Examples of extensions of Hopf algebras

We give some examples of, and raise some questions on, extensions of semisimple Hopf algebras.


Introduction
Semisimple Hopf algebras have been studied intensively in the last years because their representation categories have a rich structure-they are fusion categories [ENO1,ENO2]. Conversely, one of the more fruitful approaches to classification problems of semisimple Hopf algebras is through fusion categories. However the basic notion of extension of Hopf algebras is not categorical [GN], at least in a straightforward way. Here we explore this notion of extension. First, we propose a definition of composition series of Hopf algebras. After we shared this definition with her, S. Natale endorsed it with a suitable Jordan-Hölder theorem [N3], see Theorem 2.2; cf. also [A2,Question 2.1]. Next we discuss briefly simple Hopf algebras and then present some new examples of length 2. We raise questions along the text; some of them are very natural and perhaps not new, some of them are perhaps naive but we feel they might be useful.
The article is organized as follows. Section 1 contains preliminaries. Section 2 is devoted to composition series and thoughts on simple Hopf algebras. In Section 3, we discuss general ways to construct extensions, in particular Hopf algebras of length 2, whose specific examples are in Section 4.
Acknowledgements. We thank César Galindo for pointing out to us Remark 4.4, Alexei Davydov for pointing out to us Remark 4.5 and Sonia Natale for useful conversations and friendly remarks on a preliminar version.
Notations. Let k be an algebraically closed field of characteristic 0. All vector spaces, tensor products, and algebras are over k. If G is a group, then G := Hom(G, k × ). The notation F G means that F is a subgroup of G, while F ⊳ G means that F G is normal. The standard basis of the group algebra kG is (g) g∈G , while (δ g ) g∈G is the basis of the dual group algebra k G given by δ g (h) = δ g,h , g, h ∈ G. If ⊲ is an action of a group G on a set F , then we denote by F G the subset of F of points fixed by ⊲.
Let H be a Hopf algebra, with coproduct ∆, counit ε and antipode S. We use Heynemann-Sweedler's notation ∆(h) = h (1) ⊗ h (2) . The group of its group-like elements is G(H) and the augmentation ideal is H + = ker ε. If π : H → T is a Hopf algebra map, then its subalgebra of right coinvariants is H co π = {x ∈ H : (id ⊗π)∆(x) = x⊗1}. As usual, H H YD denotes the category of Yetter-Drinfeld modules over H. An element X = r i=1 X i ⊗ X i ∈ A ⊗ B is expressed X = X i ⊗ X i using the Einstein summation convention.
1. Preliminaries 1.1. Hopf algebra extensions. We write K ≤ H to express that K is a Hopf subalgebra of H. We denote a short exact sequence of Hopf algebras by K ι / / H π / / / / T , meaning that ι and π are Hopf algebra maps, ι is injective, π is surjective, ker π = Hι(K + ) and H co π = K. We say that H is an extension of K by T . From [M, 3.1], we get then there is an exact sequence of Hopf algebras K ι / / H π / / / / H//K , where ι is the inclusion and π the canonical projection. See [AD, M, Sch].
A finite dimensional Hopf algebra is simple if it does not contain proper normal Hopf subalgebras; then its dual is simple too.
Let K, T be Hopf algebras and ⇀: T ⊗ K → K, ρ : T → T ⊗ K, σ : T ⊗ T → K and τ : T → K ⊗ K be linear maps. If (⇀, ρ, σ, τ ) is a compatible datum, i.e., satisfies the conditions in [A1, §3] and σ, τ are invertible with respect to the convolution product, then there is a Hopf algebra structure H = K τ # σ T on the vector space K ⊗ T given by An element a ⊗ g in H is denoted by a#g. Further, H is an extension of K by T ; conversely, any cleft extension of K by T is ≃ K τ # σ T , for some compatible data (⇀, ρ, σ, τ ) [AD]. All finite-dimensional extensions are cleft [Sch]. The next is a particular case, that historically appeared first.
1.2. Abelian extensions. It is customary to call group algebras and their duals trivial Hopf algebras. The first non-trivial examples of finite-dimensional Hopf algebras are the abelian extensions [K, T, Ma]. The input for the definition consists of (i) A matched pair of finite groups (F, G, ⊳, ⊲); F are (respectively, right and left) actions, such that s⊲xy = (s⊲x)((s⊳x)⊲y) and st⊳x = (s⊳(t⊲x))(t⊳x), for all s, t ∈ G, x, y ∈ F . Given these data, there is a Hopf algebra structure H = k G τ ⊲⊳ σ kF on the vector space k G ⊗ kF by (δ s #x)(δ t #y) = δ s⊳x,t σ s (x, y)δ s #xy, Then H fits into an extension k G ֒→ H ։ kF and every extension of k G by kF arises like this. Hopf algebras of this sort are named abelian extensions.
Lemma 1.2. Hopf subalgebras of abelian extensions are abelian extensions.
Twisting. There are ways to obtain new Hopf algebras by altering the comultiplication or the multiplication. The first appears in [D] in the context of quasi-Hopf algebras; the second, dual to the first, was first studied in [DT].
1.3.1. Twisting the comultiplication. Let H be a Hopf algebra. A twist for For instance, if S is abelian, then the twist corresponding to (S, ω) is given  [ENO1]. We discuss briefly the interplay between classes of fusion categories and extensions. Two fusion categories C and D are Morita-equivalent if there exists an indecomposable C-module category M such that D is equivalent to End C (M) [ENO1]. This relation settles a basic reduction in the classification program of fusion categories and has the following counterpart: two semisimple Hopf algebras H and K are Morita-equivalent if Rep H and Rep K are Moritaequivalent. Taking M = Vec, we see that H and H * are Morita-equivalent. Thus, gauge equivalence implies Morita equivalence, but not vice versa.
A fusion category is pointed if all simple objects are invertible; all pointed fusion categories are of the form Vec G ω , that is, categories of G-graded vector spaces (for some finite group G) with associator induced by ω ∈ H 3 (G, k × ). A fusion category Morita-equivalent to a pointed one is called group-theoretical [O]. A semisimple Hopf algebra H is group-theoretical when Rep H is so; hence the class of group-theoretical Hopf algebras is stable under twisting. Abelian extensions are group-theoretical [N1]. It was conjectured that every semisimple Hopf algebra is group-theoretical [ENO1], but counterexamples were found in [Ni2]: there is a non-group-theoretical Hopf algebra H p of dimension 4p 2 for each odd prime p. Indeed, H p fits into an exact sequence k Z/2 ֒→ H p ։ (kG) J , where G = (Z/p × Z/p) ⋊ Z/2 and J is a nontrivial twist in kG. Thus the class of group-theoretical Hopf algebras is not stable under extensions. More general examples of non-group-theoretical Hopf algebras were described in [GNN].
The notions of weakly group-theoretical fusion categories and Hopf algebras were introduced in [ENO2]; [ENO2, Question 2] asks whether any semisimple Hopf algebra is weakly group-theoretical. Question 1. [N5] Is an extension of weakly group-theoretical Hopf algebras, again weakly group-theoretical?
Affirmative answers are known under some specific hypothesis [ENO2].

