of selfcompleteness and of introducing the appropriate notations, and also because we think that some of our formulations have some interest.

Generalities
We recall first the concept and the structure of the Hopf algebras arising from abelian cocentral extensions. There is a general notion of exact sequence of Hopf algebras [1], but in this paper we will only need the cocentral ones.
When B is commutative, a cocentral exact sequence as above is called an abelian cocentral extension.
is cocentral abelian extension, as well as Hence graded twists of function algebras fit into appropriate abelian cocentral extensions.
We now restrict ourselves to finite dimensional Hopf algebras. In this case the abelian cocentral extensions are of the form for some finite groups H, Γ. There are some general descriptions of the Hopf algebras A fitting into such abelian cocentral extensions using various actions and cocycles (see [1,22]). Since we only will consider the case when Γ is cyclic, there is an even simpler description, inspired by [20], that we give now. We start with a lemma.   (H, θ, a). The defining relations give that for any φ ∈ O(H), we have gφ = (φ • θ −1 )g, and since a = g m must be central, we see from this that if the above set is linearly independent, we have a • θ = a.
To prove the converse, we recall a general construction. Let R be a commutative algebra endowed with an action of a group Γ, α : Γ → Aut(R), and let σ : Γ × Γ → R × be a 2-cocycle according to this action: The crossed product algebra R# σ kΓ is then defined to be the algebra having R ⊗ kΓ as underlying vector space, and product defined by Assume furthermore that Γ = Z m = g is cyclic, consider an element a ∈ R × that is Z m -invariant, and define the algebra A to be the quotient of the free product R * k[X] by the relations Xb = α g (b)X and X m = a. Since a is invariant under the Z m -action, the classical description of the second cohomology of a cyclic group shows that there exists a 2-cocycle σ : Applying this to R = O(H), the Z m -action on it induced by θ and the assumption that a is invariant yields that {e x g i , x ∈ H, 0 i m − 1} is a linearly independent set since its image is in the crossed product algebra O(H)# σ kZ m .
We now check m-data as above produce Hopf algebras fitting into abelian cocentral extensions, and that any such Hopf algebra arises in this way.
(1) There exists a unique Hopf algebra structure on A m (H, θ, a) extending that of O(H) and such that We denote by A m (H, θ, a, τ ) the resulting Hopf algebra. (2) The Hopf algebra A m (H, θ, a, τ ) has dimension m|H| and fits into an abelian cocentral extension where p is the Hopf algebra map defined by p |O(H) = ε and p(g) = g (here g denotes any fixed generator of Z m ).
Proof. -It is a straightforward verification, using the axioms of mdata, that there indeed exists a Hopf algebra structure on A m (H, θ, a) as in the statement. That A m (H, θ, a, τ ) has dimension m|H|, follows from Lemma 4.3, while the last statement follows easily from the decomposition Then there exists an m-datum (H, θ, a, τ ) such that A A m (H, θ, a, τ ) as Hopf algebras.
Proof. -To simplify the notation, we will identify O(H) with its image in A, so that A e = O(H). The finite-dimensionality assumption ensures that the extension is cleft (see e.g. [23,Theorem 3.5] or [28,Theorem 2.4]). Here this simply means that for any h ∈ Z m , there exists an invertible element u h in A h , that we normalize so that ε(u h ) = 1, and hence p(u h ) = h, where p : A → kZ m is the given cocentral surjective Hopf algebra map. We have Fix now a generator g of Z m and u g as above. We have u m g ∈ A g m = A e , and we put a = u m g . Since ∆(A g ) ⊂ A g ⊗ A g we have ∆(u g ) = x,y∈H τ (x, y)e x u g ⊗ e y u g for scalars τ (x, y) ∈ k, these scalars all being non-zero since ∆(u g ) is invertible. The coassociativity and counit conditions give that the map τ : H × H → k × defined in this way is a 2-cocycle. We have u g A e u −1 g ⊂ A e and hence we get an automorphism α := ad(u g ) of the algebra A e , satisfying α m = id since u m g ∈ A e and A e is commutative. It is a direct verification to check that α is as well a coalgebra automorphism, and hence a Hopf algebra automorphism of A e = O(H), necessarily arising from an automorphism θ of H, Clearly α(a) = a, ε(a) = 1, and one checks that the last condition defining an m-datum is fulfilled by comparing ∆(u g ) m and ∆(a). We thus obtain an m-datum (H, θ, a, τ ) and it is straightforward to check that there exists a Hopf algebra map A m (H, θ, a, τ ) Combining Lemma 4.3 and the first paragraph in the proof, we see that this is an isomorphism.

