Lifting of Nichols Algebras of Type A_{2} and Pointed Hopf Algebras of order p^{4}

Nicolás Andruskiewitsch Universidad de Córdoba Córdoba, Argentina Hans-Jürgen Schneider Universität München München, Germany

Abstract

We compute all finite-dimensional Hopf algebras whose coradical is the group algebra of an abelian group Γ and such that gr(A) ≅ 𝔅(V)#𝕜[Γ], V a 2-dimensional Yetter-Drinfeld module over Γ of type A_{2}. We classify pointed Hopf algebras of order p^{4} up to isomorphism and show that the number of quasi-isomorphism types is finite.

0 Introduction

We shall work over an algebraically closed field 𝕜 of characteristic 0. Let p be a prime number. Several classification results are known for Hopf algebras of order p^{N}. Semisimple Hopf algebras of order p^{N} have always a non-trivial central group-like element; so they can be obtained recursively by an extension process. The explicit description of all isomorphism types for p^{2} and p^{3} is actually known [9]. We deal in this paper with pointed Hopf algebras of dimension p^{N}. A Hopf algebra with coradical 𝕜[ℤ/(2)] has dimension 2^{N} and for each N there is exactly one such Hopf algebra up to isomorphisms [11, Th. 4.2.1]. Pointed Hopf algebras of order 8, 16 and 32 are respectively classified in [14], [6], [8]. ¿From now on we assume that p is odd. The only known non-semisimple Hopf algebras of dimension p^{2} are pointed [1] and it is easy to see that they are Taft algebras. Pointed non-semisimple Hopf algebras of dimension p^{3} are classified in [2], [5], [12]. Infinite families of pointed non-semisimple Hopf algebras of dimension p^{4} were constructed in [2], [4], [7]. In this paper we complete the classification of all pointed non-semisimple Hopf algebras of dimension p^{4}. If A is a pointed Hopf algebra of dimension p^{4}, then we know by [3, §7] gr(A), the graded Hopf algebra associated to the coradical filtration. We have to describe all possible liftings A of gr(A). There are two main cases: gr(A) ≅ 𝔅(V)#𝕜[Γ], and V is a Yetter-Drinfeld module over Γ of (1) type A_{1} or A_{1} × A_{1}, or (2) type A_{2} over the cyclic group of order p. The liftings of type a product of copies of A_{1} are determined in [2]. We describe explicitly all the isomorphism types in Section 2. In Section 3 of this paper we compute all liftings of type A_{2} over any finite abelian group Γ, if the corresponding root of 1 is odd and greater than 3. We conclude that a pointed Hopf algebra of dimension p^{4} with coradical of dimension p is necessarily coradically graded, and we obtain the classification of pointed Hopf algebras of dimension p^{4} in Theorem 4.1. In the appendix we show that although the number of isomorphism types of pointed Hopf algebras of dimension p^{4} is infinite, the number of quasi-isomor-phism types is in fact finite.

1 Preliminaries

We use standard notation for Hopf algebras. Our references for the theory of Hopf algebras are [10], [13]. We shall follow the method developed in [2], [3]. Let us briefly recall the main ideas. Given a pointed Hopf algebra A with coradical A_{0} = 𝕜[Γ], the graded coalgebra associated to the coalgebra filtration is in fact a graded Hopf algebra which we denote by gr(A). In fact, gr(A) is a coradically graded Hopf algebra, and can be obtained as a bosonization R#𝕜[Γ], where R is a braided Hopf algebra in the category of Yetter-Drinfeld modules over Γ. Now we know from [3, §7] the classification of all possible coradically graded Hopf algebras of order p^{4}:

Theorem 1.1 Let p be an odd prime number. Let A be a pointed non-semisimple Hopf algebras of order p^{4} with coradical 𝕜[Γ]. Then the associated graded algebra gr(A) is isomorphic to R#𝕜[Γ], where

(A). R is a quantum line if Γ has order p^{3};

(B). R is a quantum line or plane if Γ has order p^{2}; (C). R = 𝔅(V), where V has Dynkin diagram A_{2} if Γ has order p, where

p = 3 or p − 1 is divisible by 3.