Composition series and length
2.1. Basic facts. Every finite-dimensional Hopf algebra can be constructed from simple ones by succesive extensions. More precisely, we propose the following definition.
Definition 2.1. Let H be a finite-dimensional Hopf algebra. A composition series H of H is a sequence of simple Hopf algebras H 1 , . . . , H n obtained recursively as follows.
• If H is simple, then we let n = 1 and The simple Hopf algebras H 1 , . . . , H n are the factors of the series H and n is its length. Clearly every finite-dimensional Hopf algebra admits at least one composition series. Thus, we define the length and the factors of H as the length and the factors (up to permutation) of any composition series. For instance, H is of length 1 means that it is simple; of length 2, that it is an extension of K by T where K and T are simple. Also, H is of length 3 means that it fits into an extension K ֒→ H ։ T where either K is simple and T is of length 2, or else T is simple and K is of length 2; but these two situations do not need to hold simultaneously, see [N3]. See [N3,Section 5] for the comparison of the notion of decomposition series with those of upper or lower series in [MW].
2.1.1. Length 1. Theorem 2.2 supports the quest of simple semisimple Hopf algebras as a fundamental step in the classification of semisimple Hopf algebras. Group algebras of simple groups, their twistings and duals are all simple [Ni], but there are twistings of group algebras of solvable groups that are simple as Hopf algebras [GN]. However, a twisting of the group algebra of a nilpotent group is never simple. To our understanding, all known examples of simple semisimple Hopf algebras are twistings of group algebras or their duals. It would be decisive either to prove that these are all, or else to find essentially new examples. We propose a working definition in this direction.
Definition 2.3. Let H, K be finite-dimensional Hopf algebras. We say that H is reachable from K if it can be obtained from K by a finite number of operations that are either duality or twisting. For instance, a cocycle-twist K σ is reachable from K: Thus H is reachable from K if and only if either H or H * is obtained from K by applying succesively twists and cocycle deformations. Clearly, being reachable is an equivalence relation.
Question 2. Is every simple semisimple Hopf algebra reachable from a group algebra?
In particular, we ignore the answer to: Question 3. Is a cocycle-twist of a triangular Hopf algebra again triangular? Is there any simple Hopf algebra of the form (kG J ) σ but not triangular?
By [GN], there are simple Hopf algebras reachable from the group algebra of a super-solvable group. Also, we ask: Question 4. Classify all simple Hopf algebras reachable from a group algebra.
If H is reachable from K, then they are Morita-equivalent, but it is unlikely that the converse is true. However, we ask: Question 5. Let H and K be simple semisimple Hopf algebras. If H and K are Morita-equivalent, is then H reachable from K?
Perhaps the most natural, and ambitious, question is the following: Question 6. Is every simple semisimple Hopf algebra Morita-equivalent to a group algebra?
The answer to Question 6 is negative without the simplicity hypothesis; indeed Hopf algebras Morita-equivalent to a group algebra are grouptheoretical. Question 6 can be rephrased as follows-see also Question 1: Question 7. Can every semisimple Hopf algebra be obtained as an extension of group-theoretical ones?
Finally, let us consider a Hopf algebra H together with a twist J and a cocycle σ, and let H J σ be the vector space H with multiplication m σ and comultiplication ∆ J . (This is not the same as (H J ) σ because being a cocycle with respect to ∆ is not the same as being a cocycle with respect to ∆ J ). A straightforward computation brings the conditions needed for H J σ to be a Hopf algebra; we call this a simultaneous twist.
Question 8. Find non-trivial examples of simultaneous twists.
2.1.2. Length 2. An abelian extension k G τ ⊲⊳ σ kF has length 2 if G and F are simple, but the determination of all semisimple Hopf algebras of length 2 (with known factors) is far from being clear.
Question 9. Let G and F be finite non-abelian simple groups. Find extensions of the forms that can not be presented as abelian extensions (in particular, they are nontrivial). By duality, solutions to (I) give solutions to (III).
Example 2.4. [N4, Proposition 4.10] Let G be a finite group and A be a Hopf algebra that fits into an exact sequence kG ֒→ A ։ kZ 2 . Then A fits into an abelian exact sequence.
We discuss more on length 2 in the next Sections.

