CLASSIFICATION OF SEMISIMPLE HOPF ALGEBRAS OF DIMENSION 16.

YEVGENIA KASHINA

Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^{n}, p is prime, cannot have a cyclic group of grouplikes.

1. Introduction.

Recently various classification results were obtained for finite-dimensional semisimple Hopf algebras over an algebraically closed field of characteristic 0. The smallest dimension, for which the question was still open, was 16. In this paper we completely classify all nontrivial (i.e. non-commutative and non-cocommutative) Hopf algebras of dimension 16. Moreover, we consider all possible structures of Grothendieck rings K_{0} (H) for semisimple non-commutative Hopf algebras of dimension 16.

Let H be a non-commutative semisimple Hopf algebra of dimension 16 over an algebraically closed field k of characteristic 0. Then irreducible representations of H of degree 1 are exactly the grouplike elements of H^{∗}. Let G (H^{∗}) denote the group of grouplikes of H^{∗}, then kG (H^{∗}) is a subHopfalgebra of H^{∗} and thus, by Nichols-Zoeller Theorem [23], |G (H^{∗})| = dim kG (H^{∗}) divides dim H^{∗} = dim H = 16. Therefore by the Artin-Wedderburn Theorem, as an algebra H is isomorphic to either

k^{(8)} ⊕ M_{2} (k) ⊕ M_{2} (k) or (1.1) k^{(4)} ⊕ M_{2} (k) ⊕ M_{2} (k) ⊕ M_{2} (k) (1.2)

dim Z (H) equals the number of summands in the Artin-Wedderburn decomposition of H, thus in the case (1.1) dim Z (H) = 10 and |G (H^{∗})| = 8 and in the case (1.2) dim Z (H) = 7 and |G (H^{∗})| = 4.

Our first result, which will be proved in the beginning of Section 3 , is the following:

Theorem 1.1. Let H be a semisimple Hopf algebra of dimension p^{n} over an algebraically closed field k of characteristic 0. If H ≇ kC_{p^{n}} then G (H) is not cyclic.

Our main result will be proved in Section 9:

Theorem 1.2. Let k be an algebraically closed field of characteristic 0. Then there are exactly 16 nonisomorphic nontrivial semisimple Hopf algebras of dimension 16.

1. G (H) is abelian of order 8 if and only if G (H^{∗}) is abelian of order 8. There are 11 Hopf algebras with such a group of grouplikes.

The author was supported in part by NSF grants DMS-9701755 and DMS 98-02086.

2. If H has a nonabelian group of grouplikes then G (H) = D_{8} and G (H^{∗}) = C_{2} × C_{2}. There are 2 Hopf algebras with a nonabelian group of grouplikes.

3. There are 3 Hopf algebras with G (H) = C_{2} × C_{2}. Two of them are dual to the Hopf algebras with a nonabelian group of grouplikes and one of them is selfdual.

Remark 1.1. A part of Theorem 1.2, saying that if H has a nonabelian group of grouplikes then G (H) = D_{8} and G (H^{∗}) = C_{2} × C_{2} can also be obtained as a corollary to a theorem of Natale [21], and Proposition 3.1. This theorem states that if G (H) is nonabelian then H^{∗} has 4 central grouplikes.

One method of constructing a new Hopf algebra from a known one H is to twist the comultiplication of H by a 2-pseudo-cocycle Ω ∈ H ⊗ H (or a 2-cocycle J ∈ H ⊗ H). The new Hopf algebra is denoted H_{Ω} (or H_{J}). The next theorem summarizes the results of Sections 5 and 6:

Theorem 1.3. Let H be a semisimple Hopf algebra of dimension 16 over an algebraically closed field k of characteristic 0. Then there are exactly 7 possible structures of the Grothendieck ring K_{0} (H). Moreover

1. G (H^{∗}) is abelian if and only if the Grothendieck ring of H is commutative. Then

(a) If |G (H^{∗})| = 8, as algebras K_{0} (H) ⊗_{ℤ} k ≅ k^{(10)}.

(b) If |G (H^{∗})| = 4, as algebras K_{0} (H) ⊗_{ℤ} k ≅ k^{(7)}.

(c) K_{0} (H) = K_{0} (kG), where G is one of the 9 nonabelian groups of order 16 (although only 6 of those K_{0}-rings are distinct).

(d) H is a twisting with a 2-pseudo-cocycle of some group algebra.

2. If K_{0} (H) is not commutative then (a) As algebras K_{0} (H) ⊗_{ℤ} k ≅ k^{(6)} ⊕ M_{2} (k).

(b) H is not a twisting of a group algebra.

(c) There is only one possible structure of the K_{0}-ring.

(d) All Hopf algebras with non-commutative K_{0}-ring are twistings of each other.

Remark 1.2. By Theorem 1.2 there are only 2 Hopf algebras with nonabelian G (H^{∗}).

We summarize the distinct non-commutative, non-cocommutative semisimple Hopf algebras of dimension 16 in the following table. We try to distinguish nonisomorphic examples of Hopf algebras using the groups G (H) and G (H^{∗}) and the Grothendieck rings K_{0} (H) (defined in Section 2). Here we consider twistings of group algebras kG, where G is a nonabelian group of order 16. There are exactly nine such groups, described in [2] (see Section 4). The twistings appearing here are explained in Section 7. The coproduct #^{α} is explained in Section 8. H_{8} denotes the unique nontrivial semisimple Hopf algebra of dimension 8 (see [7] and [11]).

Remark 1.3. H_{C:σ_{1}} is not triangular for the following reasons. If it were triangular then by [4, Theorem 2.1] it would be equal to a twisting with a 2-cocycle of a group algebra kG. Then by [24, Theorem 4.1] K_{0} (H_{C:σ_{1}}) = K_{0} (kG) and therefore H_{C:σ_{1}} would be a twisting of kD_{16} or of kQ_{16}. But by [16, Theorem 4.1], kQ_{16} doesn’t have nontrivial cocycle twistings and H_{C:1} ≅ (kD_{16})_{J} is the only cocycle twisting of kD_{16}.

Table 1

No. Example  G (H) G (H^{∗}) K_{0} (H)  Notes
1 H_{d:−1,1} ≅ H_{8} ⊗ kC_{2} (C_{2})^{3}  (C_{2})^{3}  K_{5.1} = K_{0} (D_{8} × C_{2}) not triang.
2 H_{d:1,1} ≅ k (D_{8} × C_{2})_{J} (C_{2})^{3}  (C_{2})^{3}  K_{5.1} = K_{0} (D_{8} × C_{2}) triangular
3 (H_{c:σ_{1}})^{∗}  (C_{2})^{3}  C_{2} × C_{4} K_{5.3} = K_{0} (G_{5})
4 (H_{b:1})^{∗}  (C_{2})^{3}  C_{2} × C_{4} K_{5.3} = K_{0} (G_{5})
5 H_{c:σ_{1}}  C_{2} × C_{4} (C_{2})^{3}  K_{5.2} = K_{0} (G_{7})
6 H_{b:1}  C_{2} × C_{4} (C_{2})^{3}  K_{5.1} = K_{0} (D_{8} × C_{2})
7 H_{c:σ_{0}}  C_{2} × C_{4} C_{2} × C_{4} K_{5.4} = K_{0} (G_{1})
8 H_{a:1}  C_{2} × C_{4} C_{2} × C_{4} K_{5.3} = K_{0} (G_{5})
9 H_{a:y}  C_{2} × C_{4} C_{2} × C_{4} K_{5.3} = K_{0} (G_{5})
10 H_{b:y}  C_{2} × C_{4} C_{2} × C_{4} K_{5.3} = K_{0} (G_{5})
11 H_{b:x^{2}y}  C_{2} × C_{4} C_{2} × C_{4} K_{5.3} = K_{0} (G_{5})
12 H_{C:1} ≅ (kD_{16})_{J}  D_{8}  C_{2} × C_{2} K_{6.1} = K_{0} (D_{16})  triangular
13 H_{E} ≅ (kG_{2})_{J}  D_{8}  C_{2} × C_{2} K_{6.2} = K_{0} (G_{2})  triangular
14 H_{B:1} ≅ ((kD_{16})_{J})^{∗}  C_{2} × C_{2} D_{8}  K_{5.5}
15 H_{B:X} ≅ ((kG_{2})_{J})^{∗}  C_{2} × C_{2} D_{8}  K_{5.5}  kQ_{8}#^{α}kC_{2}
16 H_{C:σ_{1}}  C_{2} × C_{2} C_{2} × C_{2} K_{6.1} = K_{0} (D_{16})  not triang.

Remark 1.4. The following Hopf algebras are selfdual: H_{d:−1,1} ≅ H_{8} ⊗ kC_{2} (since H_{8} is selfdual), H_{c:σ_{0}} (since comparing K_{0}-rings we see that H_{c:σ_{0}} ≅ A_{3}^{+} ≅ (A_{3}^{+})^{∗}, described in [8] and [9]), H_{d:1,1} ≅ k (D_{8} × C_{2})_{J} and H_{C:σ_{1}} (since there is no other choice for the dual).

2. Preliminaries.

First we will need the following definition, which was introduced in [28].

Definition 2.1. Let K_{0} (H)^{+} denote the abelian semigroup of all equivalence classes of representations of H with the addition given by a direct sum. Then its enveloping group K_{0} (H) has the structure of an ordered ring with involution ^{∗} and is called the Grothendieck ring.

In [22] the structure of K_{0} (H) was described for comodules; it was then translated into the language of modules in [25]. The multiplication in this ring is defined as follows: let [π_{1}] and [π_{2}] denote the classes of representations equivalent to π_{1} and π_{2}, then [π_{1}] • [π_{2}] is the class of the representation (π_{1} ⊗ π_{2}) ∘ Δ; the unit of this ring is the class [ε] and [π]^{∗} is the equivalence class of the dual representation ^{t} (π ∘ S) defined by ⟨^{t} (π ∘ S (h)) (f) , v⟩ = ⟨f, (π ∘ S (h)) (v)⟩. The equivalence classes of irreducible representations of H form a basis of K_{0} (H) and are called basic elements. If [π_{1}] , …, [π_{d}] are the basic elements then [ρ] = ∑_{i=1}^{d} deg π_{i} [π_{i}] is called the marked element. For basic elements x and y we write

x • y = ∑_{z - basic} m (z, x • y) z

where m (z, x • y) are non-negative integers. Then the following properties are true (see [22] and [25]):

(2.1)  m (z, x • y) = m (x^{∗}, y • z^{∗})
(2.2)  m (1, x • y^{∗}) = δ_{x,y}
(2.3)  ∑ m (z, x • y) deg (z) = deg (x • y)

For simplicity of notation we will write π instead of [π] for elements of K_{0} (H). We will denote the degree 2 irreducible representations of H by π_{i} and the degree 1 irreducible representations of H (i.e. elements of G (H^{∗}) or multiplicative characters of H) by χ_{i}. We denote the generators of G (H^{∗}) by χ, φ and ψ. If H = kG then G ((kG)^{∗}) is the group of multiplicative characters of G.

The following proposition can be also obtained as a corollary to [16, Proposition 2.4]:

Proposition 2.2. Let H be a nontrivial semisimple Hopf algebra of dimension 16. Assume that there exists an element χ ∈ G (H^{∗}) ∩ Z (H^{∗}) of order 2 such that χ • π = π for every 2-dimensional representation π of H. Then H^{∗} has a group algebra of dimension 8 as a quotient.

Proof. Write G = G (H^{∗}). Dualizing formulas (1.1) and (1.2) we get that as coalgebras

H^{∗} = kG ⊕ E_{1} ⊕ E_{2}  if |G (H^{∗})| = 8 or
H^{∗} = kG ⊕ E_{1} ⊕ E_{2} ⊕ E_{3} if |G (H^{∗})| = 4

where E_{i} are simple subcoalgebras of dimension 4 and χE_{i} = E_{i}. (χ − 1) H^{∗} is a normal Hopf ideal of H^{∗}. Then L = H^{∗}/ (χ − 1) H^{∗} is a Hopf algebra of dimension 8. Consider the projection p : H^{∗} → L. Since χE_{i} = E_{i}, p (E_{i}) = E_{i}/ (χ − 1) E_{i}. Therefore

L = k (G/ < χ >) ⊕ p (E_{1}) ⊕ p (E_{2})  if |G (H^{∗})| = 8 or
L = k (G/ < χ >) ⊕ p (E_{1}) ⊕ p (E_{2}) ⊕ p (E_{3})  if |G (H^{∗})| = 4

p (E_{i}) are cosemisimple coalgebras of dimension 2, therefore each of them is spanned by two grouplikes. Thus L is spanned by 8 grouplikes and L is a group algebra.

We will also need the notion of a twisting of a Hopf algebra (see [3], [31], [24]):

Definition 2.3. The twisting H_{Ω} of a Hopf algebra H is a Hopf algebra with the same algebra structure and counit and with comultiplication and antipode given by

Δ_{Ω} (h) = ΩΔ (h) Ω^{−1}
S_{Ω} (h) = uS (h) u^{−1}

for all h ∈ H, where Ω ∈ H ⊗ H and u ∈ H are invertible elements.

The new comultiplication Δ_{Ω} is coassociative if and only if Ω is a 2-pseudo-cocycle, that is ∂_{2} (Ω) lies in the centralizer of (Δ ⊗ id) Δ (H) in H ⊗ H ⊗ H, where

∂_{2} (Ω) = (id ⊗ Δ) (Ω^{−1}) (1 ⊗ Ω^{−1}) (Ω ⊗ 1) (Δ ⊗ id) (Ω) .

Ω is called a 2-cocycle if ∂_{2} (Ω) = 1 ⊗ 1 ⊗ 1 and in this case we will denote it by J.

Remark 2.1. By [24, Theorem 4.1] K_{0} (H) ≅ K_{0} (H_{Ω}) as ordered rings with marked elements, and thus G (H^{∗}) ≅ G ((H_{Ω})^{∗}).

3. Hopf algebras of dimension 16 with a commutative subHopfalgebra of dimension 8.

We apply the methods used by Masuoka in [11], [12] and [14]. Let H be a nontrivial semisimple Hopf algebra of dimension 16 with a subHopfalgebra K = (kG)^{∗} of dimension 8. Since K is a subHopfalgebra of index 2, by [10, Proposition 2] or [20, Theorem 2.1.1] K is normal in H and thus we have an exact sequence of Hopf algebras

K ↪^{i} H ↠^{π} F (3.1)

where F = k ⟨t⟩ ≅ kC_{2} and K = (kG)^{∗}, which is cleft by [26] or [17]. Such a sequence is called an extension of F by K and was first studied by Kac in [6]. The construction of extensions from cohomological data was done in [18] and [1]. K is commutative and F is cocommutative and thus (F, K) form an abelian matched pair of Hopf algebras and (G, ⟨t⟩) form an abelian matched pair of groups (see [30], [5], [29], [12, Section 1]). Therefore H becomes a bicrossed product K#_{σ}^{θ}F with an action ⇀: F ⊗ K → K, a coaction ρ : F → F ⊗ K, a cocycle σ : F ⊗ F → K and a dual cocycle θ : F → K ⊗ K. G is a normal subgroup of the group G × ⟨t⟩, arising from a matched pair (G, ⟨t⟩), since G has index 2 in G × ⟨t⟩. Thus ρ is trivial and the action by t is a Hopf algebra automorphism of K (see [12, Section 1]). ⇀ is a nontrivial action on K, since otherwise H ≅ K^{t} [C_{2}] as an algebra, and thus H is commutative.

Let v = σ (t, t) ∈ K. Then by the properties of the cocycle v is a unit and

t ⇀ v = v (3.2)

Multiplication in H gives us

(3.3)  \overline{t}^{2} = v
(3.4)  \overline{t}c = (t ⇀ c) \overline{t}

where \overline{t} = 1#t and c ∈ K.

Moreover, if a unit v ∈ K satisfies (3.2), (3.3) and (3.4), we can define a cocycle σ by σ (1, 1) = σ (1, t) = σ (t, 1) = 1 and σ (t, t) = v.

We proceed by considering the possible G, namely C_{8}, C_{4} × C_{2}, C_{2} × C_{2} × C_{2}, D_{8} and Q_{8}. Theorem 1.1 says that the first case cannot appear.

Theorem 1.1. Let’s prove the statement by induction on n. When n = 2, by [13, Theorem 2] H is a group algebra and if H ≇ kC_{p^{2}} then H ≅ k (C_{p} × C_{p}) and G (H) ≅ C_{p} × C_{p}.

Now assume the statement is true for n = m. Consider H of dimension p^{m+1}. dim (H^{∗}) = p^{m+1} and thus, by [13, Theorem 1], there exists a central grouplike of order p in H^{∗} and therefore H^{∗} contains a normal subHopfalgebra K ≅ kC_{p}. Thus we get a short exact sequence of Hopf algebras

K ↪^{i} H^{∗} ↠^{π} F (3.5)

where F = H^{∗}/K^{+}H^{∗}. Dualizing (3.5) we get another short exact sequence of Hopf algebras:

F^{∗} ↪^{π^{∗}} H ↠^{i^{∗}} K^{∗} (3.6)

where K^{∗} ≅ K ≅ kC_{p} and dim F^{∗} = dim F = p^{m}. Thus we get G (F^{∗}) ⊆ G (H) and G (F^{∗}) is not cyclic unless F^{∗} ≅ kC_{p^{m}}. In the first case we are done since it implies that G (H) is not cyclic. In the second case, since K is normal in H^{∗}, H^{∗} is isomorphic as an algebra to a twisted group ring K^{t} [F] where F ≅ F^{∗} ≅ kC_{p^{m}}. It is easy to show that, since F is a group algebra of a cyclic group, K^{t} [F] is commutative. Thus H is cocommutative and the only possible H with a cyclic group of grouplikes is kC_{p^{m+1}}.