To obtain our result we have to determine all possible Hopf algebras A with gr(A) as in (A), (B) or (C) of Theorem 2.1. For (A) and (B), the description follows from [2, Theorem 2]; see Section 2. We treat the case (C) in Section 3. In principle it is easy to describe all possible isomorphisms between liftings of 𝔅(V)#𝕜[Γ] when the Yetter-Drinfel’d module V is of finite Cartan type (once the defining relations are known). Let (a_{ij}) be a Cartan matrix of finite type of size θ × θ, Γ a finite abelian group, g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ such that for all i, j

χ_{i}(g_{j})χ_{j}(g_{i}) = χ_{i}(g_{i})^{a_{ij}}.

We define q_{i} = χ_{i}(g_{i}) and assume that the orders N_{i} of all q_{i} are odd and > 1. Let V be a Yetter-Drinfel’d module over 𝕜[Γ] with a basis x_{1}, …, x_{θ} such that

h · x_{i} = χ_{i}(h)x_{i} and δ(x_{i}) = g_{i} ⊗ x_{i}

for all h ∈ Γ and all i. Let A be a pointed Hopf algebra with gr(A) ≅ 𝔅(V)#𝕜[Γ]. We choose elements a_{i} in the first term of the coradical filtration of A such that the image of x_{i}#1 in gr(A) is the canonical image of a_{i} in A_{1}/A_{0}. Then A is generated as an algebra by Γ and the elements a_{i}. We consider another such Yetter-Drinfel’d module W over 𝕜[Γ̃] of Cartan type (ã_{ij}) of size θ̃ × θ̃ with elements h_{j} ∈ Γ̃, characters η_{j} and corresponding basis elements w_{j}. Let B be a lifting of 𝔅(W)#𝕜[Γ̃] with corresponding elements b_{j}.

Lemma 1.2 Assume for all i ≠ j, g_{i} ≠ g_{j} or χ_{i} ≠ χ_{j}. (In particular this holds if for all i ≠ j, the order of q_{i} does not divide 2 − a_{ij}). Let Φ : A → B be a Hopf algebra isomorphism. Then θ = θ̃, q_{i}^{a_{ij}} = qã_{ij} ; i for all i, j and there are a group isomorphism ϕ : Γ → Γ̃, a permutation σ and non-zero scalars α_{i} such that

Φ|Γ = ϕ, Φ(a_{i}) = α_{i}b_{σ(i)}, ϕ(g_{i}) = h_{σ(i)}, χ_{i} = η_{σ(i)}ϕ, 1 ≤ i ≤ θ.

Proof. Φ induces a Hopf algebra isomorphism gr(A) → gr(B). By [3, Proposition 6.3] the assertion holds on the graded level since by our assumption the Yetter-Drinfel’d module V is a direct sum of pairwise non-isomorphic one-dimen-sional submodules 𝕜x_{i}. Thus for all i, Φ(a_{i}) − α_{i}b_{σ(i)} is an element in the group algebra which must be zero since the group is acting on it via the non-trivial character η_{σ(i)}. □

Conversely, data ϕ, σ and α_{i} as in Lemma 1.2 define a Hopf algebra isomorphism Φ by Φ|Γ = ϕ and Φ(a_{i}) = α_{i}b_{σ(i)} if all the relations between the a_{i} are respected. This gives a condition on the scalars α_{i} which has to be worked out in each case. In the case of quantum linear spaces, that is if a_{ij} = 0 for all i ≠ j, this is very easy. By [2, Section 5], a lifting A of a quantum linear space is given by scalars μ_{i} ∈ {0, 1} for all i, λ_{ij} ∈ 𝕜 for all i < j such that μ_{i} = 0 if g_{i}^{N_{i}} = 1 or χ_{i}^{N_{i}} ≠ 1, and λ_{ij} = 0 if g_{i}g_{j} = 1 or χ_{i}χ_{j} ≠ 1. The relations we have to check are

a_{i}^{N_{i}} = μ_{i}(1 − g_{i}^{N_{i}}) for all i, a_{j}a_{i} = χ_{i}(g_{j})a_{i}a_{j} + λ_{ij}(1 − g_{i}g_{j}) for all i < j.

If B is another such lifting with data μ̃_{i}, λ̃_{ij}, then A is isomorphic to B if and only if there are σ and α_{i} as in Lemma 1.2 satisfying for all i and i < j,

α_{i}^{N_{i}} μ̃_{σ(i)} = μ_{i};

α_{i}α_{j}λ̃_{σ(i)σ(j)} = λ_{ij} if σ(i) < σ(j); − χ_{i}(g_{j})α_{i}α_{j}λ̃_{σ(j)σ(i)} = λ_{ij} if σ(i) > σ(j).

