EXAMPLES OF HOPF ALGEBRAS OF DIMENSION 2^{m}.

YEVGENIA KASHINA

Abstract. In this paper we construct two families of semisimple Hopf algebras of dimension 2^{n+1}, n ≥ 3. They are all constructed as Radford’s biproducts. For these examples and their duals we compute their grouplike elements, centers, character algebras and Grothendieck rings. Comparing these facts we are able to show that depending on the dimension, representatives of one of the families are selfdual.

1. Introduction.

In this paper we construct two families of semisimple Hopf algebras of dimension 2^{n+1} for n ≥ 3 over an algebraically closed field of characteristic 0. They are all constructed as biproducts A = R ⋆ H, as in Radford [R], where H = kℤ_{2} and R is a Hopf algebra in the Yetter-Drinfeld category _{H}^{H}Y D. This construction is similar to the one in [FMS, 5.11] and [R, Section 4]. The definitions of the Yetter-Drinfeld category and biproducts are given in Section 2.

In Section 3 we construct two families, R_{n}^{+} and R_{n}^{−}, of semisimple Hopf algebras in the Yetter-Drinfeld category _{kℤ_{2}}^{kℤ_{2}}Y D of dimension 2^{n}, n ≥ 3. In Section 4 we construct two families, A_{n}^{+} and A_{n}^{−}, of semisimple Hopf algebras of dimension 2^{n+1} as biproducts of our constructions in Section 3.

The study of these families appears to be very interesting, since their behavior depends on the dimension. For example, A_{n}^{+} and A_{n}^{−} exchange their roles with respect to groups of grouplikes in the case of n = 3 and n ≥ 4. As will be shown, the same “exchange” happens with respect to self-duality. Moreover, it follows from the results of [K1], that A_{3}^{+} ≅ (A_{3}^{+})^{∗} and (A_{3}^{−})^{∗} are the unique Hopf algebras of dimension 16 with “exceptional” Grothendieck ring structures. That is G (A_{3}^{+}) = ℤ_{4} × ℤ_{2}, G (A_{3}^{−}) = ℤ_{2} × ℤ_{2} × ℤ_{2} and 2-dimensional representations are not selfdual.

In Section 5 we describe the structure of the Hopf algebras dual to A_{n}^{±}. In Section 6 we describe character algebras of A_{n}^{±} and (A_{n}^{±})^{∗} and compute their central orthogonal idempotents. We show:

Proposition 6.1. The character algebras of A_{n}^{±} and (A_{n}^{±})^{∗} are commutative of dimension 5 · 2^{n−2}. Thus, since a character algebra of a semisimple Hopf algebra is semisimple, all these algebras are isomorphic to k^{5·2^{n−2}}.

In Section 7 we prove the following result:

Theorem 7.1. The Hopf algebra A_{3}^{+} of dimension 16 is selfdual.

Research at MSRI is supported in part by NSF grant DMS-9701755.

To construct the isomorphism between A_{3}^{+} and (A_{3}^{+})^{∗} we compare the corresponding central orthogonal idempotents of A_{3}^{+} and (A_{3}^{+})^{∗}, C (A_{3}^{+}) and C ((A_{3}^{+})^{∗}). In Section 8 we describe the Grothendieck rings of A_{n}^{±} and (A_{n}^{±})^{∗}. The computation of 2-dimensional representations of (A_{n}^{±})^{∗} uses the isomorphism between A_{3}^{+} and (A_{3}^{+})^{∗}. Combining these descriptions with the results in [K2] we prove the following theorem:

Theorem 8.3. The Hopf algebras A_{n}^{−} of dimension 2^{n+1} for n ≥ 4 are selfdual.

2. The Yetter-Drinfeld category and biproducts.

Let H be a finite-dimensional semisimple Hopf algebra. The Yetter-Drinfeld category _{H}^{H}Y D, introduced in [Y], consists of vector spaces V over a field k such that V is both a left H-module and a left H-comodule (with coaction ρ) and it satisfies the compatibility condition

∑ h_{1}v_{−1} ⊗ h_{2} · v_{0} = ∑ (h_{1} · v)_{−1} h_{2} ⊗ (h_{1} · v)_{0} (2.1)

for all h ∈ H, v ∈ V and

ρ (v) = ∑ v_{−1} ⊗ v_{0}

_{H}^{H}Y D is a braided monoidal category with the braiding τ : V ⊗ W → W ⊗ V given by

τ (v ⊗ w) = ∑ v_{−1} · w ⊗ v_{0}

If A and B are algebras in the category _{H}^{H}Y D (that is m_{A}, m_{B}, u_{A} and u_{B} are all category morphisms), then A ⊗ B is also an algebra in this category via the multiplication m_{A⊗B} = (m_{A} ⊗ m_{B}) ∘ (id_{A} ⊗ τ ⊗ id_{B}).

If C and D are coalgebras in the category _{H}^{H}Y D (that is Δ_{C}, Δ_{D}, ε_{C} and ε_{D} are all category morphisms), then C ⊗ D is also a coalgebra in this category via the comultiplication Δ_{C⊗D} = (id_{C} ⊗ τ ⊗ id_{D}) ∘ (Δ_{C} ⊗ Δ_{D}).

Definition 2.1. R is a Hopf algebra in _{H}^{H}Y D if it is both an algebra and a coalgebra in this category, the bialgebra structure maps are all category morphisms, comultiplication and counit are multiplicative in the category and it has an antipode S_{R}, which is a convolution inverse to the identity and is a morphism in the category.

The condition for Δ_{R} to be multiplicative in _{H}^{H}Y D, using τ on R ⊗ R is

Δ_{R} ∘ m_{R} = (m_{R} ⊗ m_{R}) ∘ τ ∘ (Δ_{R} ⊗ Δ_{R})

or if we denote Δ_{R} (r) = ∑ r^{1} ⊗ r^{2},

Δ_{R} (rs) = ∑ r^{1} ((r^{2})_{−1} · s^{1}) ⊗ (r^{2})_{0} s^{2} (2.2)

The antipode S_{R} is an anti-algebra morphism in the category _{H}^{H}Y D, that is

S_{R} ∘ m = (m ⊗ m) ∘ (id ⊗ τ ⊗ id) ∘ (S_{R} ⊗ S_{R})  or
(2.3)  S_{R} (rs) = ∑ (r_{−1} · S_{R} (s)) S_{R} (r_{0})

Definition 2.2. Let R be a Hopf algebra in _{H}^{H}Y D. The biproduct A = R⋆H, defined in [R, Theorem 1], is an ordinary Hopf algebra with the following unit, multiplication, counit, comultiplication and antipode:

1_{A} = 1_{R} ⋆ 1_{H}
(2.4)  (r ⋆ x) (s ⋆ y) = ∑ r (x_{1} · s) ⋆ x_{2}y
(2.5)  ε_{A} (r ⋆ x) = ε_{R} (r) ε_{H} (x)
(2.6)  Δ_{A} (r ⋆ x) = ∑ (r^{1} ⋆ (r^{2})_{−1} x_{1}) ⊗ (r^{2})_{0} ⋆ x_{2}
(2.7)  S_{A} (r ⋆ x) = ∑ (1 ⋆ S_{H} (r_{−1}x)) (S_{R} (r_{0}) ⋆ 1)

where r, s ∈ R, x, y ∈ H.

For more details on the Yetter-Drinfeld category and biproducts see [M, §10.6] and [R].