3.1. Case of G (H) = C_{4} × C_{2}. We will show that there are at most 7 possible Hopf algebras of this kind. Let H be a nontrivial semisimple Hopf algebra of dimension 16 with a subHopfalgebra K = k (C_{4} × C_{2})^{∗} ≅ k (C_{4} × C_{2}). Then G (H) = G ≅ C_{4} × C_{2}.

Let G = ⟨x⟩ × ⟨y⟩ with |x| = 4 and |y| = 2. Then the dual basis of K ≅ K^{∗} is given by

e_{pq} = 1/8 (1 + i^{p}x + i^{2p}x^{2} + i^{3p}x^{3}) (1 + (−1)^{q} y) , p = 0, 1, 2, 3; q = 0, 1

Then

Δ_{H} (e_{pq}) = Δ_{K} (e_{pq}) = ∑_{\begin{matrix}p_{1}+p_{2}≡p mod 4 \\ q_{1}+q_{2}≡q mod 2 \end{matrix}} e_{p_{1}q_{1}} ⊗ e_{p_{2}q_{2}}

Δ_{H} (\overline{t}) = θ (t) \overline{t} ⊗ \overline{t}

where \overline{t} = 1#t. Dualizing (3.1) we get another extension

F^{∗} ↪^{π^{∗}} H^{∗} ↠^{i^{∗}} K^{∗}

and as in [11, 2.4], [12, 2.11] or [15, 2.1], since k is algebraically closed, there exist units \overline{x} and \overline{y} ∈ H^{∗}, such that \overline{x}^{4} = \overline{y}^{2} = 1_{H^{∗}}, ⟨e_{pq}, \overline{x}^{i}\overline{y}^{j}⟩ = δ_{ip}δ_{jq} and α = \overline{y}^{−1}\overline{x}^{−1}\overline{y}\overline{x} ∈ F^{∗} = k \{e_{0}, e_{1}\}, where \{e_{r}\} is a dual basis of \{t^{r}\}, r = 0, 1. ε (α) = ε (\overline{y}^{−1}\overline{x}^{−1}\overline{y}\overline{x}) = 1 and therefore α = e_{0} + ξe_{1}. The right action ρ^{∗} : F^{∗} ⊗ K^{∗} → F^{∗} is trivial, thus F^{∗} lies in the center of H^{∗}.

\overline{x} = \overline{y}^{2}\overline{x} = \overline{y}\overline{x}\overline{y}α = \overline{x}\overline{y}α\overline{y}α = \overline{x}\overline{y}^{2}α^{2} = \overline{x}α^{2}

Thus α^{2} = 1 and therefore ξ = ±1.

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨\overline{t}, \overline{x}^{i}\overline{y}^{j}e_{k}\overline{x}^{p}\overline{y}^{q}e_{r}⟩ = δ_{kr} ⟨\overline{t}, \overline{x}^{i+p}\overline{y}^{j+q}α^{jp}e_{k}⟩ = ξ^{jp}δ_{k1}δ_{r1} On the other hand

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨θ (t) \overline{t} ⊗ \overline{t}, \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨θ (t) , \overline{x}^{i}\overline{y}^{j} ⊗ \overline{x}^{p}\overline{y}^{q}⟩ δ_{k1}δ_{r1}

Therefore

θ (t) = ∑_{ijpq} ξ^{jp}e_{ij} ⊗ e_{pq}

and since H should be non-cocommutative, θ (t, t) is nontrivial, and thus ξ = −1 and

θ (t) = ∑_{ijpq} (−1)^{jp} e_{ij} ⊗ e_{pq} = \frac{1}{2} ((1 + y) ⊗ 1 + (1 − y) ⊗ x^{2})

Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} then c_{0,0} = ε (v) = 1 and c_{i,j} ≠ 0, since v is a unit, and

Δ_{H} (\overline{t}^{2}) = Δ_{H} (v) = Δ_{K} (∑ c_{i,j}e_{i,j}) = ∑ c_{i+p,j+q}e_{i,j} ⊗ e_{p,q}

On the other hand, if we write

t ⇀ e_{p,q} = e_{α_{1}(p,q),α_{2}(p,q)}
Δ (\overline{t}) Δ (\overline{t}) = ∑_{ijpq} (−1)^{jp} e_{ij}\overline{t} ⊗ e_{pq}\overline{t} ∑_{ijpq} (−1)^{jp} e_{ij}\overline{t} ⊗ e_{pq}\overline{t}
= ∑_{ijpq} (−1)^{jp} e_{ij} ⊗ e_{pq} ∑_{ijpq} (−1)^{jp} e_{α_{1}(i,j),α_{2}(i,j)}\overline{t}^{2} ⊗ e_{α_{1}(p,q),α_{2}(p,q)}\overline{t}^{2}
= ∑_{ijpq} (−1)^{jp} e_{ij} ⊗ e_{pq} ∑_{ijpq} (−1)^{α_{2}(i,j)α_{1}(p,q)} e_{ij}\overline{t}^{2} ⊗ e_{pq}\overline{t}^{2}
= ∑_{ijpq} (−1)^{jp+α_{2}(i,j)α_{1}(p,q)} c_{ij}c_{pq}e_{ij} ⊗ e_{pq}

Thus for H to be a bialgebra we should have

c_{i+p,j+q} = (−1)^{jp+α_{2}(i,j)α_{1}(p,q)} c_{i,j}c_{p,q} (3.7)

Action by t is a Hopf algebra map and therefore t ⇀ G = G and f_{t} : G → G defined by f_{t} (g) = t ⇀ g is a group automorphism of order 2. There are three possibilities for such an automorphism; we consider them below:

Case a). The action is given by

t ⇀ x = xy
t ⇀ y = y

Then t ⇀ e_{i,j} = e_{i+2j,j}. Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} . By (3.2) and (3.7)

(3.8)  c_{i,j} = c_{i+2j,j}
(3.9)  c_{i+p,j+q} = c_{i,j}c_{p,q}

Conditions (3.8) and (3.9) imply that c_{1,0} = (−1)^{k} and c_{0,1} = (−1)^{l} for k, l = 0, 1 and

σ (t, t) = ∑_{p,q} (−1)^{kp+lq} e_{p,q} = ∑ (−1)^{kp} e_{p,q} ∑ (−1)^{lq} e_{p,q} = x^{2k}y^{l} k, l = 0, 1

For k, l = 0, 1 let H_{k,l} be the Hopf algebras with the structures described above with cocycles σ_{k,l} (t, t) = x^{2k}y^{l}. Define

f : H_{k,l} → H_{k+1,l+1} by
f (e_{r,s}) = e_{r,s}
f (\overline{t}) = x\overline{t}

and extend it multiplicatively to f (e_{r,s}\overline{t}). Then f is a trivial group homomorphism on G (H_{k,l}) and

f (\overline{t}) f (\overline{t}) = x\overline{t}x\overline{t} = x^{2}y\overline{t}^{2} = x^{2}yx^{2(k+1)}y^{(l+1)} = x^{2k}y^{l} = f (\overline{t}^{2})
f (\overline{t}x) = f (xy\overline{t}) = xyx\overline{t} = x\overline{t}x = f (\overline{t}) f (x)
(f ⊗ f) Δ (\overline{t}) = (f ⊗ f) (θ (t) \overline{t} ⊗ \overline{t}) = θ (t) (f (\overline{t}) ⊗ f (\overline{t})) = θ (t) (x\overline{t} ⊗ x\overline{t})
= (x ⊗ x) θ (t) (\overline{t} ⊗ \overline{t}) = Δ (x) Δ (\overline{t}) = Δ (x\overline{t}) = Δ (f (\overline{t}))

and such an f is a Hopf algebra isomorphism between H_{k,l} and H_{k+1,l+1}. Thus there are at most two nonisomorphic Hopf algebras of this type:

1. H_{a:1} = H_{0,0} with the trivial cocycle and G (H_{a:1}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(x) = i, χ(y) = χ(t) = 1, φ (x) = φ (y) = 1, φ (t) = −1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) π(y) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) π(t) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ^{2} + φ + χ^{2}φ.

2. H_{a:y} = H_{0,1} with the cocycle defined by σ (t, t) = y and G (H_{a:y}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(x) = i, χ(y) = χ(t) = 1, φ (x) = φ (y) = 1, φ (t) = −1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) π(y) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) π(t) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ^{2} + φ + χ^{2}φ.

Case b). The action is given by

t ⇀ x = x^{−1}
t ⇀ y = y

Then t ⇀ e_{i,j} = e_{−i,j}. Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} . By (3.2) and (3.7)

(3.10)  c_{i,j} = c_{−i,j}
(3.11)  c_{i+p,j+q} = c_{i,j}c_{p,q}

Conditions (3.10) and (3.11) imply that c_{1,0} = (−1)^{k} and c_{0,1} = (−1)^{l} for k, l = 0, 1 and

σ (t, t) = ∑_{p,q} (−1)^{kp+lq} e_{p,q} = ∑ (−1)^{kp} e_{p,q} ∑ (−1)^{lq} e_{p,q} = x^{2k}y^{l} k, l = 0, 1

For k, l = 0, 1 let H_{k,l} be the Hopf algebras with the structures described above with cocycles σ_{k,l} (t, t) = x^{2k}y^{l}. Define

f : H_{0,0} → H_{1,0} by
f (e_{r,s}) = e_{r,r+s}
f (\overline{t}) = \frac{1}{2} ((1 + i) 1 + (1 − i) x^{2}) \overline{t} = ∑_{p=0}^{3} ∑_{q=0}^{1} i^{p^{2}}e_{p,q}\overline{t}

and extend it multiplicatively to f (e_{r,s}\overline{t}). Then f |_{G(H_{0,0})} is a group isomorphism G (H_{0,0}) → G (H_{1,0}) with f (x) = x, f (y) = x^{2}y and

f (\overline{t}) f (\overline{t}) = \frac{1}{4} ((1 + i) 1 + (1 − i) x^{2}) \overline{t} ((1 + i) 1 + (1 − i) x^{2}) \overline{t}
= \frac{1}{4} ((1 + i) 1 + (1 − i) x^{2})^{2} \overline{t}^{2} = \frac{1}{4} (2i · 1 + 4x^{2} − 2i · 1) σ_{1,0} (t, t)
= x^{2}x^{2} = 1 = f (\overline{t}^{2})
f (\overline{t}x) = f (x^{−1}\overline{t}) = \frac{1}{2}x^{−1} ((1 + i) 1 + (1 − i) x^{2}) \overline{t}
= \frac{1}{2} ((1 + i) 1 + (1 − i) x^{2}) \overline{t}x = f (\overline{t}) f (x)
(f ⊗ f) Δ (\overline{t}) = (f ⊗ f) (θ (t) \overline{t} ⊗ \overline{t}) = (f ⊗ f) (∑ (−1)^{q_{1}p_{2}} e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t})
= ∑ (−1)^{q_{1}p_{2}} e_{p_{1},q_{1}+p_{1}} ∑ i^{p^{2}}e_{p,q}\overline{t} ⊗ e_{p_{2},q_{2}+p_{2}} ∑ i^{p^{2}}e_{p,q}\overline{t}
= ∑ (−1)^{(q_{1}+p_{1})p_{2}} i^{p_{1}^{2}+p_{2}^{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}
= ∑ (−1)^{q_{1}p_{2}} i^{p_{1}^{2}+2p_{1}p_{2}+p_{2}^{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}
= ∑ (−1)^{q_{1}p_{2}} i^{(p_{1}+p_{2})^{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}
= (∑ i^{(p_{1}+p_{2})^{2}}e_{p_{1},q_{1}} ⊗ e_{p_{2},q_{2}}) (∑ (−1)^{s_{1}r_{2}} e_{r_{1},s_{1}} \overline{t} ⊗ e_{r_{2},s_{2}} \overline{t})
= Δ (∑ i^{p^{2}}e_{p,q}) Δ (\overline{t}) = Δ (∑ i^{p^{2}}e_{p,q}\overline{t}) = Δ (f (\overline{t}))

and such an f is a Hopf algebra isomorphism between H_{0,0} and H_{1,0}. There are at most three nonisomorphic Hopf algebras of this kind:

1. H_{b:1} = H_{0,0} with the trivial cocycle and G (H_{b:1}^{∗}) = ⟨χ⟩×⟨φ⟩×⟨ψ⟩ ≅ C_{2}×C_{2}×C_{2}, where χ(x) = −1, χ(y) = χ(t) = 1, φ (x) = φ (y) = 1, φ (t) = −1, ψ (y) = −1, ψ (x) = ψ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) π(y) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & (−1)^{p} \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ + φ + χφ.

2. H_{b:y} = H_{0,1} with the cocycle defined by σ (t, t) = y and G (H_{b:y}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(x) = 1, χ(y) = −1, χ(t) = i, φ (x) = −1, φ (y) = φ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) π(y) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & (−1)^{p} \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ^{2} + φ + χ^{2}φ.

3. H_{b:x^{2}y} = H_{1,1} with the cocycle defined by σ (t, t) = x^{2}y and G (H_{b:x^{2}y}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(x) = 1, χ(y) = −1, χ(t) = i, φ (x) = −1, φ (y) = φ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) π(y) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & (−1)^{p} \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ^{2} + φ + χ^{2}φ.

Case c). The action is given by

t ⇀ x = x
t ⇀ y = x^{2}y

Then t ⇀ e_{i,j} = e_{i,j+i}. Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} . By (3.2) and (3.7)

(3.12)  c_{i,j} = c_{i,i+j}
(3.13)  c_{i+p,j+q} = (−1)^{ip} c_{i,j}c_{p,q}

Conditions (3.12) and (3.13) imply that c_{1,0}^{4} = c_{0,1} = 1, c_{2,0} = −c_{1,0}^{2} Thus c_{1,0} = i^{k} for k = 0, 1, 2, 3 and

σ_{k} (t, t) = ∑_{p,q} (−1)^{\frac{p(p−1)}{2}} i^{kp}e_{p,q} = x^{1−k} (\frac{1 + i}{2}1 + \frac{1 − i}{2}x^{2})

For k = 0, 1, 2, 3 let H_{k} be the Hopf algebras with the structures described above with cocycles σ_{k}. Define

f : H_{k+2} → H_{k}  by
f (e_{p,q}) = e_{p,q}
f (\overline{t}) = ∑_{p,q} (−1)^{p+q} e_{p,q}\overline{t} = y\overline{t}

and extend it multiplicatively to f (e_{p,q}\overline{t}). Then

f (\overline{t}) f (\overline{t}) = y\overline{t}y\overline{t} = x^{2}\overline{t}^{2} = x^{2}x^{1−(k−2)} (\frac{1 + i}{2} + \frac{1 − i}{2}x^{2})
= x^{1−k} (\frac{1 + i}{2} + \frac{1 − i}{2}x^{2}) = f (\overline{t}^{2})
f (\overline{t}y) = f (x^{2}y\overline{t}) = x^{2}yy\overline{t} = y\overline{t}y = f (\overline{t}) f (y)

(f ⊗ f) Δ (\overline{t}) = (f ⊗ f) (θ (t) \overline{t} ⊗ \overline{t}) = θ (t) (f (\overline{t}) ⊗ f (\overline{t})) = θ (t) (y\overline{t} ⊗ y\overline{t})
= (y ⊗ y) θ (t) (\overline{t} ⊗ \overline{t}) = Δ (y) Δ (\overline{t}) = Δ (y\overline{t}) = Δ (f (\overline{t}))

and such a f is a Hopf algebra isomorphism between H_{k+2} and H_{k}. Thus there are exactly 2 nonisomorphic Hopf algebras of this type:

1. H_{c:σ_{0}} = H_{0} with cocycle σ_{0} defined by σ (t, t) = \frac{1+i}{2}x + \frac{1−i}{2}x^{3} and G (H_{c:σ_{0}}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(x) = −1, χ(y) = 1, χ(t) = i, φ (y) = −1, φ (x) = φ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & i \end{matrix} ) π(y) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) π(t) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} )

with the property π^{2} = π • π = χ + χ^{3} + χφ + χ^{3}φ.

2. H_{c:σ_{1}} = H_{1} with cocycle σ_{1} defined by σ (t, t) = \frac{1+i}{2}1 + \frac{1−i}{2}x^{2} and G (H_{c:σ_{1}}^{∗}) = ⟨χ⟩ × ⟨φ⟩ × ⟨ψ⟩ ≅ C_{2} × C_{2} × C_{2}, where χ(y) = −1, χ(x) = χ(t) = 1, φ (x) = φ (y) = 1, φ (t) = −1, ψ (x) = −1, ψ (y) = ψ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}i & 0 \\ 0 & i \end{matrix} ) π(y) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) π(t) = ( \begin{matrix}ω & 0 \\ 0 & −ω \end{matrix} )

where ω is a primitive 8^{th}-root of unity, with the property π^{2} = π • π = ψ + χψ + φψ + χφψ.

3.2. Case of G (H) = C_{2} × C_{2} × C_{2}. We will show that there are at most 4 possible Hopf algebras of this kind. Let H be a nontrivial semisimple Hopf algebra of dimension 16 with a subHopfalgebra K = k (C_{2} × C_{2} × C_{2})^{∗} ≅ k (C_{2} × C_{2} × C_{2}). Then G (H) = G ≅ C_{2} × C_{2} × C_{2}.