2 The braided Hopf algebra R is a quantum linear space

Let p be an odd prime number. In order to classify pointed Hopf algebras of order p^{4} we describe all liftings of quantum linear lines over a group Γ of order p^{3} and of quantum lines or planes over Γ of order p^{2}. The following list contains all such Hopf algebras, and the remarks at the end of the last section show that Hopf algebras in two different cases of this list are non-isomorphic. We leave the details of the proof to the reader.

2.1 Index p Here Γ is a group of order p^{3}. Since R is a quantum line, there exist g ∈ Γ central and a character χ ∈ Γ̂ such that χ(g) has order p. Note that Γ is necessarily abelian: indeed, the center of a non-abelian group of order p^{3} is contained in its commutator, so χ(g) = 1 for any g ∈ Γ central, χ ∈ Γ̂. There are three abelian groups of order p^{3} which we write as ⟨u_{1}, …, u_{r}⟩, 1 ≤ r ≤ 3. Then A can be presented by generators u_{i}, 1 ≤ i ≤ r, a, and relations

u_{i}u_{j} = u_{j}u_{i} for all 1 ≤ i, j ≤ r;
u_{1}^{p^{3}} = 1 if r = 1; u_{1}^{p^{2}} = 1, u_{2}^{p} = 1 if r = 2;
u_{i}^{p} = 1 for all i if r = 3.
a^{p} = μ(1 − g^{p}), with μ either 0 or 1;
hah^{−1} = χ(h)a, h ∈ Γ.

(1)

If μ = 1, then g^{p} ≠ 1 and χ^{p} = 1. The value of μ is invariant under isomorphism. The Hopf algebra structure of A is determined by

Δ(h) = h ⊗ h, Δ(a) = a ⊗ 1 + g ⊗ a, h ∈ Γ.

Up to isomorphism there are various subcases according to the order of g.

If Γ = ⟨u_{1}⟩, we have three cases where g = u_{1}^{p^{s}} and χ(u_{1}) = ξ, ξ a root of unity of order p^{s+1}, 0 ≤ s ≤ 2. In each case, if η is another root of unity of the same order p^{s+1}, then the corresponding Hopf algebras are isomorphic if and only if there is an integer i with ξ = η^{i} and i ≡ 1 mod p^{3−s} (hence ξ = η in case s = 0, 1).

If Γ = ⟨u_{1}, u_{2}⟩, we take g = u_{1}^{p} and χ(u_{1}) = ξ, χ(u_{2}) = 1, ξ a root of unity of order p^{2} (we get an isomorphic Hopf algebra only if we replace ξ by ξ^{i}, i ≡ 1 mod p); or g = u_{2} and χ(u_{1}) = ρ, χ(u_{2}) = ξ, ξ a primitive p-th root of unity and ρ = 1 or an a priori given root of unity of order p^{2}; or g = u_{1} and χ(u_{1}) = ξ, χ(u_{2}) = 1, ξ a p-th root of unity. In the last two cases the Hopf algebras are pairwise non-isomorphic.

If Γ = ⟨u_{1}, u_{2}, u_{3}⟩, we take g = u_{1} and χ(u_{1}) = ξ, ξ a primitive p-th root of unity, χ(u_{2}) = 1, χ(u_{3}) = 1. These Hopf algebras are pairwise non-isomorphic.

2.2 Index p^{2}, R a quantum line There exist g ∈ Γ and a character χ ∈ Γ̂ such that χ(g) has order p^{2}. Necessarily, Γ is isomorphic to ℤ/(p^{2}). Then there exists a primitive p^{2}-th root of unity ξ ∈ 𝕜 such that A can be presented by generators g, a and relations

\begin{matrix}g^{p^{2}} = 1, \\ a^{p^{2}} = 0; \\ gag^{−1} = ξa. \end{matrix}  (2)

The Hopf algebra structure of A is determined by

Δ(g) = g ⊗ g, Δ(a) = a ⊗ 1 + g ⊗ a.

That is, A is isomorphic to a Taft algebra of order p^{4}. For different roots of unity the Hopf algebras are non-isomorphic.