Let H = kG, where G is an abelian group with a bicharacter ⟨ | ⟩ : G × G → k^{∗}. The category of kG-comodules ^{kG}M, which coincides with the category of G-graded modules, is contained in _{H}^{H}Y D, since any V ∈ ^{kG}M is also a left kG-module via

g · v_{h} = ⟨g | h⟩ v_{h} (2.8)

for all g, h ∈ G and v_{h} ∈ V_{h} , the h-component of V, and the compatibility condition (2.1) is satisfied. The braiding τ : V ⊗ W → W ⊗ V is given by

τ (v ⊗ w) = ⟨g | h⟩ w ⊗ v (2.9)

for all homogeneous elements v ∈ V_{g} and w ∈ W_{h} . When k contains ω, a primitive m^{th}root of 1, and G = ℤ_{m} = k ⟨g⟩ we can define the bicharacter ⟨ | ⟩ : ℤ_{m} × ℤ_{m} → k^{∗} by

⟨g^{i} | h^{j}⟩ = ω^{ij} (2.10)

R is a G-graded Hopf algebra if it is both a G-graded algebra and G-graded coalgebra, Δ_{R} is multiplicative in this category (using τ, defined by the formula 2.9) and the antipode S_{R} is a G-graded map. The G-graded Hopf algebra is a Hopf algebra in _{H}^{H}Y D.

3. Construction of Hopf algebras of dimension 2^{n} in _{kℤ_{2}}^{kℤ_{2}}Y D.

In the rest of the paper k will be an algebraically closed field of char k = 0, H = kℤ_{2} = k ⟨g⟩, n ≥ 3. In this section we construct two families, R^{−} and R^{+}, of Hopf algebras in _{H}^{H}Y D. Let’s define two k-algebras R_{n}^{−} and R_{n}^{+} of dimension 2^{n}:

R_{n}^{±} = k ⟨u_{0}, u_{1} | (u_{0})^{2^{n−1}+1} = u_{0}, (u_{1})^{2^{n−1}+1} = ±u_{1}, u_{0}u_{1} = u_{1}u_{0} = 0⟩ (3.1)

Then R_{n}^{±} is a ℤ_{2}-graded module with (R_{n}^{±})_{1} = k ⟨u_{0}^{j}, u_{1}^{2s}⟩_{j=1,...,2^{n−1};s=1,...,2^{n−2}}, (R_{n}^{±})_{g} = k ⟨u_{1}^{2s−1}⟩_{s=1,...,2^{n−2}}. Thus R_{n}^{±} ∈ _{H}^{H}Y D via the following action (see formulas (2.8) and (2.10)) and coaction: 1

1 · r = r
g · r_{1} = r_{1}
g · r_{g} = −r_{g}
ρ (r_{1}) = 1 ⊗ r_{1}
ρ (r_{g}) = g ⊗ r_{g}

where r ∈ R_{n}^{±}, r_{1} ∈ (R_{n}^{±})_{1}, r_{g} ∈ (R_{n}^{±})_{g}.

Define the comultiplication and the counit in R_{n}^{±} by

Δ_{R_{n}^{±}} (u_{0}) = u_{0} ⊗ u_{0} + u_{1} ⊗ u_{1}, ε_{R_{n}^{±}} (u_{0}) = 1
Δ_{R_{n}^{±}} (u_{1}) = u_{0} ⊗ u_{1} + u_{1} ⊗ u_{0}, ε_{R_{n}^{±}} (u_{1}) = 0

and extend to Δ_{R_{n}^{±}} (u_{0}^{j}) and Δ_{R_{n}^{±}} (u_{1}^{j}) via the formula (2.2). Then

Δ_{R_{n}^{±}} (u_{0}^{j}) = u_{0}^{j} ⊗ u_{0}^{j} + (−1)^{\frac{j(j−1)}{2}} u_{1}^{j} ⊗ u_{1}^{j}
Δ_{R_{n}^{±}} (u_{1}^{j}) = u_{0}^{j} ⊗ u_{1}^{j} + u_{1}^{j} ⊗ u_{0}^{j}

Define the antipode in R_{n}^{±} by

S_{R_{n}^{±}} (u_{0}) = u_{0}^{2^{n−1}−1}
S_{R_{n}^{±}} (u_{1}) = ±u_{1}^{2^{n−1}−1}

and extend it to S_{R_{n}^{±}} (u_{0}^{j}) and S_{R_{n}^{±}} (u_{1}^{j}) via the formula (2.3). Then

S_{R_{n}^{±}} (u_{0}^{j}) = u_{0}^{2^{n−1}−j}  j = 1, …, 2^{n−1} − 1
S_{R_{n}^{±}} (u_{1}^{j}) = ± (±1)^{\frac{j(j−1)}{2}} u_{1}^{2^{n−1}−j}  j = 1, …, 2^{n−1} − 1
S_{R_{n}^{±}} (u_{0}^{2^{n−1}}) = u_{0}^{2^{n−1}}
S_{R_{n}^{±}} (u_{1}^{2^{n−1}}) = u_{1}^{2^{n−1}}

Lemma 3.1. R_{n}^{±} with the structures defined above is a semisimple ℤ_{2}-graded Hopf algebra.

Proof. The comultiplication Δ_{R_{n}^{±}} is well defined since

Δ_{R_{n}^{±}} (u_{0}^{2^{n−1}+1}) = u_{0}^{2^{n−1}+1} ⊗ u_{0}^{2^{n−1}+1} + u_{1}^{2^{n−1}+1} ⊗ u_{1}^{2^{n−1}+1}
= u_{0} ⊗ u_{0} + (±u_{1}) ⊗ (±u_{1}) = Δ_{R_{n}^{±}} (u_{0})
Δ_{R_{n}^{±}} (u_{1}^{2^{n−1}+1}) = u_{0}^{2^{n−1}+1} ⊗ u_{1}^{2^{n−1}+1} + u_{1}^{2^{n−1}+1} ⊗ u_{0}^{2^{n−1}+1}
= ±u_{0} ⊗ u_{1} ± u_{1} ⊗ u_{0} = Δ_{R_{n}^{±}} (±u_{1})

The antipode S_{R_{n}^{±}} is well defined since

(S_{R_{n}^{±}} ∗ id) (u_{0}) = S_{R_{n}^{±}} (u_{0}) u_{0} + S_{R_{n}^{±}} (u_{1}) u_{1}
= u_{0}^{2^{n−1}} ± u_{1}^{2^{n−1}} = 1_{R_{n}^{−}} = ε (u_{0}) 1_{R_{n}^{−}}
(S_{R_{n}^{±}} ∗ id) (u_{1}) = S_{R_{n}^{±}} (u_{0}) u_{1} + S_{R_{n}^{±}} (u_{1}) u_{0} = 0 = ε (u_{1}) 1_{R_{n}^{−}}

S_{R_{n}^{±}} (u_{0}^{2^{n−1}+1}) = u_{0}^{2^{n−1}}u_{0}^{2^{n−1}−1} = u_{0}^{2^{n−1}−1} = S_{R_{n}^{−}} (u_{0})
S_{R_{n}^{±}} (u_{1}^{2^{n−1}+1}) = (g · u_{1}^{2^{n−1}}) (±u_{1}^{2^{n−1}−1}) = ±u_{1}^{2^{n−1}+2^{n−1}−1} = ±u_{1}^{2^{n−1}+1}u_{1}^{2^{n−1}−2}
= u_{1}u_{1}^{2^{n−1}−2} = u_{1}^{2^{n−1}−1} = S_{R_{n}^{±}} (−u_{1})

R_{n}^{±} has a unit 1_{R_{n}^{−}} = (u_{0})^{2^{n−1}} ± (u_{1})^{2^{n−1}}, it is easy to check that all the structure maps are ℤ_{2}-graded and comultiplication is multiplicative in _{H}^{H}Y D, since Δ_{R_{n}^{±}} was defined using the formula (2.2). Thus R_{n}^{±} is a Hopf algebra in _{H}^{H}Y D.

R_{n}^{±} is semisimple, since t = 2^{n−1} ; ∑_{j=1}u_{0}^{j} is an integral in R_{n}^{±}, with ε_{R_{n}^{−}} (t) = 2^{n−1} ≠ 0.

4. Construction of Hopf algebras of dimension 2^{n+1}.

In this section we construct two families of Hopf algebras A^{−} and A^{+}, using Hopf algebras R^{−} and R^{+} in _{H}^{H}Y D described in the previous section.