Let G (H) = ⟨x⟩ × ⟨y⟩ × ⟨z⟩, where |x| = |y| = |z| = 2. Then the dual basis of K ≅ K^{∗} is given by

e_{p,q,r} = 1/8 (1 + (−1)^{p} x) (1 + (−1)^{q} y) (1 + (−1)^{r} z) , p, q, r = 0, 1

Then

Δ_{H} (e_{p,q,r}) = Δ_{K} (e_{p,q,r}) = ∑_{p_{1}+p_{2}≡p mod 2 q_{1}+q_{2}≡q mod 2 r_{1}+r_{2}≡r mod 2} e_{p_{1},q_{1},r_{1}} ⊗ e_{p_{2},q_{2},r_{2}}

Δ_{H} (\overline{t}) = θ (t) \overline{t} ⊗ \overline{t}

where \overline{t} = 1#t. Dualizing (3.1) we get another extension

F^{∗} ↪^{π^{∗}} H^{∗} ↠^{i^{∗}} K^{∗}

and as in [11, 2.4], [12, 2.11] or [15, 2.1], since k is algebraically closed, there exist units \overline{x}, \overline{y} and \overline{z} ∈ H^{∗}, such that \overline{x}^{2} = \overline{y}^{2} = \overline{z}^{2} = 1_{H^{∗}}, ⟨e_{p,q,r}, \overline{x}^{i}\overline{y}^{j}\overline{z}^{k}⟩ = δ_{ip}δ_{jq}δ_{kr} and α = \overline{z}^{−1}\overline{y}^{−1}\overline{z}\overline{y}, β = \overline{z}^{−1}\overline{x}^{−1}\overline{z}\overline{x}, γ = \overline{y}^{−1}\overline{x}^{−1}\overline{y}\overline{x} ∈ F^{∗} = k \{e_{0}, e_{1}\}, where \{e_{r}\} is a dual basis of {t^{r}}, r = 0, 1. ε (α) = ε (β) = ε (γ) = 1 and therefore α = e_{0} + ξ_{3}e_{1}, β = e_{0} + ξ_{2}e_{1}, γ = e_{0} + ξ_{1}e_{1}. The right action ρ^{∗} : F^{∗} ⊗ K^{∗} → F^{∗} is trivial, thus F^{∗} lies in the center of H^{∗}.

\overline{x} = \overline{y}^{2}\overline{x} = \overline{y}\overline{x}\overline{y}γ = \overline{x}\overline{y}γ\overline{y}γ = \overline{x}\overline{y}^{2}γ^{2} = \overline{x}γ^{2}

Thus γ^{2} = 1 and similarly α^{2} = β^{2} = 1. Therefore ξ_{1}, ξ_{2}, ξ_{3} = ±1 and, since H^{∗} is non-commutative, they cannot be all equal to 1.

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}\overline{z}^{k}e_{l} ⊗ \overline{x}^{p}\overline{y}^{q}\overline{z}^{r}e_{s}⟩ = ⟨\overline{t}, \overline{x}^{i}\overline{y}^{j}\overline{z}^{k}e_{l}\overline{x}^{p}\overline{y}^{q}\overline{z}^{r}e_{s}⟩
= δ_{ls} ⟨\overline{t}, \overline{x}^{i+p}\overline{y}^{j+q}\overline{z}^{k+r}α^{kq}β^{kp}γ^{jp}e_{l}⟩ = ξ_{1}^{jp}ξ_{2}^{kp}ξ_{3}^{kq}δ_{l1}δ_{s1}

On the other hand

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}\overline{z}^{k}e_{l} ⊗ \overline{x}^{p}\overline{y}^{q}\overline{z}^{r}e_{s}⟩ = ⟨θ (t) \overline{t} ⊗ \overline{t}, \overline{x}^{i}\overline{y}^{j}\overline{z}^{k}e_{l} ⊗ \overline{x}^{p}\overline{y}^{q}\overline{z}^{r}e_{s}⟩
= ⟨θ (t) , \overline{x}^{i}\overline{y}^{j}\overline{z}^{k} ⊗ \overline{x}^{p}\overline{y}^{q}\overline{z}^{r}⟩ δ_{k1}δ_{r1}

Therefore

θ (t) = ∑_{ijkpqr} ξ_{1}^{jp}ξ_{2}^{kp}ξ_{3}^{kq}e_{i,j,k} ⊗ e_{p,q,r}

Action by t is a Hopf algebra map and therefore t ⇀ G = G and f_{t} : G → G defined by f_{t} (g) = t ⇀ g is a group automorphism of order 2. Then, without loss of generality there is only one possibility for such an automorphism:

t ⇀ x = y
t ⇀ y = x
t ⇀ z = z

Then t ⇀ e_{i,j,k} = e_{j,i,k}.

Write v = σ (t, t) = ∑ c_{i,j,k}e_{i,j,k} then c_{0,0,0} = ε (v) = 1 and c_{i,j,k} ≠ 0, since v is a unit. By formula (3.2)

c_{i,j,k} = c_{j,i,k} (3.14)

For H to be a bialgebra we need Δ_{H} (\overline{t}^{2}) = Δ_{H} (\overline{t}) Δ_{H} (\overline{t})

Δ_{H} (\overline{t}^{2}) = Δ_{H} (v) = Δ_{K} (∑ c_{i,j,k}e_{i,j,k}) = ∑ c_{i+p,j+q,k+r}e_{i,j,k} ⊗ e_{p,q,r}

On the other hand,

Δ_{H} (\overline{t}) Δ_{H} (\overline{t}) = (θ (t) \overline{t} ⊗ \overline{t}) (θ (t) \overline{t} ⊗ \overline{t})
= ∑_{ijpq} ξ_{1}^{jp}ξ_{2}^{kp}ξ_{3}^{kq}e_{ijk} ⊗ e_{pqr} (∑_{ijpq} ξ_{1}^{jp}ξ_{2}^{kp}ξ_{3}^{kq} (t ⇀ e_{ijk}) ⊗ (t ⇀ e_{pqr})) σ (t, t) ⊗ σ (t, t)
= ∑ ξ_{1}^{jp+iq}ξ_{2}^{kp+kq}ξ_{3}^{kq+kp}c_{i,j,k}c_{p,q,r}e_{i,j,k} ⊗ e_{p,q,r}

Therefore

c_{i+p,j+q,k+r} = ξ_{1}^{jp+iq}ξ_{2}^{kp+kq}ξ_{3}^{kq+kp}c_{i,j,k}c_{p,q,r} (3.15)

Conditions (3.14) and (3.15) imply that c_{1,0,0}^{2} = c_{0,1,0}^{2} = c_{0,0,1}^{2} = c_{1,1,0}^{2} = c_{1,1,1}^{2} = 1 and c_{0,1,0} = c_{1,0,0}

c_{1,1,0} = ξ_{1}c_{0,1,0}c_{1,0,0} = ξ_{1}c_{1,0,0}^{2} = ξ_{1}
c_{1,0,1} = c_{1,0,0}c_{0,0,1}

c_{1,0,1} = c_{0,0,1}c_{1,0,0}ξ_{2}ξ_{3}

Thus ξ_{2}ξ_{3} = 1, that is ξ_{2} = ξ_{3} and c_{1,0,0} = c_{0,1,0} = ω = ±1 and c_{0,0,1} = τ = ±1 and

θ (t) = ∑_{ijkpqr} ξ_{1}^{jp}ξ_{2}^{kp+kq}e_{i,j,k} ⊗ e_{p,q,r}

σ (t, t) = e_{0,0,0} + τe_{0,0,1} + ξ_{1}e_{1,1,0} + ξ_{1}τe_{1,1,1} + ω (e_{1,0,0} + e_{0,1,0} + τe_{1,0,1} + τe_{0,1,1})
= ∑ ω^{p+q}e_{pqr} ∑ ξ_{1}^{pq}e_{pqr} ∑ τ^{r}e_{pqr} = (xy)^{δ_{ω,−1}} (1 + x + y − xy)^{δ_{ξ_{1},−1}} z^{δ_{τ,−1}}

For ξ_{1}, ξ_{2}, τ, ω = ±1 let H_{d:ξ_{1},ξ_{2},τ,ω} be the Hopf algebras with the structures described above with cocycles σ_{ξ_{1},τ,ω}. Then σ_{ξ_{1},τ,−1} (t, t) = xyσ_{ξ_{1},τ,1} (t, t). Define

f : H_{d:ξ_{1},ξ_{2},τ,−1} → H_{d:ξ_{1},ξ_{2},τ,1} by
f (e_{p,q,r}) = e_{p,q,r}
f (\overline{t}) = x\overline{t}

and extend it multiplicatively to f (e_{p,q,r}\overline{t}). Then

f (\overline{t}) f (\overline{t}) = x\overline{t}x\overline{t} = xy\overline{t}^{2} = xyσ_{ξ_{1},τ,1} (t, t)
= σ_{ξ_{1},τ,−1} (t, t) = f (σ_{ξ_{1},τ,−1} (t, t)) = f (\overline{t}^{2})
f (\overline{t}x) = f (y\overline{t}) = yx\overline{t} = x\overline{t}x = f (\overline{t}) f (x)
f (\overline{t}y) = f (x\overline{t}) = x^{2}\overline{t} = x\overline{t}y = f (\overline{t}) f (y)
(f ∘ f) Δ (\overline{t}) = (f ∘ f) (θ (t) \overline{t} ⊗ \overline{t}) = θ (t) (f (\overline{t}) ⊗ f (\overline{t})) = θ (t) (x\overline{t} ⊗ x\overline{t})
= (x ⊗ x) θ (t) (\overline{t} ⊗ \overline{t}) = Δ (x) Δ (\overline{t}) = Δ (x\overline{t}) = Δ (f (\overline{t}))

and such a f is a Hopf algebra isomorphism between H_{d:ξ_{1},ξ_{2},τ,−1} and H_{d:ξ_{1},ξ_{2},τ,1}. Define

f′ : H_{d:−1,−1,τ,1} → H_{d:−1,1,τ,1} by
f′ (e_{p,q,r}) = e_{p+r,q+r,r}
f′ (\overline{t}) = \frac{1}{2} (1 + z + iy − iyz) \overline{t} = ∑ i^{r^{2}} (−1)^{qr} e_{p,q,r}\overline{t}

and extend it multiplicatively to f′ (e_{p,q,r}\overline{t}). Then f′ |_{G(H_{d:−1,−1,τ,1})} is a group isomorphism G (H_{d:−1,−1,τ,1}) → G (H_{d:−1,1,τ,1}) with f′ (x) = xz, f′ (y) = yz, f′ (z) = z and

f′ (\overline{t}) f′ (\overline{t}) = \frac{1}{4} ((1 + z) + iy (1 − z)) \overline{t} ((1 + z) + iy (1 − z)) \overline{t}
= \frac{1}{4} ((1 + z) + iy (1 − z)) ((1 + z) + ix (1 − z)) \overline{t}^{2}
= \frac{1}{8} (2 + 2z − xy (2 − 2z)) (1 + x + y − xy) z^{δ_{τ,−1}}
= \frac{1}{4} ((1 − xy) + z (1 + xy)) ((1 − xy) + x (1 + xy)) z^{δ_{τ,−1}}
= \frac{1}{4} (2 − 2xy + xz (2 + 2xy)) z^{δ_{τ,−1}} = \frac{1}{2} (1 + xz + yz − xy) z^{δ_{τ,−1}}

= f (\frac{1}{2} (1 + x + y − xy) z^{δ_{τ,−1}}) = f′ (\overline{t}^{2})
f′ (\overline{t}x) = f′ (y\overline{t}) = \frac{1}{2}yz (1 + z + iy − iyz) \overline{t}
= \frac{1}{2} (1 + z + iy − iyz) \overline{t}xz = f′ (\overline{t}) f′ (x)
f′ (\overline{t}y) = f′ (x\overline{t}) = \frac{1}{2}xz (1 + z + iy − iyz) \overline{t}
= \frac{1}{2} (1 + z + iy − iyz) \overline{t}yz = f′ (\overline{t}) f′ (y)
(f′ ⊗ f′) Δ (\overline{t}) = (f′ ⊗ f′) (θ (t) \overline{t} ⊗ \overline{t})
= (f′ ⊗ f′) (∑ (−1)^{bp} (−1)^{cp+cq} e_{a,b,c}\overline{t} ⊗ e_{p,q,r}\overline{t})
= (∑ (−1)^{bp} (−1)^{c(p+q)} e_{a+c,b+c,c} ⊗ e_{p+r,q+r,r})

× (∑ i^{n^{2}} (−1)^{mn} e_{l,m,n}\overline{t} ⊗ ∑ i^{n^{2}} (−1)^{mn} e_{l,m,n}\overline{t})

= (∑ (−1)^{(b+c)(p+r)} (−1)^{c(p+q)} e_{a,b,c} ⊗ e_{p,q,r})

× (∑ i^{n^{2}} (−1)^{mn} e_{l,m,n}\overline{t} ⊗ ∑ i^{n^{2}} (−1)^{mn} e_{l,m,n}\overline{t})

= ∑ (−1)^{bp+cp+br+cr} (−1)^{cp+cq} i^{c^{2}} (−1)^{bc} i^{r^{2}} (−1)^{qr} e_{a,b,c}\overline{t} ⊗ e_{p,q,r}\overline{t}
= ∑ (−1)^{bp+br+cq+bc+qr} (−1)^{cr} i^{c^{2}}i^{r^{2}}e_{a,b,c}\overline{t} ⊗ e_{p,q,r}\overline{t}
= ∑ (−1)^{bp+br+cq+bc+qr} (−1)^{cr} i^{c^{2}}i^{r^{2}}e_{a,b,c}\overline{t} ⊗ e_{p,q,r}\overline{t}
= ∑ i^{(c+r)^{2}} (−1)^{(b+q)(c+r)} (−1)^{bp} e_{a,b,c}\overline{t} ⊗ e_{p,q,r}\overline{t}
= ∑ i^{n^{2}} (−1)^{mn} ∑_{\begin{matrix}l_{1}+l_{2}=l \\ m_{1}+m_{2}=m \\ n_{1}+n_{2}=n \end{matrix}} e_{l_{1},m_{1},n_{1}} ⊗ e_{l_{2},m_{2},n_{2}}

× ∑ (−1)^{bp} e_{a,b,c}\overline{t} ⊗ e_{p,q,r}\overline{t}

= ∑ i^{n^{2}} (−1)^{mn} Δ (e_{l,m,n}) Δ (\overline{t})
= Δ (∑ i^{n^{2}} (−1)^{mn} e_{l,m,n}\overline{t}) = Δf′ (\overline{t})

and such an f′ is a Hopf algebra isomorphism between H_{d:−1,−1,τ,1} and H_{d:−1,1,τ,1}. Thus we may assume that ω = 1 and ξ_{2} = −1. Therefore there are at most four nonisomorphic Hopf algebras H_{d:ξ_{1},τ} of this kind, H_{d:1,1}, H_{d:1,−1}, H_{d:−1,1} and H_{d:−1,−1}:

1. H_{d:1,1} with the trivial cocycle and G (H_{d:1,1}^{∗}) = ⟨χ⟩ × ⟨φ⟩ × ⟨ψ⟩ ≅ C_{2} × C_{2} × C_{2}, where χ (x) = χ (y) = χ (z) = 1, χ (t) = −1, φ(x) = φ(y) = −1, φ(z) = φ(t) = 1, ψ (z) = −1, ψ (x) = ψ (y) = ψ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) π(y) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) π(z) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ + φ + χφ.

2. H_{d:1,−1} with the cocycle defined by σ (t, t) = z and G (H_{d:1,−1}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(x) = χ(y) = 1, χ(z) = −1, χ(t) = i, φ (x) = φ (y) = −1, φ (t) = φ (z) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) π(y) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) π(z) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ^{2} + φ + χ^{2}φ.

3. H_{d:−1,1} with the cocycle defined by σ (t, t) = 1 + x + y − xy and G (H_{d:−1,1}^{∗}) = ⟨χ⟩ × ⟨φ⟩ × ⟨ψ⟩ ≅ C_{2} × C_{2} × C_{2}, where χ (x) = χ (y) = χ (z) = 1, χ (t) = −1, φ(x) = φ(y) = −1, φ(z) = 1, φ(t) = i, ψ (z) = −1, ψ (x) = ψ (y) = ψ (t) = 1. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) π(y) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) π(z) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ + φ + χφ.

4. H_{d:−1,−1} with the cocycle defined by σ (t, t) = (1 + x + y − xy) z and G (H_{d:−1,−1}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ (x) = χ (y) = 1, χ (z) = −1, χ (t) = i, φ(x) = φ(y) = −1, φ(z) = 1, φ(t) = i. There is a degree 2 irreducible representation defined by

π(x) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) π(y) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) π(z) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) π(t) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

with the property π^{2} = π • π = 1 + χ^{2} + φ + χ^{2}φ.