2.3 Index p^{2}, R a quantum plane and Γ ≃ ℤ/(p) × ℤ/(p)

Then there exist g_{1}, g_{2} ∈ Γ and characters χ_{1}, χ_{2} ∈ Γ̂ such that χ_{1}(g_{1}) and χ_{2}(g_{2}) have order p, χ_{1}(g_{2})χ_{2}(g_{1}) = 1. A can be presented by generators u_{1}, u_{2}, a_{1}, a_{2} and relations

u_{i}^{p} = 1, i = 1, 2, u_{1}u_{2} = u_{2}u_{1};
a_{i}^{p} = 0, i = 1, 2;
ha_{i}h^{−1} = χ_{i}(h)a_{i}, h ∈ Γ, i = 1, 2;
a_{1}a_{2} − χ_{2}(g_{1})a_{2}a_{1} = λ(1 − g_{1}g_{2}), with λ either 0 or 1.

(3)

If λ = 1, then χ_{1}χ_{2} = 1 and g_{1}g_{2} ≠ 1. The value of λ is invariant under isomorphism. The Hopf algebra structure of A is determined by

Δ(u_{i}) = u_{i} ⊗ u_{i}, Δ(a_{i}) = a_{i} ⊗ 1 + g_{i} ⊗ a_{i}, i = 1, 2.

There are two subcases depending on whether or not g_{1}, g_{2} are linearly independent over ℤ/(p). In the first subcase we take g_{i} = u_{i} and χ_{i}(u_{i}) = ξ_{i}, ξ_{i} primitive p-th roots of unity, 1 ≤ i ≤ 2 and χ_{1}(u_{2}) = ρ, ρ a p-th root of unity. Note that necessarily χ_{2}(u_{1}) = ρ^{−1}. If we interchange ξ_{1}, ξ_{2} and replace ρ by ρ^{−1}, then we get isomorphic Hopf algebras, and this is the only possibility to get an isomorphism.

In the second subcase we take g_{1} = u_{1}, g_{2} = u_{1}^{l}, l an integer not divisible by p, χ_{1}(u_{1}) = ξ, χ_{2}(u_{1}) = ξ^{−l}, χ_{1}(u_{2}) = 1, χ_{2}(u_{2}) = ξ^{ε}, ξ a primitive p-th root of unity and ε = 0, 1. The only way to get isomorphic Hopf algebras is to replace l by l^{−1} mod p and ξ by ξ^{−l^{2}}.

2.4 Index p^{2}, R a quantum plane and Γ ≃ ℤ/(p^{2})

Then there exist g_{1}, g_{2} ∈ Γ and characters χ_{1}, χ_{2} ∈ Γ̂ such that χ_{1}(g_{1}) and χ_{2}(g_{2}) have order p, χ_{1}(g_{2})χ_{2}(g_{1}) = 1. There are further possible subcases according to the orders of g_{1}, g_{2}, χ_{1}, and χ_{2}. It is easy to see that either ord(g_{i}) = p and ord(χ_{i}) = p^{2}, i = 1, 2 (first subcase) or ord(g_{i}) = p^{2} and ord(χ_{i}) = p, i = 1, 2 (second, third and fourth subcases according to the different choices of μ_{1} and μ_{2}).

First subcase. Let ξ be a primitive p^{2}-th root of unity and integers b, c_{1}, c_{2} integers prime to p such that p|(bc_{1} + c_{2}). A is generated by h, x_{1}, x_{2} with relations

h^{p^{2}} = 1
hx_{1} = ξx_{1}h, hx_{2} = ξ^{b}x_{2}h;
x_{1}^{p} = 0, x_{2}^{p} = 0;
x_{2}x_{1} − ξ^{pbc_{1}}x_{1}x_{2} = λ (1 − h^{p(c_{1}+c_{2})}) , where λ is 0 or 1
(and it could be 1 only if b ≡ −1 mod p^{2} and c_{1} + c_{2} ≢ 0 mod p).

(4)

carries a Hopf algebra structure given by

Δ(h) = h ⊗ h, Δ(x_{i}) = x_{i} ⊗ 1 + h^{pc_{i}} ⊗ x_{i}, 1 ≤ i ≤ 2.