Define A_{n}^{±} = R_{n}^{±} ⋆ H, Radford’s biproduct. Then by [R, Theorem 1], A_{n}^{±} is an ordinary Hopf algebra of dimension 2^{n+1}. Moreover, A_{n}^{±} is semisimple since both R_{n}^{±} and H are semisimple. A_{n}^{±} is spanned by the elements {u_{i}^{k} ⋆ g^{s}}_{i=0,1,k=1,...,2^{n−1},s=0,1}. We will sometimes omit ⋆ in this notation.

Theorem 4.1. The semisimple Hopf algebras A_{n}^{±} have the following properties:

1. The unit in A_{n}^{±} is given by the formula

1_{A_{n}^{±}} = 1_{R_{n}^{±}} ⋆ 1_{H} = (u_{0})^{2^{n−1}} ⋆ 1_{H} ± (u_{1})^{2^{n−1}} ⋆ 1_{H}

and the multiplication is given by formula (2.4). Since g^{s} · u_{j}^{l} = (−1)^{jls} u_{j}^{l} we get:

(u_{i}^{k} ⋆ g^{s}) (u_{j}^{l} ⋆ g^{t}) = (−1)^{jls} (u_{i}^{k}u_{j}^{l}) ⋆ g^{s+t} (4.1)

2. The counit and the comultiplication in A_{n}^{±} are given by the formulas (2.5) and (2.6). Then

(4.2)  Δ_{A_{n}^{±}} (u_{0} ⋆ 1) = (u_{0} ⋆ 1) ⊗ (u_{0} ⋆ 1) + (u_{1} ⋆ g) ⊗ (u_{1} ⋆ 1)
Δ_{A_{n}^{±}} (u_{1} ⋆ 1) = (u_{0} ⋆ g) ⊗ (u_{1} ⋆ 1) + (u_{1} ⋆ 1) ⊗ (u_{0} ⋆ 1)
Δ_{A_{n}^{±}} (1 ⋆ g) = (1 ⋆ g) ⊗ (1 ⋆ g)
Δ_{A_{n}^{±}} (u_{0}^{j} ⋆ g^{k}) = (u_{0}^{j} ⋆ g^{k}) ⊗ (u_{0}^{j} ⋆ g^{k}) + (−1)^{\frac{j(j−1)}{2}} (u_{1}^{j} ⋆ g^{k+j}) ⊗ (u_{1}^{j} ⋆ g^{k})
Δ_{A_{n}^{±}} (u_{1}^{j} ⋆ g^{k}) = (u_{0}^{j} ⋆ g^{k+j}) ⊗ (u_{1}^{j} ⋆ g^{k}) + (u_{1}^{j} ⋆ g^{k}) ⊗ (u_{0}^{j} ⋆ g^{k})

3. The antipode S_{A_{n}^{±}} is defined by the formula (2.7).

4. A_{n}^{±} are not commutative, since

(u_{1} ⋆ g) (u_{1} ⋆ 1) = −u_{1}^{2} ⋆ g
(u_{1} ⋆ 1) (u_{1} ⋆ g) = u_{1}^{2} ⋆ g

5. A_{n}^{±} are not cocommutative (see formula (4.2)).

6. The center of A_{n}^{±} is Z(A_{n}^{±}) = k ⟨u_{0} ⋆ 1, u_{0} ⋆ g, u_{1}^{2} ⋆ 1⟩ of dimension 2^{n} + 2^{n−2} = 5 · 2^{n−2}. Here we list the central orthogonal idempotents of A_{n}^{±}:

\begin{matrix}A_{n}^{−} & A_{n}^{+} \\ \frac{1}{2^{n}} ∑_{j=1}^{2^{n−1}} ∑_{k=0}^{1} ω^{jl}u_{0}^{j}g^{k} & \frac{1}{2^{n}} ∑_{j=1}^{2^{n−1}} ∑_{k=0}^{1} ω^{jl}u_{0}^{j} ⋆ g^{k} \\ \frac{1}{2^{n−2}} ∑_{s=1}^{2^{n−2}} ω^{(2m−1)s}u_{1}^{2s} & \frac{1}{2^{n−2}} ∑_{s=1}^{2^{n−2}} ω^{2ms}u_{1}^{2s} \\ \frac{1}{2^{n}} ∑_{j=1}^{2^{n−1}} ∑_{k=0}^{1} (−1)^{k} ω^{jl}u_{0}^{j}g^{k} & \frac{1}{2^{n}} ∑_{j=1}^{2^{n−1}} ∑_{k=0}^{1} (−1)^{k} ω^{jl}u_{0}^{j}g^{k} \end{matrix}

where ω is a primitive 2^{n−1}-root of unity, l = 1, ..., 2^{n−1}, m = 1, ..., 2^{n−2}.

7. A_{n}^{−} ≇ A_{n}^{+} since they have different groups of grouplikes:

(a) When n = 3, i.e. dim A_{n}^{±} = 16, G(A_{3}^{−}) = ⟨u_{0}^{4} + u_{1}^{4}⟩ × ⟨u_{0}^{2} + iu_{1}^{2}⟩ × ⟨g⟩ ≅ ℤ_{2} × ℤ_{2} × ℤ_{2}, G(A_{3}^{+}) = ⟨u_{0}^{2} + iu_{1}^{2}⟩ × ⟨g⟩ ≅ ℤ_{4} × ℤ_{2}.

(b) When n > 3, G(A_{n}^{−}) = ⟨u_{0}^{2} + iu_{1}^{2}⟩ × ⟨g⟩ ≅ ℤ_{2^{n−1}} × ℤ_{2}, G(A_{n}^{+}) = ⟨u_{0}^{2} + iu_{1}^{2}⟩ × ⟨u_{0}^{2^{n−1}} − u_{1}^{2^{n−1}}⟩ × ⟨g⟩ ≅ ℤ_{2^{n−2}} × ℤ_{2} × ℤ_{2}.

5. The structure of the duals (A_{n}^{±})^{∗}.

5.1. The algebra structure of (A_{n}^{±})^{∗}. Both (A_{n}^{−})^{∗} and (A_{n}^{+})^{∗} have the same algebra structure. They are spanned by elements p_{i,j,k}, where i, k ∈ {0, 1} and j = 1, ..., 2^{n−1}, such that

⟨p_{i,j,k}, u_{q}^{s} ⋆ g^{t}⟩ = δ_{i,q}δ_{j,s}δ_{k,t} (5.1)

The unit in (A_{n}^{±})^{∗} is

1_{(A_{n}^{±})}^{∗} = ε_{A_{n}^{±}} = 2^{n−1} ; ∑_{j=1} 1 ; ∑_{k=0}p_{0,j,k}

Let’s determine the multiplication rules in (A_{n}^{±})^{∗}:

1. p_{i,j,k} ∗ p_{q,s,t} = 0, if j ≠ s

2. (a) ⟨p_{0,j,k} ∗ p_{0,j,t}, u_{1}^{l}g^{m}⟩ = ⟨p_{0,j,k}, u_{0}^{l} ⋆ g^{m+l}⟩ ⟨p_{0,j,t}, u_{1}^{l} ⋆ g^{m}⟩ +

⟨p_{0,j,k}, u_{1}^{l} ⋆ g^{m}⟩ ⟨p_{0,j,t}, u_{0}^{l} ⋆ g^{m}⟩ = 0

(b) ⟨p_{0,j,k} ∗ p_{0,j,t}, u_{0}^{l}g^{m}⟩ = ⟨p_{0,j,k}, u_{0}^{l} ⋆ g^{m}⟩ ⟨p_{0,j,t}, u_{0}^{l} ⋆ g^{m}⟩ +

(−1)^{\frac{j(j−1)}{2}} ⟨p_{0,j,k}, u_{1}^{l} ⋆ g^{m+l}⟩ ⟨p_{0,j,t}, u_{1}^{l} ⋆ g^{m}⟩ = δ_{j,l}δ_{k,m}δ_{t,m}