3.3. Case of G = D_{8}. Let G = D_{8} = ⟨x, y | x^{4} = y^{2} = 1, yx = x^{−1}y⟩. Let {e_{pq}}_{p=0,1,2,3;q=0,1} be the basis of K, dual to the basis {x^{p}y^{q}}_{p=0,1,2,3;q=0,1} of K^{∗} = kD_{8}. Then

Δ_{H} (e_{pq}) = Δ_{K} (e_{pq}) = ∑_{\begin{matrix}p_{1}+p_{2}+2q_{1}p_{2}≡p mod 4 \\ q_{1}+q_{2}≡q mod 2 \end{matrix}} e_{p_{1}q_{1}} ⊗ e_{p_{2}q_{2}}

and it is easy to check that elements

X = ∑_{pq} (−1)^{p} e_{pq}
Y = ∑_{pq} (−1)^{q} e_{pq}

are grouplike of order 2. For \overline{t} = 1#t

Δ_{H} (\overline{t}) = θ (t) \overline{t} ⊗ \overline{t}

Dualizing (3.1) we get another extension

F^{∗} ↪^{π^{∗}} H^{∗} ↠^{i^{∗}} K^{∗}

and as in [11, 2.4], [12, 2.11] or [15, 2.1], since k is algebraically closed, there exist units \overline{x} and \overline{y} ∈ H^{∗}, such that \overline{x}^{4} = \overline{y}^{2} = 1_{H^{∗}}, ⟨e_{pq}, \overline{x}^{i}\overline{y}^{j}⟩ = δ_{ip}δ_{jq} and α = \overline{y}\overline{x}^{2}\overline{y}\overline{x}^{2} ∈ F^{∗} = k {e_{0}, e_{1}}, where {e_{r}} is a dual basis of {t^{r}}, r = 0, 1. The right action ρ^{∗} : F^{∗} ⊗ K^{∗} → F^{∗} is trivial, thus F^{∗} lies in the center of H^{∗}.

\overline{x}^{2} = \overline{y}^{2}\overline{x}^{2} = \overline{y}\overline{x}^{2}\overline{y}α = \overline{x}^{2}\overline{y}α\overline{y}α = \overline{x}^{2}\overline{y}^{2}α^{2} = \overline{x}^{2}α^{2}

Thus α^{2} = 1.

Consider β = \overline{y}\overline{x}\overline{y}\overline{x} ∈ F^{∗} = k \{e_{0}, e_{1}\}. ε (β) = ε (\overline{y}^{−1}\overline{x}^{−1}\overline{y}\overline{x}) = 1 and therefore β = e_{0} + ξe_{1}. Moreover, \overline{y}\overline{x}\overline{y}\overline{x}^{−1} = β\overline{x}^{2} and

\overline{x} = \overline{y}^{2}\overline{x} = \overline{y}β\overline{x}^{2}\overline{x}\overline{y} = \overline{y}β\overline{x}^{2}\overline{y}β\overline{x}^{2}\overline{x} = \overline{y}\overline{x}^{2}\overline{y}\overline{x}^{3}β^{2} = \overline{y}^{2}α\overline{x}^{2}\overline{x}^{3}β^{2} = \overline{x}αβ^{2}

Thus β^{2} = α^{−1} = α, implying β^{4} = 1 and ξ = ±1 or ±i.

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨\overline{t}, \overline{x}^{i}\overline{y}^{j}e_{k}\overline{x}^{p}\overline{y}^{q}e_{r}⟩ = δ_{kr} ⟨\overline{t}, \overline{x}^{i+p}β^{jp}\overline{x}^{2jp}\overline{y}^{j+q}e_{k}⟩
= δ_{kr} ⟨\overline{t}, \overline{x}^{i+p+2jp}\overline{y}^{j+q}β^{jp}e_{k}⟩ = ξ^{jp}δ_{k1}δ_{r1}

On the other hand

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨θ (t) \overline{t} ⊗ \overline{t}, \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨θ (t) , \overline{x}^{i}\overline{y}^{j} ⊗ \overline{x}^{p}\overline{y}^{q}⟩ δ_{k1}δ_{r1}

Therefore

θ (t) = ∑_{ijpq} ξ^{jp}e_{ij} ⊗ e_{pq}

It is easy to check that if ξ = ±i then 1, X, Y and XY are the only grouplikes of H.

Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} then c_{0,0} = ε (v) = 1 and c_{i,j} ≠ 0, since v is a unit and

Δ_{H} (\overline{t}^{2}) = Δ_{H} (v) = Δ_{K} (∑ c_{i,j}e_{i,j}) = ∑ c_{p+r+2rq,q+s}e_{p,q} ⊗ e_{r,s}

On the other hand, if we write

t ⇀ e_{p,q} = e_{α_{1}(p,q),α_{2}(p,q)}
Δ (\overline{t}) Δ (\overline{t}) = ∑_{pqrs} ξ^{qr}e_{p,q}\overline{t} ⊗ e_{r,s}\overline{t} ∑_{pqrs} ξ^{qr}e_{p,q}\overline{t} ⊗ e_{r,s}\overline{t}
= ∑_{pqrs} ξ^{qr}e_{p,q} ⊗ e_{r,s} ∑_{pqrs} ξ^{qr}e_{α_{1}(p,q),α_{2}(p,q)}\overline{t}^{2} ⊗ e_{α_{1}(r,s),α_{2}(r,s)}\overline{t}^{2}
= ∑_{pqrs} ξ^{qr}e_{p,q} ⊗ e_{r,s} ∑_{pqrs} ξ^{α_{2}(p,q)α_{1}(r,s)}e_{p,q}\overline{t}^{2} ⊗ e_{r,s}\overline{t}^{2}
= ∑_{pqrs} ξ^{qr+α_{2}(p,q)α_{1}(r,s)}c_{pq}c_{rs}e_{p,q} ⊗ e_{r,s}

Thus for H to be a bialgebra we should have

c_{p+r+2rq,q+s} = ξ^{qr+α_{2}(p,q)α_{1}(r,s)}c_{pq}c_{rs} (3.16)

Action by t is a Hopf algebra map and therefore it induces a group automorphism f_{t} : G → G defined by ⟨e_{p,q}, f_{t} (g)⟩ = ⟨t ⇀ e_{p,q}, g⟩, which has order 2.

f_{t} (x) = x or x^{−1} since order of x is 4. If f_{t} (x) = x then in order for f_{t} to be of order 2 we should have f_{t} (y) = x^{2}y. If f_{t} (x) = x^{−1} then renaming generators we are down to two choices for f_{t} (y), namely f_{t} (y) = y or xy. Thus there are three possibilities for the action of t; we consider them below:

Case A). The action is given by t ⇀ e_{p,q} = e_{p+2q,q}, corresponding to

f_{t} (x) = x
f_{t} (y) = x^{2}y

Then X and Y are central grouplikes of H. Write v = σ (t, t) = ∑ c_{p,q}e_{p,q}. By (3.2) and (3.16)

(3.17)  c_{p,q} = c_{p+2q,q}
(3.18)  c_{p+r+2rq,q+s} = ξ^{qr+q(r+2s)}c_{p,q}c_{r,s}ξ^{2q(r+s)}c_{p,q}c_{r,s}

Conditions (3.17) and (3.18) imply that

ξ^{2}c_{0,1}c_{1,0} = c_{3,1} = c_{1,1} = c_{1,0}c_{0,1}
ξ^{2}c_{0,1}c_{0,1} = c_{0,0} = 1
c_{1,0}c_{1,0} = c_{2,0}
c_{2,0}c_{0,1} = c_{2,1} = c_{0,1}

Thus ξ^{2} = 1, c_{2,0} = 1 and c_{1,0}^{2} = c_{0,1}^{2} = 1. Therefore c_{1,0} = (−1)^{k} and c_{0,1} = (−1)^{l} for k, l = 0, 1 and

σ (t, t) = ∑ (−1)^{kp} (−1)^{lq} e_{p,q} = ∑ (−1)^{kp} e_{p,q} ∑ (−1)^{ls} e_{r,s} = X^{k}Y^{l},  k, l = 0, 1 (3.19)

If ξ = 1 then \overline{t} is a grouplike of H, if ξ = −1 then ∑ i^{p}e_{p,q}\overline{t} is a grouplike of H. In both cases G (H) is abelian of order 8 and H was described in Section 3.1 or Section 3.2.

Case B). The action is given by t ⇀ e_{p,q} = e_{−p,q}, corresponding to

f_{t} (x) = x^{−1}
f_{t} (y) = y

Then X and Y are central grouplikes of H. Write v = σ (t, t) = ∑ c_{p,q}e_{p,q} . By (3.2) and (3.16)

(3.20)  c_{p,q} = c_{−p,q}
(3.21)  c_{p+r+2rq,q+s} = ξ^{qr−qr}c_{p,q}c_{r,s} = c_{p,q}c_{r,s}

Conditions (3.20) and (3.21) imply that c_{1,0} = (−1)^{k} and c_{0,1} = (−1)^{l} for k, l = 0, 1 and

σ (t, t) = ∑ (−1)^{kp} (−1)^{lq} e_{p,q} = ∑ (−1)^{kp} e_{p,q} ∑ (−1)^{ls} e_{r,s} = X^{k}Y^{l},  k, l = 0, 1 (3.22)

If ξ = 1 then \overline{t} is a grouplike of H, if ξ = −1 then ∑ i^{p}e_{p,q}\overline{t} is a grouplike of H. In both cases G (H) is abelian of order 8 and H was described in Section 3.1 or Section 3.2. So now we will consider only ξ = ±i.

For k, l = 0, 1 let H_{ξ,X^{k}Y^{l}} be the Hopf algebras with the structures described above with cocycles σ_{k,l} (t, t) = X^{k}Y^{l}. Define

f : H_{ξ,X^{k}Y^{l}} → H_{−ξ,X^{k}Y^{l}}  by
f (e_{r,s}) = e_{r,s}
f (\overline{t}) = ∑ i^{p}e_{p,q}\overline{t}

and extend it multiplicatively to f (e_{r,s}\overline{t}). Then

f (\overline{t}) f (\overline{t}) = ∑ i^{p}e_{p,q}\overline{t} ∑ i^{p}e_{p,q}\overline{t} = ∑ i^{p}e_{p,q} ∑ i^{−p}e_{p,q}\overline{t}^{2} = \overline{t}^{2} = f (\overline{t}^{2})
f (\overline{t}e_{r,s}) = f (e_{−r,s}\overline{t}) = e_{−r,s} ∑ i^{p}e_{p,q}\overline{t} = ∑ i^{p}e_{p,q}\overline{t}e_{r,s} = f (\overline{t}) f (e_{r,s})
Δ (f (\overline{t})) = Δ (∑ i^{p}e_{p,q}\overline{t}) = Δ (∑ i^{p}e_{p,q}) Δ (\overline{t}) =
= (∑ i^{p_{1}+p_{2}+2q_{1}p_{2}}e_{p_{1},q_{1}} ⊗ e_{p_{2},q_{2}}) (∑ ξ^{s_{1}r_{2}}e_{r_{1},s_{1}} \overline{t} ⊗ e_{r_{2},s_{2}} \overline{t})
= (∑ i^{p_{1}+p_{2}+2q_{1}p_{2}}ξ^{q_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t})
= ∑ i^{p_{1}+p_{2}} (−ξ)^{q_{1}p_{2}} e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}
= ∑ (−ξ)^{q_{1}p_{2}} e_{p_{1},q_{1}}f (\overline{t}) ⊗ e_{p_{2},q_{2}}f (\overline{t})
= (f ⊗ f) (∑ (−ξ)^{q_{1}p_{2}} e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}) = (f ⊗ f) Δ (\overline{t})

and such an f is a Hopf algebra isomorphism between H_{ξ,X^{k}Y^{l}} and H_{−ξ,X^{k}Y^{l}}. Thus we may assume that ξ = i and write H_{i,X^{k}Y^{l}} = H_{X^{k}Y^{l}}. Define

f′ : H_{X^{k}Y} → H_{X^{k}}  by
f′ (e_{r,s}) = e_{r+2s,s}
f′ (\overline{t}) = ∑ i^{q}e_{p,q}\overline{t} = (\frac{1 + i}{2}1 + \frac{1 − i}{2}Y ) \overline{t}

and extend it multiplicatively to f′ (e_{r,s}\overline{t}). Note that restriction f′|_{(kD_{8})}^{∗} corresponds to the group automorphism f_{t} described in Case A) and f′ (X) = X, f′ (Y) = Y. Then

f′ (\overline{t}) f′ (\overline{t}) = ∑ i^{q}e_{p,q}\overline{t} ∑ i^{q}e_{p,q}\overline{t} = ∑ i^{q}e_{p,q} ∑ i^{q}e_{−p,q}\overline{t}^{2}
= ∑(−1)^{q}e_{p,q}X^{k} = Y X^{k} = f′ (Y X^{k}) = f′ (\overline{t}^{2})
f′ (\overline{t}e_{r,s}) = f′ (e_{−r,s}\overline{t}) = e_{−r+2s,s} ∑ i^{q}e_{p,q}\overline{t} = ∑ i^{q}e_{p,q}\overline{t}e_{r−2s,s} = f′ (\overline{t}) f′ (e_{r,s})

There are at most two nonisomorphic Hopf algebras of this kind:

1. H_{B:1} with trivial cocycle and G (H_{B:1}^{∗}) = ⟨χ, φ⟩ ≅ D_{8}, where φ(e_{r,s}) = δ_{r,2}δ_{s,0}, φ (\overline{t}) = 1, χ(e_{r,s}) = δ_{r,2}δ_{s,1}, χ(\overline{t}) = −1. There are two degree 2 irreducible representations defined by

\begin{matrix}π_{1} (e_{1,0}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{1} (e_{3,0}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{1} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{2} (e_{1,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{2} (e_{3,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{2} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

with the property π_{k}^{2} = π_{k} • π_{k} = 1 + χ^{2} + φ + χ^{2}φ.

2. H_{B:X} with the cocycle defined by σ_{X} (t, t) = X and G (H_{B:X}^{∗}) = ⟨χ, φ⟩ ≅ D_{8}, where χ(e_{r,s}) = δ_{r,2}δ_{s,1}, χ(\overline{t}) = −1, φ(e_{r,s}) = δ_{r,2}δ_{s,0}, φ (\overline{t}) = 1. There are two degree 2 irreducible representations defined by

\begin{matrix}π_{1} (e_{1,0}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{1} (e_{3,0}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{1} (\overline{t}) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) \\ π_{2} (e_{1,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{2} (e_{3,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{2} (\overline{t}) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) \end{matrix}

with the property π_{k}^{2} = π_{k} • π_{k} = 1 + χ^{2} + φ + χ^{2}φ.

Case C). The action is given by t ⇀ e_{p,q} = e_{−p+q,q}, corresponding to

f_{t} (x) = x^{−1}
f_{t} (y) = xy

Then Y is a central grouplike of H. Write v = σ (t, t) = ∑ c_{p,q}e_{p,q} . By (3.2) and (3.16)

(3.23)  c_{p,q} = c_{−p+q,q}
(3.24)  c_{p+r+2rq,q+s} = ξ^{qr+q(−r+s)}c_{p,q}c_{r,s} = ξ^{qs}c_{p,q}c_{r,s}

Conditions (3.23) and (3.24) imply that

c_{0,1} = c_{1,1}
c_{2,1} = c_{3,1}
c_{1,0} = c_{3,0}
c_{1,0}c_{0,1} = c_{1,1} = c_{0,1}
c_{0,1}c_{1,0} = c_{3,1}
ξc_{0,1}c_{0,1} = c_{0,0} = 1

Thus c_{1,0} = 1 and c_{0,1} = ω^{k}, where ω is a primitive 8-th root of 1 and ξ = ω^{−2k}. Therefore

σ_{k} (t, t) = ∑ ω^{kq}e_{p,q} = \frac{1 + Y}{2} + \frac{ω^{k} (1 − Y)}{2} , k = 0, …, 7

For k = 0, …, 7 let H_{k} be the Hopf algebra with the structure described above with cocycle σ_{k} (t, t). Define

f : H_{k+2} → H_{k}  by
f (e_{p,q}) = e_{p,q}

f (\overline{t}) = ∑_{p,q} i^{p}e_{p,q}\overline{t}

and extend it multiplicatively to f (e_{p,q}\overline{t}). Then

f (\overline{t}) f (\overline{t}) = ∑ i^{p}e_{p,q}\overline{t} ∑ i^{p}e_{p,q}\overline{t} = ∑ i^{p}e_{p,q} ∑ i^{p}e_{−p+q,q}\overline{t}^{2}
= ∑ i^{p}i^{−p+q}e_{p,q}σ_{k} (t, t) = ∑ i^{q}e_{p,q} ∑ ω^{kq}e_{p,q}
= ∑ ω^{kq+2q}e_{p,q} = σ_{k+2} (t, t) = f (σ_{k+2} (t, t)) = f (\overline{t}^{2})
f (\overline{t}e_{p,q}) = f (e_{−p+q,q}\overline{t}) = e_{−p+q,q} ∑ i^{p}e_{p,q}\overline{t} = ∑ i^{p}e_{p,q}\overline{t}e_{p,q} = f (\overline{t}) f (e_{p,q})
Δ (f (\overline{t})) = Δ (∑ i^{p}e_{p,q}\overline{t}) = Δ (∑ i^{p}e_{p,q}) Δ (\overline{t}) =
= (∑ i^{p_{1}+p_{2}+2q_{1}p_{2}}e_{p_{1},q_{1}} ⊗ e_{p_{2},q_{2}}) (∑ ω^{−2ks_{1}r_{2}}e_{r_{1},s_{1}} \overline{t} ⊗ e_{r_{2},s_{2}} \overline{t})
= (∑ i^{p_{1}+p_{2}+2q_{1}p_{2}}ω^{−2kq_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t})
= ∑ i^{p_{1}+p_{2}}ω^{−2(k+2)q_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}
= ∑ ω^{−2(k+2)q_{1}p_{2}}e_{p_{1},q_{1}}f (\overline{t}) ⊗ e_{p_{2},q_{2}}f (\overline{t})
= (f ⊗ f) (∑ ω^{−2(k+2)q_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}) = (f ⊗ f) Δ (\overline{t})

and such a f is a Hopf algebra isomorphism between H_{k} and H_{k+2}. Thus there are exactly two nonisomorphic Hopf algebras of this type:

1. H_{C:1} = H_{0} with a trivial cocycle and ξ = 1. Then G (H_{C:1}) = ⟨X\overline{t}, X⟩ ≅ D_{8} and G (H_{C:1}^{∗}) = ⟨χ⟩×⟨φ⟩ ≅ C_{2} ×C_{2}, where χ(e_{p,q}) = δ_{p,2}δ_{q,0}, χ(\overline{t}) = 1, φ(e_{p,q}) = δ_{p,0}δ_{q,0}, φ (\overline{t}) = −1. There are three degree 2 irreducible representations defined by

\begin{matrix}π_{1} (e_{0,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{1} (e_{1,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{1} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{2} (e_{1,0}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{2} (e_{3,0}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{2} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{3} (e_{2,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{3} (e_{3,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{3} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

with the property π_{2}^{2} = π_{2} • π_{2} = 1 + χ + φ + χφ, π_{1}^{2} = π_{3}^{2} = 1 + φ + π_{2}.