The only ways to get isomorphic Hopf algebras (except congruence mod p for c_{1}, c_{2} and mod p^{2} for b) are replacing c_{1}, c_{2}, b, ξ by sc_{1}, sc_{2}, b, η where s is an integer prime to p and η^{s} = ξ; or by sc_{2}, sc_{1}, d, η where s again is prime to p, d is an integer with bd ≡ 1 mod p^{2} and η^{s} = ξ^{b}.

Second subcase. Let q be a primitive p-th root of unity and a, d_{1}, d_{2} integers prime to p such that p|(ad_{1} + d_{2}). A is generated by h, x_{1}, x_{2} with relations

h^{p^{2}} = 1;

hx_{1} = q^{d_{1}}x_{1}h, hx_{2} = q^{d_{2}}x_{2}h;

x_{1}^{p} = 0, x_{2}^{p} = 0;

x_{2}x_{1} − q^{d_{2}}x_{1}x_{2} = λ (1 − h^{1+a}) , where λ is 0 or 1

(and it could be 1 only if d_{1} + d_{2} ≡ 0 mod p and 1 + a ≢ 0 mod p).

(5)

carries a Hopf algebra structure given by

Δ(h) = h ⊗ h, Δ(x_{1}) = x_{1} ⊗ 1 + h ⊗ x_{1}, Δ(x_{2}) = x_{2} ⊗ 1 + h^{a} ⊗ x_{2}.

The only way to get isomorphic Hopf algebras (except congruence mod p for d_{1}, d_{2} and mod p^{2} for a) is to replace d_{1}, d_{2}, a by d_{2}a, d_{1}a, b where ab ≡ 1 mod p^{2}.

Third subcase. Let q be a primitive p-th root of unity and a, d_{1}, d_{2} integers prime to p such that p|(ad_{1} + d_{2}). A is generated by h, x_{1}, x_{2} with relations

h^{p^{2}} = 1;

hx_{1} = q^{d_{1}}x_{1}h, hx_{2} = q^{d_{2}}x_{2}h;

x_{1}^{p} = 1 − h^{p}, x_{2}^{p} = 0;

x_{2}x_{1} − q^{d_{2}}x_{1}x_{2} = λ (1 − h^{1+a}) , where λ is 0 or 1

(and it could be 1 only if d_{1} + d_{2} ≡ 0 mod p and 1 + a ≢ 0 mod p).

(6)

carries a Hopf algebra structure given by

Δ(h) = h ⊗ h, Δ(x_{1}) = x_{1} ⊗ 1 + h ⊗ x_{1}, Δ(x_{2}) = x_{2} ⊗ 1 + h^{a} ⊗ x_{2}.

Here the only way to get isomorphic Hopf algebras is to replace d_{1}, d_{2} by mod p congruent elements and a by a mod p^{2} congruent element.

In the first three subcases, λ is invariant under isomorphism.

Fourth subcase. It is only here that we get an infinite family of non-isomorphic Hopf algebras. Let q be a primitive p-th root of unity and a, d_{1}, d_{2} integers prime to p such that p|(ad_{1} + d_{2}). A is generated by h, x_{1}, x_{2} with relations

h^{p^{2}} = 1;

hx_{1} = q^{d_{1}}x_{1}h, hx_{2} = q^{d_{2}}x_{2}h; x_{1}^{p} = 1 − h^{p}, x_{2}^{p} = 1 − h^{pa}; x_{2}x_{1} − q^{d_{2}}x_{1}x_{2} = λ (1 − h^{1+a}) , where λ ∈ 𝕜 (and it could be non zero only if d_{1} + d_{2} ≡ 0 mod p and 1 + a ≢ 0 mod p).

(7)

carries a Hopf algebra structure given by

Δ(h) = h ⊗ h, Δ(x_{1}) = x_{1} ⊗ 1 + h ⊗ x_{1}, Δ(x_{2}) = x_{2} ⊗ 1 + h^{a} ⊗ x_{2}.

Hopf algebras defined by d_{1}, d_{2}, a, λ and d̃_{1}, d̃_{2}, ã, λ̃ are isomorphic if and only if there exists a scalar α ∈ 𝕜 such that α^{p} = 1 and (1) λ = αλ̃, d̃_{i} ≡ d_{i} mod p, 1 ≤ i ≤ 2, ã ≡ a mod p^{2}, or (2) λ = −αλ̃, d̃_{2} ≡ d_{1}a mod p, d̃_{1} ≡ d_{2}a mod p and aã ≡ 1 mod p^{2}.