3. (a) ⟨p_{1,j,k} ∗ p_{0,j,t}, u_{0}^{l}g^{m}⟩ = ⟨p_{1,j,k}, u_{0}^{l} ⋆ g^{m}⟩ ⟨p_{0,j,t}, u_{0}^{l} ⋆ g^{m}⟩ +

(−1)^{\frac{j(j−1)}{2}} ⟨p_{1,j,k}, u_{1}^{l} ⋆ g^{m+l}⟩ ⟨p_{0,j,t}, u_{1}^{l} ⋆ g^{m}⟩ = 0

(b) ⟨p_{1,j,k} ∗ p_{0,j,t}, u_{1}^{l}g^{m}⟩ = ⟨p_{1,j,k}, u_{0}^{l} ⋆ g^{m+l}⟩ ⟨p_{0,j,t}, u_{1}^{l} ⋆ g^{m}⟩ + ⟨p_{1,j,k}, u_{1}^{l} ⋆ g^{m}⟩ ⟨p_{0,j,t}, u_{0}^{l} ⋆ g^{m}⟩ = δ_{j,l}δ_{k,m}δ_{t,m}

EXAMPLES OF HOPF ALGEBRAS OF DIMENSION 2^{m}. 7

4. (a) ⟨p_{0,j,k} ∗ p_{1,j,t}, u_{0}^{l}g^{m}⟩ = ⟨p_{0,j,k}, u_{0}^{l} ⋆ g^{m}⟩ ⟨p_{1,j,t}, u_{0}^{l} ⋆ g^{m}⟩ +

(−1)^{\frac{j(j−1)}{2}} ⟨p_{0,j,k}, u_{1}^{l} ⋆ g^{m+l}⟩ ⟨p_{1,j,t}, u_{1}^{l} ⋆ g^{m}⟩ = 0

(b) ⟨p_{0,j,k} ∗ p_{1,j,t}, u_{1}^{l}g^{m}⟩ = ⟨p_{0,j,k}, u_{0}^{l} ⋆ g^{m+l}⟩ ⟨p_{1,j,t}, u_{1}^{l} ⋆ g^{m}⟩ +

⟨p_{0,j,k}, u_{1}^{l} ⋆ g^{m}⟩ ⟨p_{1,j,t}, u_{0}^{l} ⋆ g^{m}⟩ = δ_{j,l}δ_{k,m+l}δ_{t,m}

5. (a) ⟨p_{1,j,k} ∗ p_{1,j,t}, u_{0}^{l}g^{m}⟩ = ⟨p_{1,j,k}, u_{0}^{l} ⋆ g^{m}⟩ ⟨p_{1,j,t}, u_{0}^{l} ⋆ g^{m}⟩ +

(−1)^{\frac{j(j−1)}{2}} ⟨p_{1,j,k}, u_{1}^{l} ⋆ g^{m+l}⟩ ⟨p_{1,j,t}, u_{1}^{l} ⋆ g^{m}⟩ = (−1)^{\frac{j(j−1)}{2}} δ_{j,l}δ_{k,m+l}δ_{t,m} (b) ⟨p_{1,j,k} ∗ p_{1,j,t}, u_{1}^{l}g^{m}⟩ = ⟨p_{1,j,k}, u_{0}^{l} ⋆ g^{m+l}⟩ ⟨p_{1,j,t}, u_{1}^{l} ⋆ g^{m}⟩ +

⟨p_{1,j,k}, u_{1}^{l} ⋆ g^{m}⟩ ⟨p_{1,j,t}, u_{0}^{l} ⋆ g^{m}⟩ = 0 The above computations imply the following multiplication rules:

(5.2)  p_{0,j,k} ∗ p_{0,j,k} = p_{0,j,k}
(5.3)  p_{1,j,k} ∗ p_{0,j,k} = p_{1,j,k}
(5.4)  p_{0,j,k} ∗ p_{1,j,k} = p_{1,j,k}, if j ≡ 0 (mod2)
(5.5)  p_{0,j,k+1} ∗ p_{1,j,k} = p_{1,j,k}, if j ≡ 1 (mod2)
(5.6)  p_{1,j,k} ∗ p_{1,j,k} = (−1)^{\frac{j}{2}} p_{0,j,k}, if j ≡ 0 (mod2)
(5.7)  p_{1,j,k+1} ∗ p_{1,j,k} = (−1)^{\frac{j−1}{2}} p_{0,j,k}, if j ≡ 1 (mod2)
(5.8)  p_{i,j,k} ∗ p_{q,s,t} = 0, otherwise

5.2. The centers of (A_{n}^{±})^{∗}. Proposition 5.1. (A_{n}^{±})^{∗} have the centers of dimension 5 · 2^{n−2} and the following central orthogonal idempotents:

\{ \begin{matrix}\frac{1}{2} & (p_{0,2s,0} ± i^{s}p_{1,2s,0}) \\ p_{0,2s−1,0} + p_{0,2s−1,1} \\ \frac{1}{2} & (p_{0,2s,1} ± i^{s}p_{1,2s,1}) \end{matrix} \}, where s = 1, ..., 2^{n−2}.

Proof. If ∑ α_{i,j,k}p_{i,j,k} ∈ Z ((A_{n}^{±})^{∗}), then

(∑ α_{i,j,k}p_{i,j,k}) p_{0,s,t} = α_{0,s,t}p_{0,s,t} + α_{1,s,t}p_{1,s,t}
p_{0,s,t} (∑ α_{i,j,k}p_{i,j,k}) = α_{0,s,t}p_{0,s,t} + α_{1,s,t−s}p_{1,s,t−s}
(∑ α_{i,j,k}p_{i,j,k}) p_{1,s,t} = α_{0,s,t+s}p_{1,s,t} + (−1)^{\frac{s(s−1)}{2}} α_{1,s,t+s}p_{0,s,t}
p_{1,s,t} (∑ α_{i,j,k}p_{i,j,k}) = α_{0,s,t}p_{1,s,t} + (−1)^{\frac{s(s−1)}{2}} α_{1,s,t−s}p_{0,s,t−s}

Thus

α_{1,j,k} = 0, when j ≡ 1 (mod2)
α_{0,j,0} = α_{0,j,1}, when j ≡ 1 (mod2)

Therefore Z ((A_{n}^{±})^{∗}) = k {p_{0,2s,0}, p_{0,2s,1}, p_{1,2s,0}, p_{1,2s,1}, p_{0,2s−1,0} + p_{0,2s−1,1}}, s = 1, ..., 2^{n−2} and dim Z ((A_{n}^{±})^{∗}) = 2^{n−2} + 2^{n−2} + 2^{n−2} + 2^{n−2} + 2^{n−2} = 5 · 2^{n−2}.