2. H_{C:σ_{1}} with cocycle σ_{1} defined by σ_{1} (t, t) = ∑ ω^{q}e_{p,q} and ξ = ω^{−2}, where ω is a primitive 8-th root of 1. Then G (H_{C:σ_{1}}) = ⟨X⟩ × ⟨Y ⟩ ≅ C_{2} × C_{2} and G (H_{C:σ_{1}}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2}, where χ(e_{p,q}) = δ_{p,2}δ_{q,0}, χ(\overline{t}) = 1, φ(e_{p,q}) = δ_{p,0}δ_{q,0}, φ (\overline{t}) = −1. There are three degree 2 irreducible representations defined by

\begin{matrix}π_{1} (e_{0,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{1} (e_{1,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{1} (\overline{t}) = ( \begin{matrix}0 & \sqrt{ω} \\ \sqrt{ω} & 0 \end{matrix} ) \\ π_{2} (e_{1,0}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{2} (e_{3,0}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{2} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{3} (e_{2,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{3} (e_{3,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{3} (\overline{t}) = ( \begin{matrix}0 & \sqrt{ω} \\ \sqrt{ω} & 0 \end{matrix} ) \end{matrix}

with the property π_{2}^{2} = π_{2} • π_{2} = 1 + χ + φ + χφ, π_{1}^{2} = π_{3}^{2} = 1 + φ + π_{2}.

3.4. Case of G = Q_{8}. Let G = Q_{8} = ⟨x, y | x^{4} = 1, y^{2} = x^{2}, yx = x^{−1}y⟩. Let {e_{pq}}_{p=0,1,2,3;q=0,1} be the basis of K, dual to the basis {x^{p}y^{q}}_{p=0,1,2,3;q=0,1} of K^{∗} = kQ_{8}. Then

Δ_{H} (e_{pq}) = Δ_{K} (e_{pq}) = ∑_{\begin{matrix}p_{1}+p_{2}+2q_{1}(p_{2}+q_{2})≡p mod 4 \\ q_{1}+q_{2}≡q mod 2 \end{matrix}} e_{p_{1}q_{1}} ⊗ e_{p_{2}q_{2}}

and it is easy to check that elements

X = ∑_{pq} (−1)^{p} e_{pq}
Y = ∑_{pq} (−1)^{q} e_{pq}

are grouplike of order 2. For \overline{t} = 1#t

Δ_{H} (\overline{t}) = θ (t) \overline{t} ⊗ \overline{t}

Dualizing (3.1) we get another extension

F^{∗} ↪^{π^{∗}} H^{∗} ↠^{i^{∗}} K^{∗}

and as in [11, 2.4], [12, 2.11] or [15, 2.1], since k is algebraically closed, there exist units \overline{x} and \overline{y} ∈ H^{∗}, such that such that \overline{x}^{4} = 1_{H^{∗}}, \overline{y}^{2} = \overline{x}^{2}, ⟨e_{pq}, \overline{x}^{i}\overline{y}^{j}⟩ = δ_{ip}δ_{jq} and α = \overline{x}\overline{y}\overline{x}\overline{y}^{−1} ∈ F^{∗} = k \{e_{0}, e_{1}\}, where \{e_{r}\} is a dual basis of \{t^{r}\}, r = 0, 1. ε (α) = ε (\overline{x}\overline{y}\overline{x}\overline{y}^{−1}) = 1 and therefore α = e_{0}+ξe_{1}. The right action ρ^{∗} : F^{∗}⊗K^{∗} → F^{∗} is trivial, thus F^{∗} lies in the center of H^{∗}. Moreover, \overline{x}^{2} = \overline{y}^{2} also lies in the center of H^{∗}. Then

\overline{x}\overline{y}\overline{x}^{−1}\overline{y}^{−1} = \overline{x}\overline{y}\overline{x}^{3}\overline{y}^{−1} = \overline{x}\overline{y}\overline{x}\overline{y}^{−1}\overline{x}^{2} = α\overline{x}^{2}
\overline{x}^{3} = \overline{x}\overline{x}^{2} = \overline{x}\overline{y}^{2} = α\overline{x}^{2}\overline{y}\overline{x}\overline{y} = α\overline{x}^{2}\overline{y}α\overline{x}^{2}\overline{y}\overline{x} = α^{2}\overline{x}^{4}\overline{y}^{2}\overline{x} = α^{2}\overline{x}^{3}

Thus α^{2} = 1 and ξ = ±1.

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨\overline{t}, \overline{x}^{i}\overline{y}^{j}e_{k}\overline{x}^{p}\overline{y}^{q}e_{r}⟩ = δ_{kr} ⟨\overline{t}, \overline{x}^{i+p} (α\overline{x}^{2})^{−jp} \overline{y}^{j}\overline{y}^{q}e_{k}⟩
= δ_{kr} ⟨\overline{t}, \overline{x}^{i+p+2jp+2jq}\overline{y}^{j+q−2jq}α^{jp}e_{k}⟩ = ξ^{jp}δ_{k1}δ_{r1}

On the other hand

⟨Δ_{H} (\overline{t}) , \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨θ (t) \overline{t} ⊗ \overline{t}, \overline{x}^{i}\overline{y}^{j}e_{k} ⊗ \overline{x}^{p}\overline{y}^{q}e_{r}⟩ = ⟨θ (t) , \overline{x}^{i}\overline{y}^{j} ⊗ \overline{x}^{p}\overline{y}^{q}⟩ δ_{k1}δ_{r1} Therefore

θ (t) = ∑_{ijpq} ξ^{jp}e_{ij} ⊗ e_{pq}

If ξ = 1 then \overline{t} is a grouplike of H, if ξ = −1 then ∑ i^{p+q^{2}}e_{p,q}\overline{t} is a grouplike of H. Thus G (H) has always order 8.

Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} then c_{0,0} = ε (v) = 1 and c_{i,j} ≠ 0, since v is a unit and

Δ_{H} (\overline{t}^{2}) = Δ_{H} (v) = Δ_{K} (∑ c_{i,j}e_{i,j}) = ∑ c_{p+r+2rq+2sq,q+s}e_{p,q} ⊗ e_{r,s}

Action by t is a Hopf algebra map and therefore it induces a group automorphism f_{t} : G → G defined by ⟨e_{p,q}, f_{t} (g)⟩ = ⟨t ⇀ e_{p,q}, g⟩, which has order 2. Renaming generators we are down to two choices for f_{t}; we consider them below:

Case D). The action is given by t ⇀ e_{i,j} = e_{i+2j,j}, corresponding to

f_{t} (x) = x
f_{t} (y) = x^{2}y

Then X and Y are central grouplikes of H. Thus G (H) is abelian of order 8 and H was described in Section 3.1 or Section 3.2.

Case E). The action is given by t ⇀ e_{i,j} = e_{−i+j,j}, corresponding to

f_{t} (x) = x^{−1}
f_{t} (y) = xy

Then Y is a central grouplike of H. Write v = σ (t, t) = ∑ c_{i,j}e_{i,j} . By (3.2)

c_{i,j} = c_{−i+j,j} (3.25)

On the other hand, for H to be a bialgebra

Δ_{H} (\overline{t}^{2}) = Δ_{H} (\overline{t}) Δ_{H} (\overline{t}) = (θ (t) \overline{t} ⊗ \overline{t}) (θ (t) \overline{t} ⊗ \overline{t})
= (∑_{pqrs} ξ^{rq}e_{pq} ⊗ e_{rs}) (∑_{pqrs} ξ^{rq} (t ⇀ e_{pq}) ⊗ (t ⇀ e_{rs})) σ (t, t) ⊗ σ (t, t)
= ∑ ξ^{rq}ξ^{(−r+s)q}c_{p,q}c_{r,s}e_{p,q} ⊗ e_{r,s} = ∑ ξ^{qs}c_{p,q}c_{r,s}e_{p,q} ⊗ e_{r,s}

Therefore

c_{p+r+2(r+s)q,q+s} = ξ^{qs}c_{p,q}c_{r,s} (3.26)

Conditions (3.25) and (3.26) imply that

c_{0,1} = c_{1,1}
c_{2,1} = c_{3,1}
c_{1,0} = c_{3,0}
c_{1,0}c_{0,1} = c_{1,1} = c_{0,1}
c_{0,1}c_{1,0} = c_{3,1}

ξc_{0,1}c_{0,1} = c_{2,0} = c_{1,0}c_{1,0}

Thus c_{1,0} = 1 and c_{0,1} = i^{k}, where ξ = i^{2k} and k = 0, 1, 2, 3. Therefore

σ_{k} (t, t) = ∑ i^{kq}e_{p,q} = \frac{1 + Y}{2} + \frac{i^{k} (1 − Y)}{2}

Let H_{k} be the Hopf algebra with the structure described above with cocycle σ_{k} (t, t). Define

f : H_{k} → H_{k+1} by
f (e_{p,q}) = e_{p,q}
f (\overline{t}) = ∑_{p,q} i^{p+q^{2}}e_{p,q}\overline{t}

and extend it multiplicatively to f (e_{p,q}\overline{t}). Then

f (\overline{t}) f (\overline{t}) = ∑ i^{p+q^{2}}e_{p,q}\overline{t} ∑ i^{p+q^{2}}e_{p,q}\overline{t} = ∑ i^{p+q^{2}}e_{p,q} ∑ i^{p+q^{2}}e_{−p+q,q}\overline{t}^{2}
= ∑ i^{p+q^{2}}i^{−p+q+q^{2}}e_{p,q}σ_{k+1} (t, t) = ∑ i^{3q}e_{p,q} ∑ i^{(k+1)q}e_{p,q}
= ∑ i^{(k+1)q+3q}e_{p,q} = ∑ i^{kq}e_{p,q} = σ_{k} (t, t) = f (σ_{k} (t, t)) = f (\overline{t}^{2})
f (\overline{t}e_{p,q}) = f (e_{−p+q,q}\overline{t}) = e_{−p+q,q} ∑ i^{p+q^{2}}e_{p,q}\overline{t} = ∑ i^{p+q^{2}}e_{p,q}\overline{t}e_{p,q} = f (\overline{t}) f (e_{p,q})
Δf (\overline{t}) = Δ (∑ i^{p+q^{2}}e_{p,q}\overline{t}) = Δ (∑ i^{p+q^{2}}e_{p,q}) Δ (\overline{t}) =
= (∑ i^{p_{1}+p_{2}+2q_{1}(p_{2}+q_{2})+(q_{1}+q_{2})^{2}}e_{p_{1},q_{1}} ⊗ e_{p_{2},q_{2}}) (∑ i^{2(k+1)s_{1}r_{2}}e_{r_{1},s_{1}} \overline{t} ⊗ e_{r_{2},s_{2}} \overline{t})
= (∑ i^{p_{1}+p_{2}+2q_{1}(p_{2}+q_{2})+(q_{1}+q_{2})^{2}+2(k+1)q_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t})
= ∑ i^{p_{1}+p_{2}+q_{1}^{2}+q_{2}^{2}}i^{2kq_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}
= ∑ i^{2kq_{1}p_{2}}e_{p_{1},q_{1}}f (\overline{t}) ⊗ e_{p_{2},q_{2}}f (\overline{t})
= (f ⊗ f) (∑ i^{2kq_{1}p_{2}}e_{p_{1},q_{1}} \overline{t} ⊗ e_{p_{2},q_{2}} \overline{t}) = (f ⊗ f) Δ (\overline{t})

and such an f is a Hopf algebra isomorphism between H_{k} and H_{k+1}. Thus there is exactly one Hopf algebra of this type: H_{E} = H_{0} with a trivial cocycle and ξ = 1. Then G (H_{E}) = ⟨X\overline{t}, X⟩ ≅ D_{8} and G (H_{E}^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2}, where χ(e_{p,q}) = δ_{p,2}δ_{q,0}, χ(\overline{t}) = 1, φ(e_{p,q}) = δ_{p,0}δ_{q,0}, φ (\overline{t}) = −1. There are three degree 2 irreducible representations defined by

\begin{matrix}π_{1} (e_{0,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{1} (e_{1,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{1} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{2} (e_{1,0}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{2} (e_{3,0}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{2} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{3} (e_{2,1}) = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) & π_{3} (e_{3,1}) = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) & π_{3} (\overline{t}) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

with the property π_{2}^{2} = π_{2} • π_{2} = 1 + χ + φ + χφ, π_{1}^{2} = π_{3}^{2} = χ + χφ + π_{2}.

3.5. Summary.

Proposition 3.1. Let H be a nontrivial semisimple Hopf algebra of dimension 16. Then G (H) is abelian of order 8 if and only if G (H^{∗}) is abelian of order 8.

Proof. All nontrivial Hopf algebras with abelian groups of grouplikes were described in Sections 3.1 and 3.2 and their duals have abelian groups of grouplikes.

Proposition 3.2. There are exactly 7 nonisomorphic nontrivial semisimple Hopf algebras of dimension 16 with G (H) ≅ C_{4} × C_{2}.

Proof. All nontrivial Hopf algebras with G (H) ≅ C_{4} × C_{2} were described in Section 3.1. There are at most 7 nonisomorphic Hopf algebras with G (H) ≅ C_{4} ×C_{2}, namely H_{a:1}, H_{a:y}, H_{b:1}, H_{b:y}, H_{b:x^{2}y}, H_{c:σ_{0}}, H_{c:σ_{1}}.

Assume f is a Hopf algebra isomorphism between Hopf algebras H_{1} and H_{2} with G (H_{1}) ≅ G (H_{2}) ≅ C_{4} × C_{2}. Then we get a group isomorphism

f |_{G(H_{1})}: G (H_{1}) → G (H_{2})

Write G (H_{1}) ≅ G (H_{2}) = ⟨x⟩ × ⟨y⟩, where |x| = 4 and |y| = 2. Then the dual basis of kG (H_{1}) ≅ kG (H_{2}) is given by

e_{pq} = 1/8 (1 + i^{p}x + i^{2p}x^{2} + i^{3p}x^{3}) (1 + (−1)^{q} y) , p = 0, 1, 2, 3; q = 0, 1

Write

f (e_{p,q}) = e_{α_{1}(p,q),α_{2}(p,q)}
f^{−1} (e_{p,q}) = e_{β_{1}(p,q),β_{2}(p,q)}

where α_{1} (p, q) , β_{1} (p, q) ∈ {0, 1, 2, 3} and α_{2} (p, q) , β_{2} (p, q) ∈ {0, 1}.

Write \{e_{pq}\overline{t^{r}}\}_{p=0,1,2,3;q=0,1;r=0,1} and \{e_{pq}\overline{T^{r}}\}_{p=0,1,2,3;q=0,1;r=0,1} for the bases of H_{1} and H_{2}, respectively. Write

f (\overline{t}) = ∑_{p,q,r} λ_{p,q,r}e_{pq}\overline{T^{r}}

Then

Δf (\overline{t}) = Δ (∑_{p,q,r} λ_{p,q,r}e_{pq}\overline{T^{r}}) = ∑_{p,q} λ_{p,q,0}Δ (e_{pq}) + ∑_{p,q} λ_{p,q,1}Δ (e_{pq}) Δ (\overline{T} )
= ∑ λ_{p_{1}+p_{2},q_{1}+q_{2},0}e_{p_{1}q_{1}} ⊗ e_{p_{2}q_{2}}
+ (∑ λ_{p_{1}+p_{2},q_{1}+q_{2},1}e_{p_{1}q_{1}} ⊗ e_{p_{2}q_{2}}) (∑ (−1)^{bc} e_{ab}\overline{T} ⊗ e_{cd}\overline{T} )
= ∑ λ_{p_{1}+p_{2},q_{1}+q_{2},0}e_{p_{1}q_{1}} ⊗ e_{p_{2}q_{2}} + ∑ (−1)^{p_{2}q_{1}} λ_{p_{1}+p_{2},q_{1}+q_{2},1}e_{p_{1}q_{1}} \overline{T} ⊗ e_{p_{2}q_{2}} \overline{T}
(f ⊗ f) Δ (\overline{t}) = (f ⊗ f) (∑ (−1)^{p_{2}q_{1}} e_{p_{1}q_{1}} \overline{T} ⊗ e_{p_{2}q_{2}} \overline{T} )
= ∑ (−1)^{p_{2}q_{1}} f (e_{p_{1}q_{1}}) ∑_{p,q,r} λ_{p,q,r}e_{pq}\overline{T^{r}} ⊗ f (e_{p_{2}q_{2}}) ∑_{p,q,r} λ_{p,q,r}e_{pq}\overline{T^{r}}
= ∑ (−1)^{β_{1}(p_{2},q_{2})β_{2}(p_{1},q_{1})} λ_{p_{1},q_{1},r_{1}}λ_{p_{2},q_{2},r_{2}}e_{p_{1}q_{1}} \overline{T^{r_{1}}} ⊗ e_{p_{2}q_{2}} \overline{T^{r_{2}}}

Since f is a coalgebra map,

Δf (\overline{t}) = (f ⊗ f) Δ (\overline{t})

and therefore λ_{p_{1},q_{1},0}λ_{p_{2},q_{2},1} = 0 for all p_{1}, p_{2} ∈ {0, 1, 2, 3}, q_{1}, q_{2} ∈ {0, 1}. Thus either λ_{p,q,0} = 0 for all p ∈ {0, 1, 2, 3}, q ∈ {0, 1} or λ_{p,q,1} = 0 for all p ∈ {0, 1, 2, 3}, q ∈ \{0, 1\}. In the latter case f (\overline{t}) = ∑ λ_{p,q,0}e_{pq} ∈ kG (H_{2}), which contradicts the bijectivity of f. Therefore λ_{p,q,0} = 0 for all p ∈ {0, 1, 2, 3}, q ∈ {0, 1}. Write λ_{p,q} = λ_{p,q,1}. Then

f (\overline{t}) = ∑_{p,q} λ_{p,q}e_{pq}\overline{T}

and so, applying ε, also

λ_{0,0} = ε (\overline{t}) = 1

Moreover, since ∑ (−1)^{p_{2}q_{1}} λ_{p_{1}+p_{2},q_{1}+q_{2}}e_{p_{1}q_{1}} \overline{T} ⊗ e_{p_{2}q_{2}} \overline{T} = ∑ (−1)^{β_{1}(p_{2},q_{2})β_{2}(p_{1},q_{1})} λ_{p_{1},q_{1}}λ_{p_{2},q_{2}}e_{p_{1}q_{1}} \overline{T} ⊗ e_{p_{2}q_{2}} \overline{T} we get

λ_{p_{1}+p_{2},q_{1}+q_{2}} = (−1)^{p_{2}q_{1}} (−1)^{β_{1}(p_{2},q_{2})β_{2}(p_{1},q_{1})} λ_{p_{1},q_{1}}λ_{p_{2},q_{2}} (3.27)

for any p_{1}, p_{2} ∈ {0, 1, 2, 3}, q_{1}, q_{2} ∈ {0, 1}.