3 Nichols algebras of type A_{2} and their liftings

We fix a finite abelian group Γ, elements g_{1}, g_{2} ∈ Γ and characters χ_{1}, χ_{2} ∈ Γ̂ of Γ, an odd integer p > 1 (which is not assumed to be prime in this section) and a primitive p-th root of unity q in 𝕜 such that

q = χ_{1}(g_{1}) = χ_{2}(g_{2}), χ_{1}(g_{2})χ_{2}(g_{1}) = q^{−1}.  (8)

Thus the braiding b_{ij} = χ_{j}(g_{i}), 1 ≤ i, j ≤ 2, is of type A_{2}. If C is a coalgebra, we denote by G(C) the set of group-like elements of C. If g, h ∈ G(C), then we let P_{g,h}(C) = {x ∈ C | Δ(x) = g ⊗ x + x ⊗ h} be the space of all (g, h)-primitive elements. If A is a Hopf algebra and Γ = G(A) is abelian, then Γ acts on each P_{g,h}(A) by conjugation, and if χ is a character of Γ, then we define P_{g,h}(A)^{χ} = {a ∈ P_{g,h}(A) | uau^{−1} = χ(u)a, for all u ∈ Γ}.

Lemma 3.1 Let A be a Hopf algebra with coradical 𝕜[Γ] and a_{1}, a_{2} ∈ A such that

Δ(a_{i}) = g_{i} ⊗ a_{i} + a_{i} ⊗ 1, ha_{i}h^{−1} = χ_{i}(h)a, for all h ∈ Γ, 1 ≤ i ≤ 2.

Define c = a_{1}a_{2} − χ_{2}(g_{1})a_{2}a_{1}. Then (1) Δ(c) = g_{1}g_{2} ⊗ c + c ⊗ 1 + (1 − q^{−1})a_{1}g_{2} ⊗ a_{2}.

(2) ca_{1} − χ_{1}(g_{2})a_{1}c ∈ P_{g_{1}^{2}g_{2},1}(A)^{χ_{1}^{2}χ_{2}}.

(3) a_{2}c − χ_{1}(g_{2})ca_{2} ∈ P_{g_{1}g_{2}^{2},1}(A)^{χ_{1}χ_{2}^{2}}.

(4) a_{i}^{p} ∈ P_{g_{i}^{p},1}(A)^{χ_{i}^{p}}, 1 ≤ i ≤ 2.

(5) Assume the Serre relations

ca_{1} − χ_{1}(g_{2})a_{1}c = 0, and a_{2}c − χ_{1}(g_{2})ca_{2} = 0.

Then

Δ(c^{p}) = g_{1}^{p}g_{2}^{p} ⊗ c^{p} + c^{p} ⊗ 1 + (q − 1)^{p}χ_{1}(g_{2})^{\frac{p(p−1)}{2}} a_{1}^{p}g_{2}^{p} ⊗ a_{2}^{p}.

Proof. (1) follows easily from the assumptions. (2) and (3) follow by direct computation using also (8).

(4) follows from the q-binomial formula: let u_{1} = a_{i} ⊗ 1, u_{2} = g_{i} ⊗ a_{i}. Then

u_{2}u_{1} = g_{i}a_{i} ⊗ a_{i} = qa_{i}g_{i} ⊗ a_{i} = qu_{1}u_{2}.

Hence by the q-binomial formula,

Δ(a_{i})^{p} = (u_{1} + u_{2})^{p} = u_{1}^{p} + u_{2}^{p} = g_{i}^{p} ⊗ a_{i}^{p} + a_{i}^{p} ⊗ 1,

since q is a p-th root of unity.

(5) Let u_{1} = c ⊗ 1, u_{2} = (1 − q^{−1})a_{1}g_{2} ⊗ a_{2} and u_{3} = g_{1}g_{2} ⊗ c. Then

u_{2}u_{1} = (1 − q^{−1})a_{1}g_{2}c ⊗ a_{2}
= (1 − q^{−1})χ_{1}(g_{2})χ_{2}(g_{2})a_{1}cg_{2} ⊗ a_{2},
= (1 − q^{−1})χ_{1}(g_{2})qχ_{1}(g_{2})^{−1}ca_{1}g_{2} ⊗ a_{2},
since a_{1}c = χ_{1}(g_{2})^{−1}ca_{1},
= qu_{1}u_{2}.