Examples
Here we discuss some particular cases of extensions, see e. g. [AN,1.1,1.2], and specific sources of examples. In this Section, K, R and T be Hopf algebras, while F , G, Γ, L and N are finite groups.
Remark 3.1. [AN, 1.1.5] The hypothesis (1), (2) and (3) mean that R is a Hopf algebra in the category K K YD, with coaction ρ and trivial action. Also, R ⋊ K coincides with the bosonization R#K.
3.1.1. Extensions of group algebras. Here we assume that R and K are group algebras. Let ϕ : kF → kG be an algebra isomorphism and ψ : G → L a group homomorphism. Set a g = ϕ −1 (g), g ∈ G. Then kF has a G-grading of algebras kF = ⊕ g∈G (kF ) g , where (kF ) g = ka g , and an L-grading of algebras kF = l∈L (kF ) l , where (kF ) l = g∈G:ψ(g)=l ka g . Let ρ G , ρ L be the associated coactions and ⇀ the trivial action of kF on either kG or kL.
• If ϕ comes from an isomorphism between F and G, then ρ L is trivial; hence kF ⋊ kL ≃ kF ⊗ kL. Indeed, if g ∈ G, then a g ∈ F , so ε(a g ) = 1 and a g ∈ (kF ) e , by Remark 3.4. Therefore kF = (kF ) e . Thus we have to consider isomorphisms of group algebras not arising from group homomorphisms. These have been intensively studied.
• In particular, we may assume that F , G are abelian groups of the same order n. Set e χ = 1 n g∈G χ(g −1 )g, for χ ∈ G; these are the primitive idempotents of kG and g = χ∈ G χ(g)e χ . Thus every isomorphism of algebras ϕ : kF → kG is determined by a bijection π : G → F ; precisely, ϕ −1 : kG → kF is given by Then ρ L is trivial and kF ⋊ kL ≃ kF ⊗ kL.
Proof. For each g ∈ G, ε(a g ) = 1 n χ∈ G x∈F Thus a g ∈ (kF ) e and ρ is trivial.