Equivalence of m-data and the isomorphism problem
The main question then is to classify the Hopf algebras A m (H, θ, a, τ ) up to isomorphism. For this, the following equivalence relation on m-data will arise naturally.
It is not completely obvious that the above relation is an equivalence relation, but this follows from the following basic result, which is a partial answer for the classification problem of the Hopf algebras A m (H, θ, a, τ ). (1) There exists a Hopf algebra isomorphism F : A m (H, θ, a, τ ) → A m (H , θ , a , τ ) and a group automorphism u ∈ Aut(Z m ) making the following diagram commute: m-data (H, θ, a, τ ) and (H , θ , a , τ ) are equivalent.
Proof. -Assume that F and u as above are given, and put A = A m (H, θ, a, τ ) and B = A m (H , θ , a , τ ). The commutativity of the diagram yields, at the level of gradings, that F (A h ) = B u(h) for any h ∈ Z m . Then  (1) Let f ∈ Aut(H) and let l 1 be prime to m.
Proof. -The first assertion is easily obtained via the previous proposition. For the second one, let µ : H → k × be such that τ = τ ∂(µ). The result is again a direct consequence of the previous proposition, taking a = a  -518 -Hence, by Corollary 4.9, the m-datum (H, θ, a, τ ) is equivalent to an mdatum (H, θ, a , τ ) with a ∈ H. Such a datum with a ∈ H will be said to be normalized. It is therefore tempting to work only with normalized data, but this forces to change the cocycle for each choice of automorphism θ, and can be inconvenient in practice if we have "nice" representatives for 2-cocycles over H. We will therefore work with the general notion of an m-datum, as given in Definition 4.4.  [13] can be described as the set of pairs (a, τ ) ∈ H × Z 2 (H, k × ) such that (H, θ, a, τ ) is a normalized m-datum modulo the equivalence relation defined by (a, τ ) ∼ (a , τ ) ⇐⇒ ∃ ϕ : The group law is by the ordinary multiplication on the components. The group Opext θ (kZ m , O(H)) is known to be possibly difficult to compute (see [21], and [11] for a recent contribution), hence the problem of the description of m-data up to equivalence is a fortiori a non-obvious one as well.
Proposition 4.8 is in general not sufficient to classify the Hopf algebras A m (H, θ, a, τ ) up to isomorphism. However, in the context of Lemma 2.3, it can be sufficient. Thus we need to analyse furthermore the Hopf algebras A m (H, θ, a, τ ) to determine when Lemma 2.3 is applicable. For this we introduce a number of groups associated to an m-datum.
We now proceed to analyse the structure of the Hopf algebras A m (H, θ, a, τ ), with first the following basic result.
(1) The Hopf algebra A m (H, θ, a, τ ) is commutative if and only if θ = id H . More generally, the abelianisation of A m (H, θ, a, τ ) is the Hopf algebra O (G(H, θ, a, τ )). Proof. -The assertions regarding the commutativity or cocommutativity of A m (H, θ, a, τ ) are easily seen using Lemma 4.3. An algebra map χ : A m (H, θ, a, τ ) → k corresponds to a pair (x, λ) ∈ H × k × , with χ(φ) = φ(x) for any φ ∈ O(H) and φ(a) = λ. The compatibility of χ with the defining relations of A m (H, θ, a, τ ) is easily seen to be equivalent to the condition that (x, λ) ∈ G (H, θ, a, τ ), and an immediate calculation shows that the group law in Alg (A m (H, θ, a, τ ), k) corresponds to the group law in G(H, θ, a, τ ). Thus the abelianization of A m (H, θ, a, τ ), which is the algebra of functions on Alg (A m (H, θ, a, τ ), k), is isomorphic to O (G(H, θ, a, τ )).
We now discuss when the universal grading group of A m (H, θ, a, τ ) is cyclic.