Let u ∈ kG (H_{1}). Then

f (t ⇀_{1} u) f (\overline{t}) = f ((t ⇀_{1} u) \overline{t}) = f (\overline{t}u) = f (\overline{t}) f (u) = ∑ λ_{p,q}e_{p,q}\overline{T} f (u)
= (t ⇀_{2} f (u)) ∑ λ_{p,q}e_{p,q}\overline{T} = (t ⇀_{2} f (u)) f (\overline{t})

Thus, since \overline{t} is a unit (\overline{t}^{2} = σ (t, t) is a unit),

f (t ⇀_{1} u) = t ⇀_{2} f (u) (3.28)

Let’s show that Hopf algebras from types H_{a}, H_{b} and H_{c} cannot be isomorphic to each other. K_{0} (H_{c}) ≇ K_{0} (H_{a}) or K_{0} (H_{b}), thus H_{c} ≇ H_{a} or H_{b}. If f : H_{a} → H_{b} then by formula (3.28)

f (x) f (y) = f (xy) = f (t ⇀_{1} x) = t ⇀_{2} f (x) = f (x)^{−1}

and therefore f (y) = f (x^{2}), which is impossible if f is an isomorphism.

H_{b:1} ≇ H_{b:y} or H_{b:x^{2}y} and H_{c:σ_{0}} ≇ H_{c:σ_{1}} since their duals have nonisomorphic groups of grouplikes. Thus there are at least 5 nonisomorphic Hopf algebras with G (H) ≅ C_{4} × C_{2}, namely H_{a:1}, H_{b:1}, H_{b:y}, H_{c:σ_{0}} and H_{c:σ_{1}}.

Now we prove that H_{a:1} ≇ H_{a:y} and H_{b:y} ≇ H_{b:x^{2}y}. If f is a Hopf algebra isomorphism as before, then since f |_{G(H_{1})} is a group isomorphism f (x) ∈ {x, x^{−1}, xy, x^{−1}y}, f (y) ∈ {y, x^{2}y} and f (x^{2}) = x^{2}.

If f (x) = x^{2k+1}y^{l} and f (y) = y, where k, l = 0, 1 then

f^{−1} (e_{p,q}) = f (e_{p,q}) = e_{(2k+1)p+2lq,q}

and by formula (3.27)

λ_{p_{1}+p_{2},q_{1}+q_{2}} = (−1)^{p_{2}q_{1}} (−1)^{((2k+1)p_{2}+2lq_{2})q_{1}} λ_{p_{1},q_{1}}λ_{p_{2},q_{2}} = λ_{p_{1},q_{1}}λ_{p_{2},q_{2}}

26 YEVGENIA KASHINA

If f (x) = x^{2k+1}y^{l} and f (y) = x^{2}y, where k, l = 0, 1 then

f (e_{p,q}) = e_{(2k+2l+1)p+2lq,p+q}
f^{−1} (e_{p,q}) = e_{(2k+1)p+2lq,p+q}

and by formula (3.27)

λ_{p_{1}+p_{2},q_{1}+q_{2}} = (−1)^{p_{2}q_{1}} (−1)^{((2k+1)p_{2}+2lq_{2})(q_{1}+p_{1})} λ_{p_{1},q_{1}}λ_{p_{2},q_{2}} = (−1)^{p_{1}p_{2}} λ_{p_{1},q_{1}}λ_{p_{2},q_{2}} Now assume f is a Hopf algebra isomorphism

f : H_{a:1} → H_{a:y}

If f (y) = x^{2}y then by formula (3.28)

f (x) y = t ⇀_{2} f (x) = f (t ⇀_{1} x) = f (xy) = f (x) f (y) = f (x) x^{2}y

that is x^{2} = 1, which contradicts the fact that |x| = 4. Thus f (y) = y and therefore

λ_{p_{1}+p_{2},q_{1}+q_{2}} = λ_{p_{1},q_{1}}λ_{p_{2},q_{2}}

and thus

λ_{1,0}^{4} = λ_{2,0}^{2} = λ_{0,1}^{2} = λ_{0,0} = 1

Then

f (\overline{t}) f (\overline{t}) = ∑ λ_{p,q}e_{pq}\overline{T} ∑ λ_{r,s}e_{rs}\overline{T}
= ∑ λ_{p,q}e_{pq} ∑ λ_{r,s}e_{r+2s,s}\overline{T}^{2} = ∑ λ_{p,q}λ_{p+2q,q}e_{pq}σ_{H_{a:y}} (t, t)
= ∑ λ_{2(p+q),0}e_{pq}σ_{H_{a:y}} (t, t) = ∑ λ_{2,0}^{p+q}e_{pq}σ_{H_{a:y}} (t, t)

If λ_{2,0} = 1, f (\overline{t}) f (\overline{t}) = σ_{H_{a:y}} (t, t) = y ≠ f (\overline{t}^{2}) = 1. If λ_{2,0} = −1, f (\overline{t}) f (\overline{t}) = x^{2}yσ_{H_{a:y}} (t, t) = x^{2} ≠ f (\overline{t}^{2}) = 1.

Therefore, there is no Hopf algebra isomorphism between H_{a:1} and H_{a:y}.

Now assume f is a Hopf algebra isomorphism

f : H_{b:y} → H_{b:x^{2}y}

Then

f (\overline{t}) f (\overline{t}) = ∑ λ_{p,q}e_{pq}\overline{T} ∑ λ_{r,s}e_{rs}\overline{T}
= ∑ λ_{p,q}e_{pq} ∑ λ_{r,s}e_{−r,s}\overline{T}^{2} = ∑ λ_{p,q}λ_{−p,q}e_{pq}σ_{H_{b:x^{2}y}} (t, t)

f (y) = y is not possible, since then we have

λ_{p,q}λ_{−p,q} = λ_{0,0} = 1

and thus

f (\overline{t}) f (\overline{t}) = (∑ e_{pq}) σ_{H_{b:x^{2}y}} (t, t) = σ_{H_{b:x^{2}y}} (t, t) = x^{2}y ≠ y = f (y) = f (\overline{t}^{2}) . f (y) = x^{2}y is not possible, since then we get

λ_{p_{1}+p_{2},q_{1}+q_{2}} = (−1)^{p_{1}p_{2}} λ_{p_{1},q_{1}}λ_{p_{2},q_{2}}

so

λ_{p,q}λ_{−p,q} = (−1)^{p^{2}} λ_{0,0} = (−1)^{p^{2}}

and

f (\overline{t}) f (\overline{t}) = (∑ (−1)^{p^{2}} e_{pq}) σ_{H_{b:x^{2}y}} (t, t) = x^{2}x^{2}y = y = f (x^{2}y) ≠ f (y) = f (\overline{t}^{2}) .

Therefore, there is no Hopf algebra isomorphism between H_{b:y} and H_{b:x^{2}y}.

Thus there are exactly 7 nonisomorphic Hopf algebras with G (H) ≅ C_{4} × C_{2}, namely H_{a:1}, H_{a:y}, H_{b:1}, H_{b:y}, H_{b:x^{2}y}, H_{c:σ_{0}} and H_{c:σ_{1}}.

Proposition 3.3. There are at least 2 and at most 4 nonisomorphic nontrivial semisimple Hopf algebras of dimension 16 with G (H) ≅ C_{2} × C_{2} × C_{2}.

Proof. All nontrivial Hopf algebras with G (H) ≅ C_{2} × C_{2} × C_{2} were described in Section 3.2. There are at most 4 nonisomorphic Hopf algebras with G (H) ≅ C_{2} × C_{2} × C_{2}, namely H_{d:1,1}, H_{d:1,−1}, H_{d:−1,1} and H_{d:−1,−1}. At least two of them are not isomorphic, since G (H_{d:1,1}^{∗}) ≇ G (H_{d:1,−1}^{∗}).

Proposition 3.4. There are exactly 2 nonisomorphic nontrivial semisimple Hopf algebras of dimension 16 with a commutative subHopfalgebra of dimension 8 and nonabelian G (H^{∗}). In this case G (H^{∗}) ≅ D_{8}, G (H) ≅ C_{2} × C_{2} and H^{∗} also has a commutative subHopfalgebra of dimension 8.

Proof. All nontrivial Hopf algebras with a commutative subHopfalgebra of dimension 8 and nonabelian G (H^{∗}) were described in Section 3.3, Case B). There are at most 2 of them, namely H_{B:1} and H_{B:X}.

Let’s compute all the possible 8-dimensional Hopf quotients of H_{B}. There is a oneto-one correspondence between hereditary subrings of K_{0}(H) and Hopf quotients of H (see [22, Theorem 6] or [24, Proposition 3.11]). Thus H_{B} has 3 quotients of dimension 8 corresponding to the hereditary subrings R_{1} = {a1+bφ+cχ^{2}+dχ^{2}φ+eπ_{1} ∈ K_{0}(H) : a, b, c, d, e ∈ ℤ}, R_{2} = {a1 + bφ + cχ^{2} + dχ^{2}φ + eπ_{2} ∈ K_{0}(H) : a, b, c, d, e ∈ ℤ} and R_{3} = {∑_{i=0}^{4} ∑_{j=0}^{2} a_{i,j}χ^{i}φ^{j} ∈ K_{0}(H) : a_{i,j} ∈ ℤ}. They are obtained by factoring modulo normal ideals (Y − 1) H, (X − 1) H and (XY − 1) H, where X, Y and XY are central grouplikes of H. It is easy to see that H/ (Y − 1) H is cocommutative (in fact, H_{B:1}/ (Y − 1) H_{B:1} ≅ kD_{8} and H_{B:X}/ (Y − 1) H_{B:X} ≅ kQ_{8}), H/ (X − 1) H is commutative (therefore H/ (X − 1) H ≅ (kD_{8})^{∗} since G (H^{∗}) ≅ D_{8}) and H/ (XY − 1) H is neither commutative nor cocommutative (therefore H/ (X − 1) H ≅ H_{8}). Therefore we see that H_{B:1} ≇ H_{B:X} since they have different sets of quotients. Both H_{B:1} and H_{B:X} have cocommutative Hopf quotients of dimension 8, kD_{8} and kQ_{8} respectively. Thus their duals were described in Section 3.4. In particular, H_{B:1} ≅ H_{C:1}^{∗}, H_{B:X} ≅ H_{E}^{∗} and G (H_{C:1}^{∗}) ≅ G (H_{E}^{∗}) ≅ C_{2} × C_{2}.

4. Nonabelian groups of order 16.

There are nine nonabelian groups of order 16 (see [2, 118]). The first four of them are of exponent 8, the last five of exponent 4 (we denote the quaternion group of order 8 by Q_{8} and the quasiquaternion group of order 16 by Q_{16}):

1. G_{1} = ⟨a, b : a^{8} = b^{2} = 1, ba = a^{5}b⟩ G ((kG_{1})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(a) = i, χ(b) = 1, φ (a) = 1, φ (b) = −1. Degree 2 irreducible representations of G_{1} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}ω & 0 \\ 0 & −ω \end{matrix} ) & π_{1}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}ω^{3} & 0 \\ 0 & −ω^{3} \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

where ω is a primitive 8^{th}-root of unity and π_{1}^{2} = χ + χ^{3} + χφ + χ^{3}φ = π_{2}^{2}.

2. G_{2} = ⟨a, b : a^{8} = b^{2} = 1, ba = a^{3}b⟩ G ((kG_{2})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2}, where χ(a) = −1, χ(b) = 1, φ (a) = 1, φ (b) = −1. Degree 2 irreducible representations of G_{2} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}ω & 0 \\ 0 & ω^{3} \end{matrix} ) & π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{3}(a) = ( \begin{matrix}ω^{5} & 0 \\ 0 & ω^{7} \end{matrix} ) \\ π_{1}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{3}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

where ω is a primitive 8^{th}-root of unity, and representations satisfy the properties

\begin{matrix}π_{1}^{2} = χ + χφ + π_{2} = π_{3}^{2} & χ • π_{1} = π_{3} & χ • π_{3} = π_{1} \\ π_{2}^{2} = 1 + χ + φ + χφ & φ • π_{1} = π_{1} & φ • π_{3} = π_{3} \end{matrix}

3. G_{3} = ⟨a, b : a^{8} = b^{2} = 1, ba = a^{−1}b⟩ = D_{16}, the dihedral group G ((kG_{3})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2}, where χ(a) = −1, χ(b) = 1, φ (a) = 1, φ (b) = −1. Degree 2 irreducible representations of G_{3} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}ω & 0 \\ 0 & ω^{7} \end{matrix} ) & π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{3}(a) = ( \begin{matrix}ω^{3} & 0 \\ 0 & ω^{5} \end{matrix} ) \\ π_{1}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{3}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

where ω is a primitive 8^{th}-root of unity, and representations satisfy the properties

\begin{matrix}π_{1}^{2} = 1 + φ + π_{2} = π_{3}^{2} & χ • π_{1} = π_{3} & χ • π_{3} = π_{1} \\ π_{2}^{2} = 1 + χ + φ + χφ & φ • π_{1} = π_{1} & φ • π_{3} = π_{3} \end{matrix}

4. G_{4} = ⟨a, b : a^{8} = 1, b^{2} = a^{4}, ba = a^{−1}b⟩ = Q_{16}, the quasiquaternion group G ((kG_{4})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2}, where χ(a) = −1, χ(b) = 1, φ (a) = 1, φ (b) = −1. Degree 2 irreducible representations of G_{4} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}ω & 0 \\ 0 & ω^{7} \end{matrix} ) & π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{3}(a) = ( \begin{matrix}ω^{3} & 0 \\ 0 & ω^{5} \end{matrix} ) \\ π_{1}(b) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{3}(b) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) \end{matrix}

where ω is a primitive 8^{th}-root of unity, and representations satisfy the properties

\begin{matrix}π_{1}^{2} = 1 + φ + π_{2} = π_{3}^{2} & χ • π_{1} = π_{3} & χ • π_{3} = π_{1} \\ π_{2}^{2} = 1 + χ + φ + χφ & φ • π_{1} = π_{1} & φ • π_{3} = π_{3} \end{matrix}

5. G_{5} = ⟨a, b : a^{4} = b^{4} = 1, ba = a^{−1}b⟩ G ((kG_{5})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(a) = 1, χ(b) = i, φ (a) = −1, φ (b) = 1. Degree 2 irreducible representations of G_{5} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{1}(b) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) & π_{2}(b) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) \end{matrix}

with the property π_{1}^{2} = 1 + χ^{2} + φ + χ^{2}φ = π_{2}^{2}.

6. G_{6} = ⟨a, b, c : a^{4} = b^{2} = c^{2} = 1, bab = ac⟩ G ((kG_{6})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2}, where χ(a) = i, χ(b) = χ(c) = 1, φ (a) = φ (c) = 1, φ (b) = −1. Degree 2 irreducible representations of G_{6} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) & π_{1}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{1}(c) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{2}(c) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) \end{matrix}

with the property π_{1}^{2} = 1 + χ^{2} + φ + χ^{2}φ = π_{2}^{2}.

7. G_{7} = ⟨a, b, c : a^{4} = b^{2} = c^{2} = 1, cbc = a^{2}b⟩ G ((kG_{7})^{∗}) = ⟨χ⟩×⟨φ⟩×⟨ψ⟩ ≅ C_{2}×C_{2}×C_{2}, where χ(a) = −1, χ(b) = χ(c) = 1, φ (a) = φ (b) = −1, φ (c) = 1, ψ (a) = ψ (b) = 1, ψ (c) = −1. Degree 2 irreducible representations of G_{7} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{1}(b) = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) & π_{1}(c) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{2}(b) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) & π_{2}(c) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

with the property π_{1}^{2} = χ + χφ + χψ + χφψ = π_{2}^{2}.