Smash product.
Here the input is a left action of K on T such that (4) ∆ T : T → T ⊗ T , ε : T → k, are K-modules maps; That is, T is a Hopf algebra in K K YD, with the given action and trivial coaction. Then the smash product T #K is the bosonization, i. e. the tensor product coalgebra T ⊗ K with the multiplication and antipode for all t ∈ T , k ∈ K; here t#k denotes again t ⊗ k. The Hopf algebra T #K fits into an exact sequence of Hopf algebras (with obvious maps ι and π) Remark 3.5. Assume that dim K < ∞. Then R is a left K-comodule that satisfies (1), (2) and (3), if and only if R a left K * -module that satisfies (4), (5) and (6), with action α · r = α, S(r (−1) ) r (0) , α ∈ K * , r ∈ R. Hence, assuming also that dim R < ∞ and combining with Remark 3.2, we have Clearly, an action of kF on T satisfying (4), (5) and (6), is equivalent to a morphism θ : F → Aut Hopf T . We describe all Hopf algebra sections in then s ϕ : kF → T #kF , s ϕ (g) = ϕ(g −1 )#g, g ∈ F is a Hopf algebra section of π and K ϕ := Im s ϕ ⊳ T #kF . Moreover, any Hopf algebra section s such that Im s⊳T #kF is like this. If π admits a Hopf algebra section with normal image, then T #kF ≃ T ⊗ kF as Hopf algebras.

3.3.
Examples where R is a twisting of a finite group. Let J be a twist of the finite group algebra kN corresponding to a pair (S, ω). Let θ : G → Aut Hopf (kN ) J be a morphism and H := (kN ) J ⋊ k G . Note that ℘ : H → k G , ℘(r#k) = ε(r)k, is a Hopf algebra retraction of ι : k G → H. Proof. By (i), k G , and a fortiori H, is not quasitriangular (use ℘). By (ii), kM ≤ H; as M is non-abelian, kM is not coquasitriangular, ditto H. Lemma 3.9. (