a, τ ) is cyclic and the restriction of θ to Z τ,θ (H) is trivial (resp. if the group Z τ,θ (H) is trivial).
We get our most useful result for the classification of Hopf algebras of type A m (H, θ, a, τ ).    (H, θ, a, τ ) → A m (H, θ, a, τ ) induces a bijection between the following sets: (1) equivalence classes of cyclic (resp. reduced) m-data having H as underlying group; (2) isomorphism classes of Hopf algebras A fitting into an abelian cocentral extension and having a cyclic universal grading group (resp. having Z m as universal grading group).

Classification results
We now apply Theorem 4.18 and Corollary 4.19 to obtain effective classification results for Hopf algebras fitting into abelian cocentral extensions, under various assumptions.
The set of equivalence classes of m-data has a very simple description under some strong assumptions on H, and then the previous result takes the following simple form, where we use the following notation: if G is a group and m 1, the set CC • m (G) is the set of elements of G such that x m = 1 and x = 1, modulo the equivalence relation defined by x ∼ y ⇐⇒ there exists l prime to m such that x l is conjugate to y. When m = 2, CC • 2 (G) is just the set of conjugacy classes of elements of order 2 in G. The key point, to be used freely, is that, since H 2 (H, k × ) Z 2 , for any θ ∈ Aut(H) and τ ∈ Z 2 (H, k × ), we have First assume that m is odd. Let (H, θ, a, τ )  Assume now that m is even, and start with a pair (θ, τ ) where θ ∈ Aut(H) satisfies θ m = id θ = id, and τ ∈ Z 2 (H, k × ). The assumption The assumption H = {1} implies that such a map a is unique and satisfies a • θ = a, so to (θ, τ ) we can unambiguously associate an m-datum (H, θ, a, τ ). Conversely if θ = f • θ l • f −1 , for f ∈ Aut(H) and l prime to m, then we have, by Corollary 4.9 The cocycle on the right is cohomologous to τ l , hence to τ , and if we assume that τ is cohomologous to τ , we have (again thanks to Corollary 4.9) (H, θ, a, τ ) ∼ (H, θ , b, τ ) ∼ (H, θ , c, τ ) for some maps b, c, with necessarily c = a by the discussion at the beginning of the proof. This concludes the proof.
Another useful consequence of Theorem 4.18 is the following one, again under strong assumptions.

, there is a bijection between the set of isomorphism classes of noncommutative Hopf algebras A fitting into an abelian cocentral extension
Proof. -Corollary 4.19 ensures that we have a bijection between the set of isomorphism classes of noncommutative Hopf algebras as above and the set of equivalence classes of m-data (H, θ, a, τ ) with θ m = id. Then, since H 2 (H, k × ) = {1}, Corollary 4.9 ensures that all such data are equivalent to data of type (H, θ, a, 1) (hence with a ∈ H). Now using that | H| 2, so that Aut(H) acts trivially on H, we see that two m-data (H, θ, a, 1) and  (H, θ , a , 1) are equivalent if and only if there exists f ∈ Aut(H), ϕ ∈ H and l prime to m such that If m is even, we have ϕ m = 1 and the last condition amounts to a = a (l being then necessarily odd), again since | H| 2. If m is odd, we have ϕ m = ϕ, and such a ϕ always exists if l does. This concludes the proof.
To prove our next classification result, we will use the following lemma. Proof. -We first assume that our data are normalized: τ · τ • θ × θ = 1 (and a, a ∈ H). Let f ∈ Aut(H) and ϕ : H → k × be such that Writing θ = ad(x) and f −1 = ad(y), we then have xy = yx since Z(H) = {1} and where µ y is as in Lemma 2.17. Hence ϕ = χµ y for some χ ∈ H, and Since | H| 2 and θ is inner, we obtain ϕ · ϕ • θ = µ y · µ y • θ = µ y · µ y • ad(x). For z ∈ H, we have where we have used the fact that our datum is normalized and that xy = yx.
Z 2 ). Then Corollary 4.9 ensures that any 2-data with non-trivial underlying isomorphism is equivalent to one in the list { (H, θ i , a, 1), i = 1, . . . , r, a ∈ H}, {(H, θ i , a, τ i ), i = 1, . . . , r, a ∈ H}. Any two different data inside one of the two sets are not equivalent by Lemma 4.22, while two data taken from the two different sets are easily seen not to be equivalent either. This concludes the proof.