8. G_{8} = ⟨a, b, c : a^{4} = b^{2} = c^{2} = 1, ba = a^{−1}b⟩ = D_{8} × C_{2} G ((kG_{8})^{∗}) = ⟨χ⟩×⟨φ⟩×⟨ψ⟩ ≅ C_{2}×C_{2}×C_{2}, where χ(a) = χ(b) = 1, χ(c) = −1, φ (a) = −1, φ (b) = φ (c) = 1, ψ (a) = ψ (c) = 1, ψ (b) = −1. Degree 2 irreducible representations of G_{8} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{1}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{1}(c) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) & π_{2}(c) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) \end{matrix}

with the property π_{1}^{2} = 1 + φ + ψ + φψ = π_{2}^{2}.

9. G_{9} = ⟨a, b, c : a^{4} = c^{2} = 1, b^{2} = a^{2}, ba = a^{−1}b⟩ = Q_{8} × C_{2} G ((kG_{9})^{∗}) = ⟨χ⟩×⟨φ⟩×⟨ψ⟩ ≅ C_{2}×C_{2}×C_{2}, where χ(a) = χ(b) = 1, χ(c) = −1, φ (a) = −1, φ (b) = φ (c) = 1, ψ (a) = ψ (c) = 1, ψ (b) = −1. Degree 2 irreducible representations of G_{9} are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{1}(b) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) & π_{1}(c) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) & π_{2}(c) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) \end{matrix}

with the property π_{1}^{2} = 1 + φ + ψ + φψ = π_{2}^{2}.

5. Computations in K_{0}(H) in the case of |G (H^{∗})| = 8.

In this case we have 8 one-dimensional irreducible representations χ_{1} = 1_{K_{0}(H)}, …, χ_{8} ∈ G (H^{∗}) and two 2-dimensional ones π_{1} and π_{2}. Then, since χ_{i} • χ_{i}^{−1} = 1_{K_{0}(H)}, χ_{i}^{∗} = χ_{i}^{−1}.

deg (χ_{i} • π_{k}) = 2 thus there are two possibilities:

i) χ_{i} • π_{k} = χ_{j} + χ_{l} ii) χ_{i} • π_{k} = π_{l} Case i) cannot happen, since otherwise π_{k} = χ_{i}^{−1} • χ_{j} + χ_{i}^{−1} • χ_{l} is not irreducible.

Thus χ_{i} • π_{k} = π_{l}. Then it is impossible to have χ_{i} • π_{k} = π_{k} and χ_{i} • π_{l} = π_{k} for k ≠ l, since otherwise π_{l} = χ_{i}^{8} • π_{l} = χ_{i}^{7} • π_{k} = π_{k}. Thus χ_{i} either fixes both π_{1} and π_{2} or interchanges them. It is easy to check that either all χ_{i} fix π_{k}, k = 1, 2, or half of χ_{i} fixes π_{k} and half of χ_{i} interchanges them.

Suppose χ_{i} • π_{k} = π_{k}, for i = 1, …, 8 and k = 1, 2. Then

1 = m (π_{k}, χ_{i} • π_{k}) = m (χ_{i}^{∗}, π_{k} • π_{k}^{∗}) = m (χ_{i}^{−1}, π_{k} • π_{k}^{∗})

Thus

π_{k} • π_{k}^{∗} = ∑_{i=1}^{8} χ_{i}^{−1} = ∑_{i=1}^{8} χ_{i}

but deg (π_{k} • π_{k}^{∗}) = 4 and deg (∑_{i=1}^{8} χ_{i}) = 8. Thus all χ_{i} cannot fix π_{k}.

Now let half of χ_{i} fix π_{k} and half of χ_{i} interchange them, say

χ_{i} • π_{k} = π_{k} for i odd
χ_{i} • π_{k} = π_{l} for i even

if k ≠ l. It is clear that χ_{i} and χ_{i}^{∗} = χ_{i}^{−1} fix or interchange π_{k} simultaneously. Then for k ≠ l

1 = m (π_{k}, χ_{i} • π_{k}) = m (χ_{i}^{∗}, π_{k} • π_{k}^{∗}) = m (χ_{i}^{−1}, π_{k} • π_{k}^{∗}) for i odd
1 = m (π_{l}, χ_{i} • π_{k}) = m (χ_{i}^{∗}, π_{k} • π_{l}^{∗}) = m (χ_{i}^{−1}, π_{k} • π_{l}^{∗}) for i even

and therefore

π_{k} • π_{k}^{∗} = χ_{1} + χ_{3} + χ_{5} + χ_{7}
π_{k} • π_{l}^{∗} = χ_{2} + χ_{4} + χ_{6} + χ_{8}

There are two possibilities for the involution: either π_{1}^{∗} = π_{1} and π_{2}^{∗} = π_{2}, or π_{1}^{∗} = π_{2} and π_{2}^{∗} = π_{1}. It is easy to check that when G (H^{∗}) is isomorphic to C_{2} × C_{2} × C_{2} or Q_{8} it does not matter which generators we choose to fix π_{k} and which to interchange them. In the case of G (H^{∗}) ≅ D_{8} or C_{4} × C_{2} it matters and should give us two more nonisomorphic structures for K_{0}(H) for each of them, but due to the results of the Section 3 we can see that in the case of G (H^{∗}) ≅ C_{4} × C_{2}, π_{k} can be fixed only by elements of order 1 or 2 (since (π_{k})^{2} is either the sum of all elements of order 1 or 2, or the sum of all elements of order 4).

Now assume that G (H^{∗}) = ⟨χ, φ : χ^{4} = 1, φ^{2} = 1, φχ = χ^{−1}φ⟩ ≅ D_{8} or G (H^{∗}) = ⟨χ, φ : χ^{4} = 1, φ^{2} = χ^{2}, φχ = χ^{−1}φ⟩ ≅ Q_{8}. Then χ^{2} is the only nontrivial central element of G (H^{∗}). Since by [13, Theorem1] H^{∗} has a nontrivial central grouplike, this grouplike should be equal to χ^{2}. Since χ^{2} is a central grouplike of order 2, which fixes all π_{k}, by Proposition 2.2 H has a commutative subHopfalgebra of dimension 8. Therefore, it should have the same K_{0}-ring as one of the Hopf algebras described in Section 3. Thus the only possible K_{0}-ring structure corresponds to G (H^{∗}) ≅ D_{8} with (π_{k})^{2} = 1 + χ^{2} + φ + χ^{2}φ and π_{i}^{∗} = π_{i}.

Now let’s list all the possible ring structures of K_{0}(H).

5.1. G (H^{∗}) = ⟨χ⟩ × ⟨φ⟩ × ⟨ψ⟩ ≅ C_{2} × C_{2} × C_{2} where π_{1}^{∗} = π_{1} and π_{2}^{∗} = π_{2} and

\begin{matrix}χ • π_{1} = π_{2} = π_{1} • χ & ψ • π_{1} = π_{1} = π_{1} • ψ \\ χ • π_{2} = π_{1} = π_{2} • χ & ψ • π_{2} = π_{2} = π_{2} • ψ \\ φ • π_{1} = π_{1} = π_{1} • φ & π_{1}^{2} = 1 + φ + ψ + φψ = π_{2}^{2} \\ φ • π_{2} = π_{2} = π_{2} • φ & π_{1} • π_{2} = χ + χφ + χψ + χφψ = π_{2} • π_{1} \end{matrix}

Examples: H_{b:1}, H_{d:1,1}, H_{d:−1,1}, k (D_{8} × C_{2}), k (Q_{8} × C_{2}), H_{8} ⊗ kC_{2}.

5.2. G (H^{∗}) = ⟨χ⟩ × ⟨φ⟩ × ⟨ψ⟩ ≅ C_{2} × C_{2} × C_{2} where π_{1}^{∗} = π_{2} and π_{2}^{∗} = π_{1} and

\begin{matrix}χ • π_{1} = π_{2} = π_{1} • χ & ψ • π_{1} = π_{1} = π_{1} • ψ \\ χ • π_{2} = π_{1} = π_{2} • χ & ψ • π_{2} = π_{2} = π_{2} • ψ \\ φ • π_{1} = π_{1} = π_{1} • φ & π_{1}^{2} = χ + χφ + χψ + χφψ = π_{2}^{2} \\ φ • π_{2} = π_{2} = π_{2} • φ & π_{1} • π_{2} = 1 + φ + ψ + φψ = π_{2} • π_{1} \end{matrix}

Examples: H_{c:σ_{1}} and kG_{7}, where G_{7} = ⟨a, b, c : a^{4} = b^{2} = c^{2} = 1, cbc = a^{2}b⟩.

5.3. G (H^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2} where π_{1}^{∗} = π_{1} and π_{2}^{∗} = π_{2} and

\begin{matrix}χ • π_{1} = π_{2} = π_{1} • χ & φ • π_{1} = π_{1} = π_{1} • φ & π_{1}^{2} = 1 + χ^{2} + φ + χ^{2}φ = π_{2}^{2} \\ χ • π_{2} = π_{1} = π_{2} • χ & φ • π_{2} = π_{2} = π_{2} • φ & π_{1} • π_{2} = χ + χ^{3} + χφ + χ^{3}φ = π_{2} • π_{1} \end{matrix}

Examples: H_{a:1}, H_{a:y}, H_{b:y}, H_{b:x^{2}y}, H_{d:1,−1}, H_{d:−1,−1}, kG_{5} and kG_{6}, where G_{5} = ⟨a, b : a^{4} = b^{4} = 1, b^{−1}ab = a^{−1}⟩ and G_{6} = ⟨a, b, c : a^{4} = b^{2} = c^{2} = 1, bab = ac⟩.

5.4. G (H^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{4} × C_{2} where π_{1}^{∗} = π_{2} and π_{2}^{∗} = π_{1} and

\begin{matrix}χ • π_{1} = π_{2} = π_{1} • χ & φ • π_{1} = π_{1} = π_{1} • φ & π_{1}^{2} = 1 + χ^{2} + φ + χ^{2}φ = π_{2}^{2} \\ χ • π_{2} = π_{1} = π_{2} • χ & φ • π_{2} = π_{2} = π_{2} • φ & π_{1} • π_{2} = χ + χ^{3} + χφ + χ^{3}φ = π_{2} • π_{1} \end{matrix}

Examples: H_{c:σ_{0}}, kG_{1}, where G_{1} = ⟨a, b : a^{8} = b^{2} = 1, bab = a^{5}⟩.

5.5. G (H^{∗}) = ⟨χ, φ : χ^{4} = 1, φ^{2} = 1, φχ = χ^{−1}φ⟩ ≅ D_{8}, where π_{1}^{∗} = π_{1} and π_{2}^{∗} = π_{2} and

\begin{matrix}χ • π_{1} = π_{2} = π_{1} • χ & φ • π_{1} = π_{1} = π_{1} • φ & π_{1}^{2} = 1 + χ^{2} + φ + χ^{2}φ = π_{2}^{2} \\ χ • π_{2} = π_{1} = π_{2} • χ & φ • π_{2} = π_{2} = π_{2} • φ & π_{1} • π_{2} = χ + χ^{3} + χφ + χ^{3}φ = π_{2} • π_{1} \end{matrix}

Examples: H_{B:1}, H_{B:X} and kQ_{8}#^{α}kC_{2}.

Remark 5.1. Noncommutative K_{0}(H) should have the structure 5.5.

6. Computations in K_{0}(H) in the case of |G (H^{∗})| = 4.

In this case by Theorem 1.1 G (H^{∗}) ≅ C_{2} × C_{2} and we have 4 one-dimensional irreducible representations χ_{1} = 1_{K_{0}(H)}, …, χ_{4} ∈ G (H^{∗}) and three two-dimensional ones π_{1}, π_{2} and π_{3}. Then, since χ_{i} • χ_{i} = 1_{K_{0}(H)}, χ_{i}^{∗} = χ_{i}. The involution is an antihomomorphism of K_{0}(H) of order 2, thus it either fixes all π_{k} or interchanges two of them and fixes the third one. Assume that we always have π_{2}^{∗} = π_{2}.

χ_{i} •π_{k} ≠ χ_{j} +χ_{l} as in the case of |G (H^{∗})| = 8. Thus multiplication by χ_{i} permutes π_{k}. Since o (χ_{i}) = 1 or 2 then each χ_{i} either fixes all π_{k} or interchanges two of them and fixes the third one. There are two possible cases:

i) χ_{i} • π_{k} = π_{k} for i = 1, …, 4 and k = 1, 2, 3. Then

m (χ_{i}, π_{k} • π_{k}^{∗}) = m (π_{k}, χ_{i} • π_{k}) = 1
(6.1)  π_{k} • π_{k}^{∗} = ∑_{i=1}^{4} χ_{i} for k = 1, 2, 3

By [13, Theorem 1], one of the χ_{i} is central of order 2 and therefore by Proposition 2.2, H has a commutative subHopfalgebra of order 8. Therefore, it should have the same K_{0}-ring as one of the Hopf algebras described in Section 3. But none of these K_{0}-rings satisfies (6.1). Therefore this case is not possible.

ii) χ_{i} • π_{k} ≠ π_{k} for some i ∈ {1, …, 4} and k ∈ {1, 2, 3}. Then, say,

χ_{1} • π_{k} = χ_{3} • π_{k} = π_{k} for k = 1, 2, 3

but χ_{2} • π_{k} ≠ π_{k}, χ_{4} • π_{k} ≠ π_{k} for some k ∈ {1, 2, 3}.

Assume that π_{1}^{∗} = π_{3}, π_{3}^{∗} = π_{1} and χ_{2} • π_{2} ≠ π_{2}. Then

1 = m (π_{2}, χ_{i} • π_{2}) = m (χ_{i}, π_{2} • π_{2}) for i = 1, 3
0 = m (π_{2}, χ_{i} • π_{2}) = m (χ_{i}, π_{2} • π_{2}) for i = 2, 4

Therefore

π_{2} • π_{2}^{∗} = χ_{1} + χ_{3} + π_{r} = π_{2} • π_{2}

Since (π_{k} • π_{k}^{∗})^{∗} = π_{k} • π_{k}^{∗} , we get (π_{r})^{∗} = π_{r} and and thus π_{r} = π_{2}, that is

π_{2} • π_{2}^{∗} = χ_{1} + χ_{3} + π_{2}

Therefore R = {aχ_{1} + bχ_{3} + cπ_{2} ∈ K_{0}(H) : a, b, c ∈ ℤ} is a hereditary subring of K_{0}(H) (see [24, Definition 3.10]). There is a one-to-one correspondence between hereditary subrings of K_{0}(H) and Hopf quotients of H, that is between hereditary subrings of K_{0}(H) and subHopfalgebras of H^{∗} (see [22, Theorem 6] or [24, Proposition 3.11]). Thus H^{∗} has a subHopfalgebra of dimension 1+1+4 = 6, which contradicts Nichols-Zoeller Theorem [23].

Thus without loss of generality χ_{2} • π_{2} = π_{2}. Then

χ_{i} • π_{2} = π_{2} for i = 1, …, 4
χ_{i} • π_{1} = π_{1} for i = 1, 3
χ_{i} • π_{1} = π_{3} for i = 2, 4
χ_{i} • π_{3} = π_{3} for i = 1, 3
χ_{i} • π_{3} = π_{1} for i = 2, 4

and therefore

1 = m (π_{2}, χ_{i} • π_{2}) = m (χ_{i}, π_{2} • π_{2}) for i = 1, …, 4
0 = m (π_{2}, χ_{i} • π_{k}) = m (χ_{i}, π_{k} • π_{2}^{∗}) for i = 1, …, 4, k ≠ 2
1 = m (π_{k}, χ_{i} • π_{k}) = m (χ_{i}, π_{k} • π_{k}^{∗}) for i = 1, 3

0 = m (π_{k}, χ_{i} • π_{k}) = m (χ_{i}, π_{k} • π_{k}^{∗}) for i = 2, 4
0 = m (π_{3}, χ_{i} • π_{1}) = m (χ_{i}, π_{1} • π_{3}^{∗}) for i = 1, 3
1 = m (π_{3}, χ_{i} • π_{1}) = m (χ_{i}, π_{1} • π_{3}^{∗}) for i = 2, 4

Thus we get

π_{2} • π_{2}^{∗} = ∑_{i=1}^{4} χ_{i}
π_{1} • π_{1}^{∗} = χ_{1} + χ_{3} + π_{2} = π_{3} • π_{3}^{∗}
π_{1} • π_{3}^{∗} = χ_{2} + χ_{4} + π_{t}
π_{k} • π_{2}^{∗} = α_{1}π_{1} + α_{2}π_{2} + α_{3}π_{3} for k ≠ 2

and then

1 = m (π_{2}, π_{k}^{∗} • π_{k}) = m (π_{k}, π_{k} • π_{2}^{∗}) for k ≠ 2
0 = m (π_{k}, π_{l}^{∗} • π_{l}) = m (π_{l}, π_{l} • π_{k}^{∗}) for k ≠ 2

Therefore

π_{1} • π_{2}^{∗} = π_{1} + π_{3}
π_{3} • π_{2}^{∗} = π_{1} + π_{3}

and

1 = m (π_{3}, π_{1} • π_{2}^{∗}) = m (π_{2}, π_{3}^{∗} • π_{1}) = m (π_{2}, π_{1} • π_{3}^{∗})

So, finally,

π_{1} • π_{3}^{∗} = χ_{2} + χ_{4} + π_{2}

Now let’s list all the possible ring structures of K_{0}(H).