5.3. The coalgebra structures of (A_{n}^{−})^{∗} and (A_{n}^{+})^{∗} . In (A_{n}^{−})^{∗} the counit and comultiplication are given by the formulas:

ε_{(A_{n}^{−})}^{∗} (p_{i,j,k}) = ⟨p_{i,j,k}, 1_{A}⟩ = δ_{i,0}δ_{2^{n−1},0}δ_{k,0} − δ_{i,1}δ_{2^{n−1},0}δ_{k,0}

Δ_{(A_{n}^{−})}^{∗} (p_{0,j,k}) = ∑_{s+t≡j(2^{n−1}),l+m≡k(2)}p_{0,s,l} ⊗ p_{0,t,m}

Δ_{(A_{n}^{−})}^{∗} (p_{1,j,k}) = ∑_{s+t≡j(2^{n−1}),l+m≡k(2)} (−1)^{δ_{s+t−j,2^{n−1}}+lt} p_{1,s,l} ⊗ p_{1,t,m}

In (A_{n}^{+})^{∗} the counit and comultiplication are given by the formulas:

ε_{(A_{n}^{+})}^{∗} (p_{i,j,k}) = ⟨p_{i,j,k}, 1_{A_{n}^{+}}⟩ = δ_{i,0}δ_{2^{n−1},0}δ_{k,0} + δ_{i,1}δ_{2^{n−1},0}δ_{k,0}

Δ_{(A_{n}^{+})}^{∗} (p_{0,j,k}) = ∑_{s+t≡j(2^{n−1}),l+m≡k(2)}p_{0,s,l} ⊗ p_{0,t,m}

(5.9)  Δ_{(A_{n}^{+})}^{∗} (p_{1,j,k}) = ∑_{s+t≡j(2^{n−1}),l+m≡k(2)} (−1)^{lt} p_{1,s,l} ⊗ p_{1,t,m}

The following result could be verified by straightforward computations:

Proposition 5.2. The group of grouplike elements of (A_{n}^{±})^{∗} is

G ((A_{n}^{±})^{∗}) = ⟨2^{n−1} ; ∑_{j=1} 1 ; ∑_{k=0} (−1)^{k} p_{0,j,k}⟩ × ⟨2^{n−1} ; ∑_{j=1} 1 ; ∑_{k=0}ω^{j}p_{0,j,k}⟩ ≅ ℤ_{2} × ℤ_{2^{n−1}}, where ω is a

primitive 2^{n−1}-root of unity.

6. The character algebras of A_{n}^{±} and (A_{n}^{±})^{∗}.

In this section we will describe the character algebras of A_{n}^{±} and (A_{n}^{±})^{∗}. In particular, the following proposition will follow from this description:

Proposition 6.1. The character algebras of A_{n}^{±} and (A_{n}^{±})^{∗} are commutative of dimension 5 · 2^{n−2}. Thus, since a character algebra of a semisimple Hopf algebra is semisimple, all these algebras are isomorphic to k^{5·2^{n−2}}.

Definition 6.2. The character algebra C (A) of a Hopf algebra A is a subalgebra of A^{∗} which is generated by the characters of representations of A. Equivalently, C (A) = {f ∈ A^{∗} | ⟨f, xy⟩ = ⟨f, yx⟩ for any x, y ∈ A}.

6.1. The character algebra of A_{n}^{±}. p ∈ C (A_{n}^{±}) if ⟨p, ab⟩ = ⟨p, ba⟩ for any a, b ∈ A_{n}^{±}. Write p = ∑ α_{i,j,k}p_{i,j,k}. By the formula (4.1) we have the following multiplication rules in A_{n}^{±}:

u_{0}^{m}g^{q}u_{r}^{n}g^{t} = u_{r}^{n}g^{t}u_{0}^{m}g^{q}  for r ∈ {0, 1}
u_{1}^{m}g^{q}u_{1}^{n}g^{t} = (−1)^{qn} u_{1}^{m}u_{1}^{n}g^{q+t}
u_{1}^{n}g^{t}u_{1}^{m}g^{q} = (−1)^{mt} u_{1}^{m}u_{1}^{n}g^{q+t}

Then p = ∑ α_{i,j,k}p_{i,j,k} ∈ C (A_{n}^{±}) if and only if

(−1)^{qn} ∑ α_{i,j,k} ⟨p_{i,j,k}, u_{1}^{m+n}g^{q+t}⟩ = (−1)^{mt} ∑ α_{i,j,k} ⟨p_{i,j,k}, u_{1}^{m+n}g^{q+t}⟩

and we get C (A_{n}^{±}) = k {p_{0,j,k}, p_{1,2s,0}}, where j = 1, ..., 2^{n−1}, s = 1, ..., 2^{n−2}, k = 0, 1. Then C (A_{n}^{±}) is commutative and C (A_{n}^{±}) ≅ k^{5·2^{n−2}}.

The orthogonal idempotents of C (A_{n}^{±}) are

\{ \begin{matrix}\frac{1}{2} & (p_{0,2s,0} ± i^{s}p_{1,2s,0}) \\ p_{0,2s−1,k} \\ p_{0,2s,1} \end{matrix} \}

where s = 1, ..., 2^{n−2}, k = 0, 1.

Note that the orthogonal idempotents of C (A_{n}^{−}) and C (A_{n}^{+}) are the same since the character algebras are the same and (A_{n}^{−})^{∗} and (A_{n}^{+})^{∗} have the same algebra structure.

6.2. The character algebra of (A_{n}^{±})^{∗}. a ∈ C ((A_{n}^{±})^{∗}) if ⟨φψ, a⟩ = ⟨ψφ, a⟩ for any φ, ψ ∈ (A_{n}^{±})^{∗}. Write a = ∑ α_{i,j,k}u_{i}^{j}g^{k}. Then by formulas (5.2) - (5.8):

⟨p_{1,2s−1,k}, a⟩ = ⟨p_{1,2s−1,k}p_{0,2s−1,k}, a⟩ = ⟨p_{0,2s−1,k}p_{1,2s−1,k}, a⟩ = 0
(−1)^{s−1} ⟨p_{0,2s−1,0}, a⟩ = ⟨p_{1,2s−1,1}p_{1,2s−1,0}, a⟩ = ⟨p_{1,2s−1,0}p_{1,2s−1,1}, a⟩
= (−1)^{s−1} ⟨p_{0,2s−1,1}, a⟩

Thus α_{1,2s−1,0} = α_{1,2s−1,1} = 0, α_{0,2s−1,0} = α_{0,2s−1,1} and C ((A_{n}^{±})^{∗}) = k {u_{i}^{2s}g^{k}, u_{0}^{2s−1} + u_{0}^{2s−1}g}, where s = 1, ..., 2^{n−2}, i = 0, 1. Then C ((A_{n}^{±})^{∗}) is commutative and C ((A_{n}^{±})^{∗}) ≅ k^{5·2^{n−2}}.

The orthogonal idempotents of C ((A_{n}^{±})^{∗}) are

\begin{matrix}C ((A_{n}^{−})^{∗}) & C ((A_{n}^{+})^{∗}) \\ \frac{1}{2^{n}} ∑_{j=1}^{2^{n−1}} ∑_{k=0}^{1} ω^{jl}u_{0}^{j} ⋆ g^{k} & \frac{1}{2^{n}} ∑_{j=1}^{2^{n−1}} ∑_{k=0}^{1} ω^{jl}u_{0}^{j} ⋆ g^{k} \\ \frac{1}{2^{n−1}} ∑_{s=1}^{2^{n−2}} ∑_{k=0}^{1} (−1)^{rk} ω^{(2m−1)s}u_{1}^{2s} ⋆ g^{k} & \frac{1}{2^{n−1}} ∑_{s=1}^{2^{n−2}} ∑_{k=0}^{1} (−1)^{rk} ω^{2ms}u_{1}^{2s} ⋆ g^{k} \\ \frac{1}{2^{n−1}} ∑_{s=1}^{2^{n−2}} ∑_{k=0}^{1} (−1)^{k} ω^{2sm}u_{0}^{2s} ⋆ g^{k} & \frac{1}{2^{n−1}} ∑_{s=1}^{2^{n−2}} ∑_{k=0}^{1} (−1)^{k} ω^{2sm}u_{0}^{2s} ⋆ g^{k} \end{matrix}

where ω is a primitive 2^{n−1}-root of unity, l = 1, ..., 2^{n−1}, m = 1, ..., 2^{n−2}, r = 0, 1.

7. The Hopf algebras A_{3}^{+} and (A_{3}^{+})^{∗} of dimension 16.

We now specialize to the case n = 3. Let’s write A_{3}^{+} = A^{+}. In this section we prove the following result:

Theorem 7.1. The Hopf algebra A_{3}^{+} of dimension 16 is selfdual.

7.1. Central orthogonal idempotents of C (A^{+}), (A^{+})^{∗}, C ((A^{+})^{∗}) and A^{+}. We would like to determine the correspondence between orthogonal idempotents of C (A^{+}) and C ((A^{+})^{∗}) and between central orthogonal idempotents of (A^{+})^{∗} and A^{+}. We will use this correspondence to construct an isomorphism between A^{+} and (A^{+})^{∗}. All the properties of orthogonal idempotents here can be easily checked by straightforward computations.