i) H is commutative if and only if N is abelian. (ii) H is cocommutative if and only if (kN ) J is cocommutative, G is abelian and ρ is trivial.
See Lemma 1.4 for the cocommutativity of (kN ) J .
We now show that under some assumptions, H is not an abelian extension.
Proposition 3.10. Assume that (i) G is a simple non-abelian group; (ii) (kN ) J is a simple Hopf algebra; (iii) if C ⊳ N is abelian, then C = {e} (in particular N is non-abelian), (iv) J is a non-trivial twist. Then H = (kN ) J ⋊ k G has length 2 and is not an abelian extension.
Proof. H is neither commutative nor cocommutative by Lemma 3.9. Let B ⊳ H. If B = k G , then H//B ≃ (kN ) J is not cocommutative by Lemma 1.4 and hypotheses (iii) and (iv). If B ≃ (kN ) J , then it is not commutative (or alternatively H//B ≃ k G is not cocommutative). By Lemma 3.7, these are the possible non-trivial normal Hopf subalgebras of H. Hence (kN ) J ⋊ k G is not an abelian extension.
3.4. The basic construction [AN,2.1.5]. We consider the following data • a 2-cocycle [σ] ∈ H 2 (G, Γ). Let A := R ⋊ k Γ # σ kG be a Hopf algebra with underlying vector space R ⊗ k Γ ⊗ kG (where r#f #g := r ⊗ f ⊗ g) with product, coproduct and antipode given by for all r, s ∈ R, f, f ′ ∈ k Γ , γ, u, v ∈ Γ, g, g ′ ∈ G. It fits into an exact sequence Remark 3.11. If Γ, G are finite simple groups (we may assume that σ is trivial) and R is a simple semisimple Hopf algebra, then the Hopf algebra R ⋊ k Γ #kG is of length 3.

Examples where R a twisting of a finite group.
Here we consider a basic construction A = (kN ) J ⋊k Γ #kG, where J is a twist of kN corresponding to a pair (S, ω), and θ : Γ → Aut Hopf (kN ) J and µ : G → Aut Hopf (kN ) J are group homomorphisms such that [µ(G), θ(Γ)] = 1.
Proposition 3.12. Assume that: (iv) J is a non-trivial twist. Then A is not an abelian extension.
Proof. By Lemma 1.2, since (kN ) J ⋊ k Γ is not so by Proposition 3.10.
Lemma 3.13. Assume that (i) Γ is simple non-abelian, (ii) G is simple, (iii) R is a simple semisimple Hopf algebra, (iv) there exists M N Γ non-abelian such that [M, S] = 1 and (v) µ(g) ∈ Aut(N ) and (µ(g) ⊗ µ(g))(J) = J. Then A is neither triangular nor cotriangular, but it fits in the exact sequence Proof. By Lemma 3.8, R ⋊ k Γ A is not coquasitriangular; then neither is A. Note that η : A → k Γ , η(r#f #g) = ε(r)f , is an epimorphism of Hopf algebras. Since Γ is non-abelian, A is not quasitriangular. It is not difficult to see that the sequence is exact.

Concrete examples
In the Examples below, we consider a finite group N and a twist J of kN corresponding to a pair (S, ω). Then If Z(N ) = 1, then Ad : N → Aut N induces a monomorphism from the centralizer C N (S) to C(N, S), and this is an isomorphism if Ad : N → Aut N is so. In any case, if Γ C N (S) G and Γ ∩ Z(N ) = 1 = G∩ Z(N ), then we denote by θ : Γ → Aut Hopf (kN ) J , µ : G → Aut Hopf (kN ) J the compositions of the corresponding monomorphisms.
Remark 4.5. (Davydov). Let J be a twist of kN , K ∈ kN ⊗ kN N -invariant and invertible and φ ∈ Aut N such that (φ ⊗ φ)(J) = JK. Then φ ∈ Aut Hopf (kN ) J . In our cases, N is simple and non-abelian or N = S n with n > 2, so K ⊆ Z(N ) = 1. The existence of others elements of Aut Hopf (kN ) J could provides new examples of extensions of length 2.