6.1. G (H^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2} where π_{1}^{∗} = π_{1}, π_{2}^{∗} = π_{2} and π_{3}^{∗} = π_{3} and

χ • π_{1} = π_{3} = π_{1} • χ φ • π_{1} = π_{1} = π_{1} • φ π_{1} • π_{2} = π_{1} + π_{3} = π_{2} • π_{1}
χ • π_{2} = π_{2} = π_{2} • χ φ • π_{2} = π_{2} = π_{2} • φ π_{1} • π_{3} = χ + χφ + π_{2} = π_{3} • π_{1}
χ • π_{3} = π_{1} = π_{3} • χ φ • π_{3} = π_{3} = π_{3} • φ π_{2} • π_{3} = π_{1} + π_{3} = π_{3} • π_{2}
π_{1}^{2} = 1 + φ + π_{2} = π_{3}^{2} π_{2}^{2} = 1 + χ + φ + χφ

Examples: H_{C:1}, H_{C:σ_{1}}, kD_{16} and kQ_{16}.

6.2. G (H^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ C_{2} × C_{2} where π_{1}^{∗} = π_{3}, π_{2}^{∗} = π_{2} and π_{3}^{∗} = π_{1} and

χ • π_{1} = π_{3} = π_{1} • χ  φ • π_{1} = π_{1} = π_{1} • φ π_{1} • π_{2} = π_{1} + π_{3} = π_{2} • π_{1}
χ • π_{2} = π_{2} = π_{2} • χ  φ • π_{2} = π_{2} = π_{2} • φ π_{1} • π_{3} = 1 + φ + π_{2} = π_{3} • π_{1}
χ • π_{3} = π_{1} = π_{3} • χ  φ • π_{3} = π_{3} = π_{3} • φ π_{2} • π_{3} = π_{1} + π_{3} = π_{3} • π_{2}
π_{1}^{2} = χ + χφ + π_{2} = π_{3}^{2} π_{2}^{2} = 1 + χ + φ + χφ

Examples: H_{E} and kG_{2}, where G_{2} = ⟨a, b : a^{8} = b^{2} = 1, bab = a^{3}⟩.

We can now prove Theorem 1.3:

Proof. In Sections 5 and 6 we have described all possible Grothendieck ring structures of non-commutative semisimple Hopf algebras of dimension 16 and there are exactly 7 of them. Only one of these K_{0}-rings is not commutative, namely K_{5.5}, which corresponds to nonabelian G (H^{∗}) ≅ D_{8}. Therefore, by [24, Theorem 4.1] all Hopf algebras with non-commutative K_{0}-ring are twistings of each other with a 2-pseudo-cocycle. Moreover, by [24, 4.5], Hopf algebras with non-commutative K_{0}-rings are not twistings of group algebras.

If G (H^{∗}) is abelian then there are 6 possibilities for the K_{0}-ring structure, all of which are commutative, namely K_{5.1} = K_{0} (k (D_{8} × C_{2})), K_{5.2} = K_{0} (kG_{7}), K_{5.3} = K_{0} (kG_{5}), K_{5.4} = K_{0} (kG_{1}), K_{6.1} = K_{0} (kD_{16}) and K_{6.2} = K_{0} (kG_{2}). Thus by [24, Theorem 4.1] H is a twisting of one of these group algebras with a 2-pseudo-cocycle. Since H is semisimple, K_{0} (H) ⊗_{ℤ} k is also semisimple by [32, Lemma 2]. Therefore, if K_{0} (H) is commutative, as algebras K_{0} (H) ⊗_{ℤ} k ≅ k^{(10)} when |G (H^{∗})| = 8 and K_{0} (H) ⊗_{ℤ} k ≅ k^{(7)} when |G (H^{∗})| = 4. If K_{0} (H) is not commutative, that is K_{0} (H) = K_{5.5}, it is easy to see that dim Z (K_{0} (H)) = 7 and thus K_{0} (H) ⊗_{ℤ} k ≅ k^{(6)} ⊕ M_{2} (k).

7. Twistings of group algebras with a 2-cocycle.

All nonabelian groups G, considered in this section, have an abelian subgroup F = {1, c, b, cb} ≅ C_{2} × C_{2}. kF ≅ (kF)^{∗} thus we can identify δ_{x} ∈ kF with the elements of the dual basis. Now define J ∈ kF ⊗ kF as follows:

J = δ_{1} ⊗ δ_{1} + δ_{1} ⊗ δ_{c} + δ_{1} ⊗ δ_{b} + δ_{1} ⊗ δ_{cb}
+δ_{c} ⊗ δ_{1} + δ_{c} ⊗ δ_{c} + iδ_{c} ⊗ δ_{b} − iδ_{c} ⊗ δ_{cb}
+δ_{b} ⊗ δ_{1} − iδ_{b} ⊗ δ_{c} + δ_{b} ⊗ δ_{b} + iδ_{b} ⊗ δ_{cb}

+δ_{cb} ⊗ δ_{1} + iδ_{cb} ⊗ δ_{c} − iδ_{cb} ⊗ δ_{b} + δ_{cb} ⊗ δ_{cb} (7.1)

where

δ_{1} = \frac{1}{4} (1 + c + b + cb)
δ_{c} = \frac{1}{4} (1 + c − b − cb)
δ_{b} = \frac{1}{4} (1 − c + b − cb)
δ_{cb} = \frac{1}{4} (1 − c − b + cb)

We can rewrite J as

J = \frac{1}{8}(5 · 1 ⊗ 1 + c ⊗ 1 + b ⊗ 1 + cb ⊗ 1
+1 ⊗ c + c ⊗ c + (−1 − 2i) b ⊗ c + (−1 + 2i) cb ⊗ c
+1 ⊗ b + (−1 + 2i) c ⊗ b + b ⊗ b + (−1 − 2i) cb ⊗ b
+1 ⊗ cb + (−1 − 2i) c ⊗ cb + (−1 + 2i) b ⊗ cb + cb ⊗ cb)

Such a J is a 2-cocycle for kF and since J ∈ kG ⊗ kG, it is also a 2-cocycle for kG. Thus we can form (kG)_{J} which is a Hopf algebra by [31], [24, 2.8]. By [31, 6.4], (kG)_{J} is non-cocommutative if and only if J^{−1} (τJ) does not lie in the centralizer of Δ (kG) in kG ⊗ kG. Moreover, by [24, Theorem 4.1] K_{0} ((kG)_{J}) ≅ K_{0} (kG). Since J is a 2-cocycle, then by [4] (kG)_{J} is triangular.

We now discuss Examples 2, 12 and 13 in the tables. We used GAP to compute G (H) in Examples 12 and 13. Example 2. H = (k (D_{8} × C_{2}))_{J}, where F = {1, c, b, cb} ≅ C_{2} × C_{2} is a subgroup of D_{8} × C_{2} = ⟨a, b, c : a^{4} = b^{2} = c^{2} = 1, ba = a^{−1}b⟩ and J is given by the formula (7.1). Then

1. G (H) = ⟨a^{2}, b, c⟩ ≅ C_{2} × C_{2} × C_{2}.

2. G (H^{∗}) ≅ G (k (D_{8} × C_{2})) ≅ C_{2} × C_{2} × C_{2} and K_{0} (H) ≅ K_{0} (k (D_{8} × C_{2})) ≅

K_{5.1}. Example 12. H = (kD_{16})_{J}, where F = {1, a^{4}, b, a^{4}b} ≅ C_{2} × C_{2} is a subgroup of D_{16} = ⟨a, b : a^{8} = b^{2} = 1, ba = a^{−1}b⟩ and J is given by the formula (7.1). Then

1. G (H) = ⟨b, g = \frac{1}{2} (−a^{2} + a^{2}b + a^{6} + a^{6}b) : g^{4} = b^{2} = 1, bgb = g^{−1}⟩ ≅ D_{8}.

2. G (H^{∗}) ≅ G ((kD_{16})^{∗}) ≅ C_{2} × C_{2} and K_{0} (H) ≅ K_{0} (kD_{16}) ≅ K_{6.1}. Example 13. H = (kG_{2})_{J}, where F = {1, a^{4}, b, a^{4}b} ≅ C_{2} × C_{2} is a subgroup of G_{2} = ⟨a, b : a^{8} = b^{2} = 1, ba = a^{3}b⟩ and J is given by the formula (7.1). Then

1. G (H) ≅ D_{8}.

2. G (H^{∗}) ≅ G ((kG_{2})^{∗}) ≅ C_{2} × C_{2} and K_{0} (H) ≅ K_{0} (kG_{2}) ≅ K_{6.2}.

8. A construction using smash coproducts.

Let H = kQ_{8}#^{α}kC_{2}, a smash coproduct of kQ_{8} and kC_{2} (see [19, 10.6.1] or [24, Proposition 3.8]), where Q_{8} = ⟨a, b : a^{4} = 1, b^{2} = a^{2}, ba = a^{−1}b⟩, C_{2} = {1, g}. H has the algebra structure of kQ_{8} ⊗ kC_{2} and the following comultiplication, antipode and counit:

Δ (x#δ_{g^{k}}) = ∑_{r+t=k} (x_{1}#δ_{g^{r}}) ⊗ (α_{g^{r}} (x_{2}) #δ_{g^{t}})
S (x#δ_{g^{k}}) = α_{g^{k}} (S (x)) #δ_{g^{−k}}
ε (x#δ_{g^{k}}) = ε (x) δ_{g^{k},1}

where δ_{1} = \frac{1}{2} (1 + g), δ_{g} = \frac{1}{2} (1 − g), x ∈ kQ_{8} and α : G → Aut (kQ_{8}) is defined by

α_{1} (x) = x
α_{g} (a) = b
α_{g} (b) = a,

see [24, Erratum]. It follows from the above that

Δ (a#1) = Δ (a#δ_{1}) + Δ (a#δ_{g})
= a#δ_{1} ⊗ a#δ_{1} + a#δ_{g} ⊗ b#δ_{g} + a#δ_{1} ⊗ a#δ_{g} + a#δ_{g} ⊗ b#δ_{1}
= a#δ_{1} ⊗ a#1 + a#δ_{g} ⊗ b#1
(8.1)  = \frac{1}{2} (a#1 ⊗ a#1 + a#g ⊗ a#1 + a#1 ⊗ b#1 − a#g ⊗ b#1)

and

Δ (b#1) = \frac{1}{2} (b#1 ⊗ b#1 + b#g ⊗ b#1 + b#1 ⊗ a#1 − b#g ⊗ a#1) (8.2)

Let’s describe G (H), G (H^{∗}) and K_{0} (H). By straightforward computations, using (8.1) and (8.2), G (H) = ⟨a^{2}#1⟩ × ⟨1#g⟩. G (H^{∗}) is generated by the multiplicative characters χ and φ, defined by χ(a) = χ(g) = −1, χ(b) = 1 and φ (a) = −1, φ (g) = φ (b) = 1. Then χ^{−1}(a) = 1, χ^{−1}(b) = χ^{−1}(g) = −1 and φχφ = χ^{−1}. Therefore G (H^{∗}) ≅ D_{8}. Degree 2 irreducible representations of H are defined by

\begin{matrix}π_{1}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{1}(b) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) & π_{1}(g) = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) \\ π_{2}(a) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) & π_{2}(b) = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) & π_{2}(g) = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) \end{matrix}

with the property π_{1}^{2} = π_{2}^{2} = 1 + χ^{2} + φ + χ^{2}φ and therefore K_{0} (H) ≅ K_{5.5}.

9. Main results.

Theorem 9.1. Every nontrivial semisimple Hopf algebra H of dimension 16 has a commutative subHopfalgebra of dimension 8.

Proof. By [13, Theorem 1] H^{∗} has a central grouplike g of order 2. Thus we get a short exact sequence of Hopf algebras

k ⟨g⟩ ↪^{i} H^{∗} ↠^{π} K (9.1)

If K is cocommutative, K^{∗} ⊂ H is commutative and we are done. If K is commutative, but not cocommutative, then K^{∗} = kG (H) ⊂ H and G (H) is nonabelian of order 8. Applying Proposition 2.2 and results of Section 5 we get that H^{∗} has a commutative subHopfalgebra of dimension 8. Therefore H^{∗} was described in Section 3.3, case B), and it has a group algebra of dimension 8 as a quotient. Therefore H has a commutative subHopfalgebra of dimension 8.

Now assume that K is neither commutative nor cocommutative. Then K ≅ K^{∗} ≅ H_{8} and as algebras H^{∗} ≅ k ⟨g⟩ #_{σ}K, a crossed product of Hopf algebras with an action ⇀: K⊗k ⟨g⟩ → k ⟨g⟩ and a cocycle σ : K⊗K → k ⟨g⟩. Since g is central in H^{∗}, the action ⇀ is trivial. Let’s prove that σ is also trivial. By [16, Theorem 4.8] K ≅ H_{8} doesn’t have nontrivial right Galois objects, thus for any 2-cocycle α : K ⊗ K → k the crossed product _{α}K = k#_{α}K is trivial, that is ab = ∑ α (a_{1}, b_{1}) a_{2}b_{2}. Write k ⟨g⟩ = k \frac{1+g}{2} ⊕k \frac{1−g}{2} and σ (a ⊗ b) = α_{1} (a ⊗ b) \frac{1+g}{2} +α_{2} (a ⊗ b) \frac{1−g}{2}. Then for j = 1, 2, α_{j} : K ⊗ K → k are 2-cocycles and

(1#a) (1#b) = ∑ σ (a_{1}, b_{1}) #a_{2}b_{2} = \frac{1 + g}{2} # ∑ α_{1} (a_{1}, b_{1}) a_{2}b_{2}

+\frac{1 − g}{2} # ∑ α_{2} (a_{1}, b_{1}) a_{2}b_{2} = \frac{1 + g}{2} #ab + \frac{1 − g}{2} #ab = 1#ab

Therefore as algebras H^{∗} ≅ k ⟨g⟩ ⊗ K. As coalgebras H ≅ k ⟨g⟩ ⊗ K^{∗} ≅ k ⟨g⟩ ⊗ H_{8}, H has 8 grouplikes and we are done by the previous argument.

We now prove Theorem 1.2:

Proof. 1. Assume G (H) is abelian of order 8. By Theorem 1.1, G (H) ≅ C_{2} × C_{2} × C_{2} or G (H) ≅ C_{4} × C_{2} and by Proposition 3.1 in this case G (H^{∗}) is also abelian of order 8. In Propositions 3.2 and 3.3 we have shown that there are exactly 7 nonisomorphic Hopf algebras with G (H) ≅ C_{4} × C_{2} and at most 4 nonisomorphic Hopf algebras with G (H) ≅ C_{2} × C_{2} × C_{2}. Now we show that there are 4 distinct Hopf algebras with G (H) ≅ C_{2} × C_{2} × C_{2}. There are 2 nonisomorphic examples of Hopf algebras with G (H) ≅ G (H^{∗}) ≅ C_{2} × C_{2} × C_{2}, namely H_{8} ⊗ kC_{2}, which is not triangular (if it were triangular, so would be H_{8}), and (k (D_{8} × C_{2}))_{J}, which is triangular (see Section 7), and 2 more nonisomorphic Hopf algebras with G (H) ≅ C_{2} × C_{2} × C_{2} and G (H^{∗}) ≅ C_{4} × C_{2}, namely (H_{b:1})^{∗} and (H_{c:σ_{1}})^{∗}. Comparing the structures of H_{8} (see [11, 2.3, 2.4, 2.8]) and H_{d:−1,1} we see that H_{8} ⊗ kC_{2} ≅ H_{d:−1,1}, and therefore H_{d:1,1} ≅ (k (D_{8} × C_{2}))_{J}.

2. Assume that G (H) is nonabelian. Then, by Theorem 9.1, H has a commutative subHopfalgebra of dimension 8. By Proposition 3.4 G (H) ≅ D_{8}, G (H^{∗}) = C_{2}×C_{2} and there are exactly 2 such Hopf algebras, H_{C:1} ≅ H_{B:1}^{∗} and H_{E} ≅ H_{B:X}^{∗}. Comparing their K_{0}-rings with K_{0}-rings of examples described in Section 7 we see that H_{C:1} ≅ (kD_{16})_{J} and H_{E} ≅ (kG_{2})_{J}.

3. Assume that G (H) is abelian of order 4. By Theorem 1.1, G (H) ≅ C_{2}×C_{2}. By Theorem 9.1, H has a commutative subHopfalgebra of dimension 8 and therefore it was described in Section 3. There are exactly 3 Hopf algebras with this group of grouplikes: two of them, H_{B:1} ≅ H_{C:1}^{∗} and H_{B:X} ≅ H_{E}^{∗}, have G (H)^{∗} ≅ D_{8} and one of them, H_{C:σ_{1}} has G (H)^{∗} ≅ C_{2} × C_{2} and therefore should be selfdual. Comparing the quotients of H_{B} and kQ_{8}#^{α}kC_{2} we see that H_{B:X} ≅ kQ_{8}#^{α}kC_{2}.

ACKNOWLEDGMENTS

I would like to thank my Ph.D. advisor Professor Susan Montgomery for numerous discussions, suggestions and comments about this paper.

Part of this paper is contained in my Ph.D. thesis in the University of Southern California. The rest of the work was done while I was a postdoctoral fellow at MSRI. I am grateful to MSRI and the organizers of the Noncommutative Algebra program for the support.

I would also like to thank the referee for Proposition 2.2 and for the suggestion to use [16] in the proof of Theorem 9.1.

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Yevgenia Kashina, Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070

E-mail address: kashina@msri.org