The orthogonal idempotents of C ((A^{+})^{∗}) are f_{1} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g + u_{0}^{2} + u_{0}^{2}g + u_{0} + u_{0}g + u_{0}^{3} + u_{0}^{3}g) f_{2} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g + u_{0}^{2} + u_{0}^{2}g − u_{0} − u_{0}g − u_{0}^{3} − u_{0}^{3}g) f_{3} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g − u_{0}^{2} − u_{0}^{2}g + iu_{0} + iu_{0}g − iu_{0}^{3} − iu_{0}^{3}g) f_{4} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g − u_{0}^{2} − u_{0}^{2}g − iu_{0} − iu_{0}g + iu_{0}^{3} + iu_{0}^{3}g) f_{5} = \frac{1}{4} (u_{1}^{4} + u_{1}^{4}g + u_{1}^{2} + u_{1}^{2}g) f_{6} = \frac{1}{4} (u_{1}^{4} − u_{1}^{4}g + u_{1}^{2} − u_{1}^{2}g) f_{7} = \frac{1}{4} (u_{1}^{4} − u_{1}^{4}g − u_{1}^{2} + u_{1}^{2}g) f_{8} = \frac{1}{4} (u_{1}^{4} + u_{1}^{4}g − u_{1}^{2} − u_{1}^{2}g) f_{9} = \frac{1}{4} (u_{0}^{4} − u_{0}^{4}g + u_{0}^{2} − u_{0}^{2}g) f_{10} = \frac{1}{4} (u_{0}^{4} − u_{0}^{4}g − u_{0}^{2} + u_{0}^{2}g)

The central orthogonal idempotents of A^{+} are F_{1} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g + u_{0}^{2} + u_{0}^{2}g + u_{0} + u_{0}g + u_{0}^{3} + u_{0}^{3}g) F_{2} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g + u_{0}^{2} + u_{0}^{2}g − u_{0} − u_{0}g − u_{0}^{3} − u_{0}^{3}g) F_{3} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g − u_{0}^{2} − u_{0}^{2}g + iu_{0} + iu_{0}g − iu_{0}^{3} − iu_{0}^{3}g) F_{4} = \frac{1}{8} (u_{0}^{4} + u_{0}^{4}g − u_{0}^{2} − u_{0}^{2}g − iu_{0} − iu_{0}g + iu_{0}^{3} + iu_{0}^{3}g) F_{5} = \frac{1}{2} (u_{1}^{4} + u_{1}^{2}) F_{6} = \frac{1}{2} (u_{1}^{4} − u_{1}^{2}) F_{7} = \frac{1}{8} (u_{0}^{4} − u_{0}^{4}g + u_{0}^{2} − u_{0}^{2}g + u_{0} − u_{0}g + u_{0}^{3} − u_{0}^{3}g) F_{8} = \frac{1}{8} (u_{0}^{4} − u_{0}^{4}g − u_{0}^{2} + u_{0}^{2}g + iu_{0} − iu_{0}g − iu_{0}^{3} + iu_{0}^{3}g) F_{9} = \frac{1}{8} (u_{0}^{4} − u_{0}^{4}g + u_{0}^{2} − u_{0}^{2}g − u_{0} + u_{0}g − u_{0}^{3} + u_{0}^{3}g) F_{10} = \frac{1}{8} (u_{0}^{4} − u_{0}^{4}g − u_{0}^{2} + u_{0}^{2}g − iu_{0} + iu_{0}g + iu_{0}^{3} − iu_{0}^{3}g)

We have the following relations between the orthogonal idempotents of C ((A^{+})^{∗}) and the central orthogonal idempotents of A^{+}: F_{1} = f_{1}, F_{2} = f_{2}, F_{3} = f_{3}, F_{4} = f_{4}, F_{5} = f_{5} + f_{7}, F_{6} = f_{6} + f_{8}, F_{7} + F_{9} = f_{9}, F_{8} + F_{10} = f_{10}.

Moreover, C ((A^{+})^{∗}) ∩ Z (A^{+}) = k {u_{0}^{2}, u_{0}^{2}g, u_{0}^{4}, u_{0}^{4}g, u_{0} + u_{0}g, u_{0}^{3} + u_{0}^{3}g, u_{1}^{2}, u_{1}^{4}} and dim (C (A^{+}) ∩ Z ((A^{+})^{∗})) = 8.

The orthogonal idempotents of C (A^{+}) are

e_{1} = \frac{1}{2} (p_{0,4,0} + p_{1,4,0}) e_{5} = p_{0,1,0} e_{9} = p_{0,4,1}
e_{2} = \frac{1}{2} (p_{0,4,0} − p_{1,4,0}) e_{6} = p_{0,3,0} e_{10} = p_{0,2,1}
e_{3} = \frac{1}{2} (p_{0,2,0} + ip_{1,2,0}) e_{7} = p_{0,1,1}
e_{4} = \frac{1}{2} (p_{0,2,0} − ip_{1,2,0}) e_{8} = p_{0,3,1}

The central orthogonal idempotents of (A^{+})^{∗} are

E_{1} = \frac{1}{2} (p_{0,4,0} + p_{1,4,0}) E_{5} = p_{0,1,0} + p_{0,1,1} E_{7} = \frac{1}{2} (p_{0,4,1} + p_{1,4,1})
E_{2} = \frac{1}{2} (p_{0,4,0} − p_{1,4,0}) E_{6} = p_{0,3,0} + p_{0,3,1} E_{8} = \frac{1}{2} (p_{0,2,1} + ip_{1,2,1})
E_{3} = \frac{1}{2} (p_{0,2,0} + ip_{1,2,0})  E_{9} = \frac{1}{2} (p_{0,4,1} − p_{1,4,1})
E_{4} = \frac{1}{2} (p_{0,2,0} − ip_{1,2,0})  E_{10} = \frac{1}{2} (p_{0,2,1} − ip_{1,2,1})

We have the following relations between the orthogonal idempotents of C (A^{+}) and the central orthogonal idempotents of (A^{+})^{∗}: E_{1} = e_{1}, E_{2} = e_{2}, E_{3} = e_{3}, E_{4} = e_{4}, E_{5} = e_{5} + e_{7}, E_{6} = e_{6} + e_{8}, E_{7} + E_{9} = e_{9}, E_{8} + E_{10} = e_{10}.

Moreover, C (A^{+}) ∩ Z ((A^{+})^{∗}) = k{p_{0,2,0}, p_{0,4,0}, p_{0,2,1}, p_{0,4,1}, p_{1,2,0}, p_{1,4,0}, p_{0,1,0} + p_{0,1,1}, p_{0,3,0} + p_{0,3,1}} and dim (C (A^{+}) ∩ Z ((A^{+})^{∗})) = 8.