Back to graded twisting
To finish the section, we go back to graded twistings.  : H → µ m such that (H, θ, a, τ ) is an m-datum  and O(G) i,α A m (H, θ, a, τ ).
Proof. -We can assume without loss of generality that G = H × τ0 Z m and that i is the canonical injection. Indeed, consider the isomorphism F : G → H × τ0 Z m making the following diagram commutative where π is the canonical surjection, and i 0 and π 0 denote the canonical injection and surjection. Using the Hopf algebra isomorphism Recall from Subsection 2.4 (particularly the proof of Lemma 2.15) that α = α g has the form α = (θ, µ) with θ ∈ Aut(H) and µ : H → Z m satisfying Define now a map a 0 : H → Z m : We then have Defining then a : H → µ m by a(x) = a 0 (x)(g), we get an m-datum (H, θ, a, τ ) satisfying the announced conditions, and we have to show that For this, first note that the Z m -grading on O(H × τ0 Z m ) i,α is given by -526 -Put, for x ∈ H, Using the product in O(H × τ0 Z m ) i,α , we see that Hence there exists an algebra map A m (H, θ, a, τ ) → O(H × τ0 Z m ) i,α sending e x to e x and g to u g , which is, exactly as in the proof of Proposition 4.6, a Hopf algebra isomorphism.  (H, θ, a, τ ) for some m-datum (H, θ, a, τ ) of graded twist type.

The previous result (and its proof) says that if (i, α) is a cocentral action of Z m on a finite group G, then letting
Conversely, it is not difficult to show that if (H, θ, a, τ ) is an m-datum of graded twist type, then A m (H, θ, a, τ ) is a graded twist of O(H × τ µ m ).
We now use the previous considerations to get another isomorphism result for graded twists of function algebras on finite groups by Z p , where p is a prime number. We start with a lemma. Lemma 4.26. -Let (H, θ, a, τ ) be a p-datum, with p a prime number.
Proof. -We can assume that τ is nontrivial, hence that [τ ] is a generator of H 2 (H, k × ).The group Aut(H) acts on the cyclic group H 2 (H, k × ) by automorphisms, hence there exits l prime to p such that The assumption that we have a p-datum now gives Since p is prime and [τ ] has order p, we get l ≡ 1 [p], and hence [τ ] = [τ •θ×θ] in H 2 (H, k × ).
We arrive at our expected isomorphism result.
-527 - Assume that (1) holds. To prove (2), we can safely assume that G = H × τ0 Z p for a 2-cocycle τ 0 : H × H → Z p and that i and j are the canonical injections. Indeed, recall from the beginning of the proof of Proposition 4.24, of which we retain the notation, that fixing an appropriate isomorphism where i 0 is the canonical injection. The cocentral actions (i, α) and (j, β) then are equivalent if and only if the cocentral actions (i 0 , F αF −1 ) and (i 0 , F βF −1 ) are.
Since A p (H, θ, a, τ ) A p (H, θ , a , τ ), Theorem 4.18, which is applicable by Lemma 2.8, provides a group automorphism f ∈ Aut(H), ϕ : H → k × and l prime to p such that The previous lemma ensures that [ in H 2 (H, k × ). Our assumptions ensure, by the universal coefficient theorem, that H 2 (H, Z p ) Z p and that the natural map H 2 (H, µ p ) → H 2 (H, k × ) is an isomorphism, because of the exact sequence induced by the in H 2 (H, Z p ). Hence by Lemma 2.16 there exists F ∈ Aut(G) such that β l g = F −1 α g F and F | Zp = ( · ) l , therefore means that our cocentral actions are equivalent. The latter set is, by Lemma 2.14, in bijection with X • p (G) (see the end of Subsection 2.3).

Examples
In this section we apply the previous results to examine the examples announced in the introduction.

Special linear groups over finite fields
We begin by examining graded twistings of linear groups over finite fields.
-529 -Theorem 5.1. -Let q = p α , with p 3 a prime number and α 1, and let n 2 be even. There is a bijection between the set of isomorphism classes of noncommutative Hopf algebras that are graded twistings of O(SL n (F q )) by Z 2 and the set X • 2 (SL n (F q )). Proof. -The center of SL n (F q ) is cyclic and has even order, the character group of SL n (F p )/{±1} is trivial, and H 2 (PSL n (F q ), k × ) is always cyclic under our assumptions (see [14,Chapter 7], for example), hence Theorem 3.3 and Remark 4.28 provide the announced bijection.
Theorem 5.2. -Let q = p α , with p a prime number and α 1, let n 2, and assume that m = gcd(n, q − 1) is prime and that (n, q) ∈ { (2,9), (3,4)}. Then there is a bijection between the set of isomorphism classes of noncommutative Hopf algebras that are graded twistings of O(SL n (F q )) by Z m and the set X • m (SL n (F q )). Proof. -The center of SL n (F q ) is µ n (F q ) and is cyclic of order m = gcd(n, q −1), the group Hom(PSL n (F p ), Z m ) is trivial, and we have moreover Z m under our assumptions (see [14,Chapter 7], for example). Hence Theorem 4.27 and Remark 4.28 provide the announced bijection.
In the case n = 2, we have results for abelian cocentral extensions as well.  Proof. -Theorem 5.1 ensures that there is a bijection between the set isomorphism classes of noncommutative Hopf algebras that are graded twistings of O(SL 2 (F p )) and X • 2 (SL 2 (F p )). All the automorphisms of SL 2 (F p ) are obtained by conjugation of a matrix in GL 2 (F p ) (see e.g. [8]), and we see that there are two equivalence classes of elements in X • 2 (SL 2 (F p ))), represented by the automorphisms where λ is a chosen element such that λ ∈ (F * p ) 2 . This proves the first assertion.

Alternating and symmetric groups
We now discuss examples involving alternating and symmetric groups. We begin with alternating groups and their Schur covers (see e.g. [14]). Proof. -In all cases Z 2 ⊂ Z( A n ), the center Z( A n ) is cyclic, the group H 2 (A n , k × ) is cyclic (isomorphic to Z 6 for n = 6, 7 and to Z 2 otherwise) and we have Hom(A n , Z 2 ) = {1}, so the first statement is a direct consequence of Theorem 3.3. We have CC • 2 (Aut(A n )) = CC • 2 (Aut(S n )), and when n = 6 this coincides with CC • 2 (S n ), which has n 2 elements.
For n = 5 or n 8, we have moreover H 2 (A n , k × ) Z 2 , and A n = {1}, and since Z(A n ) = {1}, the statement follows from Theorem 4.20. Proof. -Every automorphism of S n is inner when n = 6, and we have S n Z 2 H 2 (S n , k × ), so the first assertion follows from Theorem 4.23.
Let G be a group as in the statement. By Proposition 4.24, a graded twisting of O(G) is isomorphic to A 2 (S n , θ, a, τ ) for a cocycle τ : S n × S n → Z 2 canonically build from the central extension 1 → Z 2 → G → S n → 1. Hence Lemma 4.22 ensures that there are at most 2 n 2 isomorphism classes of noncommutative graded twistings of O(G).
Conversely, start with a 2-datum (S n , θ, a, τ ), with τ as before. We wish to prove that A 2 (S n , θ, a, τ ) is isomorphic to a graded twist of O(G). By Lemma 2.17, since any automorphism of S n is inner, there exists µ : S n → µ 2 such that τ · τ • θ × θ = ∂(µ). Then a −1 and µ differ by an element of S n , and hence a 2 = 1. Our 2-datum (S n , θ, a, τ ) is then of graded twist type as in Remark 4.25, and then we know that A 2 (S n , θ, a, τ ) is a graded twist of O(S n × τ µ 2 ) O(G). This concludes the proof.