7.2. Proof of Theorem 7.1. If there exists a Hopf algebra isomorphism φ between A^{+} and (A^{+})^{∗} then central orthogonal idempotents of A^{+} should be sent to central orthogonal idempotents of (A^{+})^{∗} and orthogonal idempotents of C ((A^{+})^{∗}) should be sent to orthogonal idempotents of C (A^{+}). We can express u_{0} and u_{0}g ∈ A^{+} as a linear combination of the central orthogonal idempotents F_{i}:

u_{0} = F_{1} − F_{2} − iF_{3} + iF_{4} + F_{7} − iF_{8} − F_{9} + iF_{10}
u_{0}g = F_{1} − F_{2} − iF_{3} + iF_{4} − F_{7} + iF_{8} + F_{9} − iF_{10}

Let’s define φ (F_{i}) = E_{i}. Then

(7.1)  φ (u_{0}) = p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1}
φ (u_{0}g) = p_{1,4,0} + p_{1,2,0} − p_{1,4,1} − p_{1,2,1}

The map φ should be a coalgebra map, that is

(φ ⊗ φ) ∘ Δ_{A^{+}} = Δ_{(A^{+})}^{∗} ∘ φ

Thus, using formulas (4.2) and (5.9), we get only two possibilities for φ (u_{1}) and φ (u_{1}g): φ (u_{1}) = p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1} and φ (u_{1}g) = p_{1,1,0} + p_{1,3,0} − p_{1,1,1} − p_{1,3,1} or φ (u_{1}) = −p_{1,1,0} − p_{1,3,0} − p_{1,1,1} − p_{1,3,1} and φ (u_{1}g) = −p_{1,1,0} − p_{1,3,0} + p_{1,1,1} + p_{1,3,1}. Let’s define

(7.2)  φ (u_{1}) = p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1}
(7.3)  φ (u_{1}g) = p_{1,1,0} + p_{1,3,0} − p_{1,1,1} − p_{1,3,1}

g is a noncentral grouplike of order 2 in A^{+}, so it should be sent to a noncentral grouplike of order 2 in (A^{+})^{∗}. Thus there are only two possibilities for φ (g) and only one of them is compatible with formulas (7.2) and (7.3), namely

φ (g) = p_{0,1,0} + p_{0,2,0} + p_{0,3,0} + p_{0,4,0} − p_{0,1,1} − p_{0,2,1} − p_{0,3,1} − p_{0,4,1} (7.4)

It can be easily checked, that the map φ : A_{3}^{+} → (A_{3}^{+})^{∗}, defined by formulas (7.1), (7.2) and (7.4) and extended multiplicatively, is a Hopf algebra isomorphism. For example

φ (u_{0}^{5}) = (p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1})^{5}

= (p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1})^{4} (p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1})
= (p_{0,4,0} − p_{0,2,0} + p_{0,4,1} − p_{0,2,1})^{2} (p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1})
= (p_{0,4,0} + p_{0,2,0} + p_{0,4,1} + p_{0,2,1}) (p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1})
= (p_{1,4,0} + p_{1,2,0} + p_{1,4,1} + p_{1,2,1}) = φ (u_{0})
φ (u_{1}^{5}) = (p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1})^{5}
= (p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1})^{4} (p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1})
= (p_{0,1,0} − p_{0,3,0} + p_{0,1,1} − p_{0,3,1})^{2} (p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1})
= (p_{0,1,0} + p_{0,3,0} + p_{0,1,1} + p_{0,3,1}) (p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1})
= (p_{1,1,0} + p_{1,3,0} + p_{1,1,1} + p_{1,3,1}) = φ (u_{1})
8. The Grothendieck rings of A_{n}^{±} and (A_{n}^{±})^{∗}.

In this section we completely describe the Grothendieck rings of A_{n}^{±} and (A_{n}^{±})^{∗}. First we will need the following definition, which was introduced in [S].

Definition 8.1. Let K_{0} (A)^{+} denote the abelian semigroup of all equivalence classes of representations of a semisimple Hopf algebra A with the addition given by a direct sum. Then its enveloping group K_{0} (A) has the structure of an ordered ring with involution ^{∗} and is called the Grothendieck ring.

In [NR] the structure of K_{0} (A) was described for comodules; it was then translated into the language of modules in [N]. 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 A form a basis of K_{0} (A) 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 [NR] and [N]):

(8.1)  m (z, x • y) = m (x^{∗}, y • z^{∗})
(8.2)  m (1, x • y^{∗}) = δ_{x,y}
(8.3)  ∑ 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} (A). The 1-dimensional irreducible representations of A are exactly the elements of G (H^{∗}), or multiplicative characters of A.

Lemma 8.2. The Hopf algebras A_{n}^{±} and (A_{n}^{±})^{∗} have only 1-dimensional and 2dimensional irreducible representations.

Proof. Let A be one of these Hopf algebras. Since |G (A^{∗})| = 2^{n} and dim Z (A) = 5 · 2^{n−2}, there are exactly 2^{n} irreducible representations of dimension 1 and 5 · 2^{n−2} − 2^{n} = 2^{n−2} irreducible representations of dimension greater or equal then 2. Sum of the squares of these dimensions should be equal to dim A − |G (A^{∗})| = 2^{n}, thus all of them should be of dimension 2.

Theorem 8.3. The Hopf algebras A_{n}^{−} of dimension 2^{n+1} for n ≥ 4 are selfdual.

Proof. We will show in Propositions 8.4 and 8.8 that for n ≥ 4 A_{n}^{−} and (A_{n}^{−})^{∗} have exactly the same structure of Grothendieck ring as an ordered ring with involution and marked element. In [K2] we show that there are exactly three nonisomorphic Hopf algebras of dimension 2^{n+1} with G (H) ≅ G ((H)^{∗}) ≅ ℤ_{2^{n−1}} × ℤ_{2}. Only one of them, H_{c_{1}: \frac{1+i}{2} x+ \frac{1−i}{2} x^{2^{n−2}+1}}, has the Grothendieck ring with the same structure as A_{n}^{−} and (A_{n}^{−})^{∗}. Thus A_{n}^{−} ≅ H_{c_{1}: \frac{1+i}{2} x+ \frac{1−i}{2} x^{2^{n−2}+1}} ≅ (A_{n}^{−})^{∗}.

8.1. The Grothendieck ring of A_{n}^{+}. G ((A_{n}^{+})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ ℤ_{2^{n−1}} × ℤ_{2}, where χ (u_{0}) = ω, χ (u_{1}) = 0, χ (g) = 1 and φ (u_{0}) = 1, φ (u_{1}) = 0, φ (g) = −1, where ω is a primitive 2^{n−1}-th root of unity. There are 2^{n−2} degree 2 irreducible representations π_{m} for m = 0, …, 2^{n−2} − 1, defined by

π_{m} (u_{0}) = 0 π_{m} (u_{1}) = ω^{m} (\sqrt{i})^{1−δ_{n,3}} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) π_{m} (g) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

The product of two 2-dimensional representations π = π_{l} • π_{m} = (π_{l} ⊗ π_{m}) ∘ Δ is defined by

π (u_{0}) = ω^{l+m}i^{1−δ_{n,3}} ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ) π (u_{1}) = 0 π (g) = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} )

and is equivalent to representation χ^{l+m} + χ^{l+m+2^{n−2}} + χ^{l+m}φ + χ^{l+m+2^{n−2}}φ. This together with formulas (8.1) and (8.2) completely determines the structure of K_{0} (A_{n}^{+}) as an ordered ring with involution and marked element.

Proposition 8.4. K_{0} (A_{n}^{+}) = ℤ ⟨χ, φ, π_{m}⟩_{m=0,... ,2^{n−2}−1} is a commutative ordered ring with involution, a marked element and following relations:

χ^{∗} = χ^{−1} χ^{2^{n−1}} = 1  χ • π_{m} = π_{m+1mod2^{n−2}}
φ^{∗} = φ φ^{2} = 1 φ • π_{m} = π_{m}
π_{0}^{∗} = π_{0} π_{m}^{∗} = π_{2^{n−2}−m},  m = 1, …, 2^{n−2} − 1
π_{l} • π_{m} = χ^{l+m} + χ^{l+m+2^{n−2}} + χ^{l+m}φ + χ^{l+m+2^{n−2}}φ

8.2. The Grothendieck ring of A_{n}^{−}. G((A_{n}^{−})^{∗}) = ⟨χ⟩ × ⟨φ⟩ ≅ ℤ_{2^{n−1}} × ℤ_{2}, where χ (u_{0}) = ω, χ (u_{1}) = 0, χ (g) = 1 and φ (u_{0}) = 1, φ (u_{1}) = 0, φ (g) = −1, where ω is a primitive 2^{n−1}-th root of unity. There are 2^{n−2} degree 2 irreducible representations π_{m} for m = 0, …, 2^{n−2} − 1, defined by

π_{m} (u_{0}) = 0 π_{m} (u_{1}) = ξ^{2m+1} (\sqrt{i})^{1−δ_{n,3}} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) π_{m} (g) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )

where ξ is a primitive 2^{n}-th root of unity and ξ^{2} = ω.