The alternating group A 5
Examples with the alternating group A 5 fall into the series studied in the last two subsections, but there is a special interest in A 5 , because of the following result from [3]: any finite-dimensional cosemisimple Hopf algebra Of course, the above theorem does not give any information about the realizability of one of the above Hopf algebras as Hopf algebras having a faithful 2-dimensional comodule.

Dihedral groups D n
In this subsection we discuss Hopf algebras fitting into an abelian cocentral extension with D n the dihedral group of order 2n. While the group structure of D n is certainly less rich than the one of the groups of the previous sections, the situation with Hopf algebra extensions as above is in fact much more involved.

Notation
As usual, the group D n is presented by generators r, s and relations r n = 1 = s 2 , sr = r n−1 s, and its automorphisms all are of the form Ψ k,l , (k, l) ∈ Z/nZ × U (Z/nZ), with Ψ k,l (r) = r l , Ψ k,l (s) = sr k .

The case when n is odd
Here the situation is very simple, since we are in the situation of Corol- (1) If n = p r with p odd prime and r 1, then e n = 2.
(2) If n = p r q s , with p, q distinct odd primes and r, s 1, then e n = 6.
Proof. -The previous statement ensures that e n is twice the number of conjugacy classes of elements of order 2 in Aut(D n ), that we compute in the above two cases. In the first case there is precisely one such conjugacy class, represented by Ψ 0,−1 . In the second situation, fix integers a, b such that p r a + q s b = 1, and such that a, b become invertible in Z/p r q s Z. One checks that there are 3 conjugacy classes of elements of order 2 in Aut(D p r q s ), represented by Ψ 0,−1 , Ψ 0,2q s b−1 and Ψ 0,2p r a−1 .
Remark 5.8. -For n = 3, the two non-isomorphic Hopf algebras of the previous theorem are the two non-isomorphic noncommutative and noncocommutative Hopf algebras of dimension 12, classified by Fukuda [10].