The product of two 2-dimensional representations π = π_{l} • π_{m} = (π_{l} ⊗ π_{m}) ∘ Δ is defined by

π (u_{0}) = ω^{l+m+1}i^{1−δ_{n,3}} ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \end{matrix} ) π (u_{1}) = 0 π (g) = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} )

and is equivalent to representation χ^{l+m+1} + χ^{l+m+2^{n−2}+1} + χ^{l+m+1}φ + χ^{l+m+2^{n−2}+1}φ. This together with formulas (8.1) and (8.2) completely determines the structure of K_{0} (A_{n}^{−}) as an ordered ring with involution and a marked element.

Proposition 8.5. K_{0} (A_{n}^{−}) = ℤ ⟨χ, φ, π_{m}⟩_{m=0,... ,2^{n−2}−1} is a commutative ordered ring with involution, marked element and following relations:

χ^{∗} = χ^{−1} χ^{2^{n−1}} = 1  χ • π_{m} = π_{m+1mod2^{n−2}}
φ^{∗} = φ φ^{2} = 1 φ • π_{m} = π_{m}
π_{m}^{∗} = π_{2^{n−2}−m−1}  π_{l} • π_{m} = χ^{l+m+1} + χ^{l+m+2^{n−2}+1} + χ^{l+m+1}φ + χ^{l+m+2^{n−2}+1}φ

8.3. The Grothendieck rings of (A_{n}^{±})^{∗}. There are 2^{n−2} degree 2 irreducible representations π_{m} for m = 0, …, 2^{n−2} − 1, defined by

π_{m} (p_{0,2m+1,0}) = \frac{1}{2} ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) = \frac{1}{2}X π_{m} (p_{1,2m+1,0}) = \frac{1}{2}i^{m^{2}} ( \begin{matrix}1 & 1 \\ −1 & −1 \end{matrix} )
π_{m} (p_{0,2m+1,1}) = \frac{1}{2} ( \begin{matrix}1 & −1 \\ −1 & 1 \end{matrix} ) = \frac{1}{2}Y π_{m} (p_{1,2m+1,1}) = \frac{1}{2}i^{m^{2}} ( \begin{matrix}1 & −1 \\ 1 & −1 \end{matrix} )
π_{m} (p_{a,b,c}) = 0, otherwise

The product of two 2-dimensional representations π_{l,m} = π_{l} • π_{m} = (π_{l} ⊗ π_{m}) ∘ Δ is defined by

π_{l,m} (p_{0,2l+2m+2,0}) = \frac{1}{2} ( \begin{matrix}X & 0 \\ 0 & X \end{matrix} ) π_{l,m} (p_{1,2l+2m+2,0}) = \frac{i^{m^{2}+l^{2}}}{2} (−1)^{d} ( \begin{matrix}0 & X \\ −X & 0 \end{matrix} )
π_{l,m} (p_{0,2l+2m+2,1}) = \frac{1}{2} ( \begin{matrix}Y & 0 \\ 0 & Y \end{matrix} ) π_{l,m} (p_{1,2l+2m+2,1}) = \frac{i^{m^{2}+l^{2}}}{2} (−1)^{d} ( \begin{matrix}0 & −Y \\ Y & 0 \end{matrix} )
π_{l,m} (p_{a,b,c}) = 0, otherwise

where in case of A_{n}^{+} d is always 0 and in case of A_{n}^{−} d = 0 if 2l + 2m + 2 ≤ 2^{n−1} and d = 1 if 2l + 2m + 2 > 2^{n−1}. The product π_{l,m} is equivalent to the representation π defined by

π (p_{0,2l+2m+2,0}) = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) π (p_{1,2l+2m+2,0}) = i^{m^{2}+l^{2}} ( \begin{matrix}i & 0 & 0 & 0 \\ 0 & −i & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

π (p_{0,2l+2m+2,1}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) π (p_{1,2l+2m+2,1}) = i^{m^{2}+l^{2}} ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & i & 0 \\ 0 & 0 & 0 & −i \end{matrix} )

π (p_{a,b,c}) = 0, otherwise

Proposition 8.6. K_{0} (A_{3}^{+})^{∗} = ℤ ⟨χ = u_{0}^{2} + iu_{1}^{2}, φ = g, π_{0}, π_{1}⟩ is a commutative ordered ring with involution, marked element and following relations:

χ^{∗} = χ^{−1}  χ^{4} = 1  χ • π_{m} = π_{m+1mod2}
φ^{∗} = φ  φ^{2} = 1  φ • π_{m} = π_{m}
π_{0}^{∗} = π_{1}  π_{l} • π_{m} = χ^{l+m+1} + χ^{l+m+3} + χ^{l+m+1}φ + χ^{l+m+3}φ

Proposition 8.7. K_{0} (A_{3}^{−})^{∗} = ℤ ⟨χ = u_{0}^{2} + iu_{1}^{2}, φ = g, ψ = u_{0}^{4} + u_{1}^{4}, π_{0}, π_{1}⟩ is a commutative ordered ring with involution, marked element and following relations:

χ^{∗} = χ  χ^{2} = 1  χ • π_{m} = π_{m+1mod2}
φ^{∗} = φ  φ^{2} = 1  φ • π_{m} = π_{m}
ψ^{∗} = ψ  ψ^{2} = 1  ψ • π_{m} = π_{m}
π_{0}^{∗} = π_{1}  π_{l} • π_{m} = χ^{l+m+1} + χ^{l+m+1}φ + χ^{l+m+1}ψ + χ^{l+m+1}φψ

Proposition 8.8. K_{0} (A_{n}^{−})^{∗} = ℤ ⟨χ = u_{0}^{2} + iu_{1}^{2}, φ = g, π_{m}⟩_{m=0,... ,2^{n−2}−1} for n ≥ 4 is a commutative ordered ring with involution, marked element and following relations:

χ^{∗} = χ^{−1} χ^{2^{n−1}} = 1  χ • π_{m} = π_{m+1mod2^{n−2}}
φ^{∗} = φ φ^{2} = 1 φ • π_{m} = π_{m}
π_{m}^{∗} = π_{2^{n−2}−m−1}  π_{l} • π_{m} = χ^{l+m+1} + χ^{l+m+2^{n−2}+1} + χ^{l+m+1}φ + χ^{l+m+2^{n−2}+1}φ

Proposition 8.9. K_{0} (A_{n}^{+})^{∗} = ℤ ⟨χ = u_{0}^{2} + iu_{1}^{2}, φ = g, ψ = u_{0}^{2^{n−1}} − u_{1}^{2^{n−1}}, π_{m}⟩_{m=0,... ,2^{n−2}−1}

for n ≥ 4 is a commutative ordered ring with involution, marked element and following relations:

χ^{∗} = χ^{−1} χ^{2^{n−1}} = 1  χ • π_{m} = π_{m+1mod2^{n−2}}
φ^{∗} = φ  φ^{2} = 1  φ • π_{m} = π_{m}
ψ^{∗} = ψ  ψ^{2} = 1  ψ • π_{m} = π_{m}
π_{m}^{∗} = π_{2^{n−2}−m−1}  π_{l} • π_{m} = χ^{l+m+1} + χ^{l+m+1}φ + χ^{l+m+1}ψ + χ^{l+m+1}φψ

ACKNOWLEDGMENTS

Part of this work is contained in the author’s Ph.D. thesis at the University of Southern California. I am very grateful to my Ph.D. advisor Professor Susan Montgomery for numerous discussions and helpful comments.

The work on this paper was finished when the author was a postdoctoral fellow at MSRI. I would like to thank MSRI and the organizers of the Noncommutative Algebra program for their support and hospitality.

References

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

  E-mail address: kashina@msri.org