The case when n is even
We now assume, throughout the subsection, that n is even. None of our previous classification results apply here and we have to perform a specific analysis. We obtain a pretty satisfactory result in Table 5.1, which, on the other hand, indicates that, in full generality, it is probably hopeless to get compact classification results, such as in Theorems 4.20, 4.21, 4.23.
Proof. -This is a direct verification, using the well-known fact that β is trivial if and only if there exists an algebra map k β D n → k, where k β D n is the twisted group algebra. Such an algebra map then furnishes a map µ with β = ∂(µ).
We now exhibit a convenient explicit non-trivial 2-cocycle over D n .
Proof. -It is a straightforward verification that τ ω is a 2-cocycle, and the triviality condition follows from Lemma 5.9. The last assertion follows from the previous one and the fact that H 2 (D n , k × ) Z 2 .
We now proceed to describe the possible 2-data over D n . We begin with a preliminary lemma.
Let ω ∈ k × with ω = −1 if n/2 is odd, and with ω a primitive nth root of unity if n/2 is even. Let τ ω ∈ Z 2 (D n , k × ) be the non trivial cocycle of Lemma 5.10. Let x, y ∈ k × be such that x n = 1, y 2 = ω u and x,y ), and any map satisfying this identity is of the form a ±x,±y . Moreover we have a x,y • Ψ u,v = a x,y if and only if x u = 1 = x v−1 .
Assume furthermore that Ψ u,v has order 2. Then a x,y • Ψ u,v = a x,y if and only if we are in one of the following situations.
(2) n/2 is odd, u is odd, x = 1 and y = ±ξ, with ξ a primitive fourth root of unity. (3) n/2 is even, u is even, v 2 = 1 + kn, u(1 + v) = ln with k, l even, and x = ±ω , y = ±ω u 0 , with ω 2 0 = ω. (4) n/2 is even, u is odd, v 2 = 1 + kn, u(1 + v) = ln with k, l even, and , y = ±ω u 0 , with ω 2 0 = ω. (5) n/2 is even, u is odd, v 2 = 1 + kn, u(1 + v) = ln with k even and l odd, and is necessarily trivial since H 2 (D n , k × ) has order 2, and Lemma 5.9 yields the identity x,y ). Any map D n → k satisfying the previous identity differs from a x,y by the multiplication of an element in D n , and hence is of the form a ±x,±y . The previous lemma ensures that a x,y • Ψ u,v = a x,y if and only if a x,y (Ψ u,v (r)) = a x,y (r) and a x,y (Ψ u,v (s)) = a x,y (s). We have Hence we have a x,y • Ψ u,v = a x,y if and only if x v−1 = 1 and x u = 1. The result is then obtained via a case by case discussion and the previous lemma.
Lemma 5.12 describes the automorphisms Ψ u,v that fit into a 2-datum (D n , Ψ u,v , a, τ ω ) with the description of the possible maps a. We now have to classify them up to equivalence: this is done in our next lemma.
Lemma 5.13. -Let Ψ u,v ∈ Aut(D n ) be an element of order 2, and retain the notation of Lemma 5.12.
Hence there is only one equivalence class of 2-data over D n having Ψ u,v as underlying automorphism.
Lemma 5.13 enables one to classify the reduced 2-data over D n , as soon as the representative elements for the conjugacy classes of order 2 elements in Aut(D n ) have been found. We record the result in Table 5.1, where Ψ u,v is an order 2 automorphism of D n (hence with v 2 ≡ 1 [n] and u(v + 1) ≡ 0 [n]), and N (u, v) denotes the number of equivalence classes of reduced 2-data over D n having Ψ u,v as underlying automorphism. We now apply the results in Table 5.1 to enumerate the Hopf algebras fitting into a universal cocentral extension k → O(D n ) → A → kZ 2 → k in a number of particular cases.
Theorem 5.14. -Let n 4 be even and let e n be the number of isomorphism classes of noncommutative Hopf algebras A fitting into a universal cocentral extension k → O(D n ) → A → kZ 2 → k.
(2) If n = 2p r , with r 1 and p odd prime, then e n = 5.
(3) If n = 4p r , with r 1 and p odd prime, then e n = 9. (4) If n = 2 s p r , with s 3, r 1 and p odd prime, then e n = 10.
Proof. -A 2-datum (D n , θ, a, τ ) is not reduced if τ is a trivial cocycle, because Z(D n ) is non trivial, and is reduced if τ is the non-trivial 2-cocycle in Lemma 5.10. Hence, by Corollary 4.9, Theorem 4.18 and Proposition 4.14, e n equals the number of equivalence classes of 2-data (D n , θ, a, τ ω ) with θ = id, which now will be determined in each case using Table 5.1.
Remark 5.15. -Part (1) of the above theorem contributes to the classification of semisimple Hopf algebra of dimension 2 r , studied in [16,17].

Hopf algebras of dimension p 2 q r
To conclude the paper, we look at an example where the group H is abelian, one of the most studied situation in the literature [19,22,25]. We wish to prove the following result, for which the case r = 1 was obtained in [25]. .
The rest of the section is devoted to the proof of Theorem 5.16. We begin with some generalities. Recall from Subsection 4.3 that if G is a group and m 1, the set CC • m (G) is the set of elements of G such that x m = 1 and x = 1, modulo the equivalence relation defined by x ∼ y ⇐⇒ there exists l prime to m such that x l is conjugate to y. For d > 1 a divisor of m, denote by CC • m,d (G) the set of equivalence classes of elements having order d in G (clearly the order of an element is well-defined in CC • m (G)). We get a decomposition For each such d, we have an obvious well-defined surjective map CC • m,d (G) → CC • d,d (G) which is injective if m is a power of a prime. Thus identifying the two sets when m = q r with q a prime number, we obtain a decomposition CC • q r (G) = r s=1 CC • q s ,q s (G).
The group we are interested in is Aut(Z 2 p ), that we identify with GL 2 (Z/pZ), and for which we have the following result.