In the same way we see u_{3}u_{2} = qu_{2}u_{3} using the relation a_{2}c = χ_{1}(g_{2})ca_{2}. Finally

u_{3}u_{1} = g_{1}g_{2}c ⊗ c = (χ_{1}χ_{2})(g_{1}g_{2})cg_{1}g_{2} ⊗ c = qu_{1}u_{3},

since (χ_{1}χ_{2})(g_{1}g_{2}) = χ_{1}(g_{1})χ_{1}(g_{2})χ_{2}(g_{1})χ_{2}(g_{2}) = q by (8). Hence by the q-binomial formula,

Δ(c^{p}) = (u_{1} + u_{2} + u_{3})^{p} = (u_{1} + u_{2})^{p} + u_{3}^{p} = u_{1}^{p} + u_{2}^{p} + u_{3}^{p}.

This proves (5) since

(a_{1}g_{2})^{p} = χ_{1}(g_{2})^{\frac{p(p−1)}{2}} a_{1}^{p}g_{2}^{p}.
□

To compute all liftings of type A_{2} we first list the following properties of our elements g_{1}, g_{2}, χ_{1}, χ_{2} satisfying (8).

Lemma 3.2 (1) g_{1}^{2}g_{2} ≠ g_{i} and g_{1}g_{2}^{2} ≠ g_{i}, 1 ≤ i ≤ 2.

(2) If p > 3, then χ_{1}^{2}χ_{2} ≠ ε, χ_{1}χ_{2}^{2} ≠ ε.

(3) If Γ is cyclic of order 3 and χ_{1}^{2}χ_{2} = ε or χ_{1}χ_{2}^{2} = ε, then g_{1}^{2}g_{2} = 1 = g_{1}g_{2}^{2}.

Proof. (1) Assume g_{1}^{2}g_{2} = g_{2}. Then g_{1}^{2} = 1, and we get the contradiction χ_{1}(g_{1})^{2} = 1. If g_{1}^{2}g_{2} = g_{1}, then g_{1}g_{2} = 1, and we see from (8) that χ_{2}(g_{1}) = 1, and χ_{1}(g_{2}) = 1. By (8) this implies q = 1 which is impossible. In the same way we see that g_{1}g_{2}^{2} ≠ g_{i}.

(2) Assume χ_{1}^{2}χ_{2} = ε. Then χ_{1}χ_{2} = χ_{1}^{−1}, and χ_{1}^{−1}(g_{1})χ_{1}(g_{2}) = 1 and χ_{1}^{−1}(g_{2})χ_{2}(g_{1}) = 1 by (8). Hence q = χ_{1}(g_{2}) = χ_{2}(g_{1}) which implies by (8) the contradiction q^{3} = 1. In the same way we prove χ_{1}χ_{2}^{2} ≠ ε.

(3) We consider the case χ_{1}^{2}χ_{2} = ε. Since Γ̂ ≅ Γ is of order 3, we get χ_{1} = χ_{2}. Then χ_{1}(g_{1}^{2}g_{2}) = χ_{1}(g_{1})χ_{2}(g_{2})χ_{1}(g_{2}) = 1 by (8). Therefore g_{1}^{2}g_{2} = 1. Hence g_{1} = g_{2}, and g_{1}g_{2}^{2} = 1. □

We also need

Lemma 3.3 Let Γ be any group, g, h ∈ Γ, α ∈ 𝕜 and x ∈ 𝕜[Γ]. Then the following are equivalent:

(1) Δ(x) = gh ⊗ x + x ⊗ 1 + α(1 − g)h ⊗ (1 − h).

(2) x = α(1 − h) + λ(1 − gh) for some λ ∈ 𝕜.

Proof. Both statements mean that x − α(1 − h) is (gh, 1)-primitive in the group algebra 𝕜[Γ]. □

Lemma 3.4 Let x, y, z be elements in an algebra, α, β scalars in 𝕜 and n a natural number. Assume yx = αxy + z.

(1) Asssume zx = βxz. Then yx^{n} = α^{n}x^{n}y + (∑_{i=0}^{n−1} α^{i}β^{n−1−i})x^{n−1}z, and if α ≠ β and α^{n} = β^{n}, then (αxy + z)x^{n−1} = α^{n}x^{n}y.

(2) Assume yz = βzy. Then y^{n}x = α^{n}xy^{n} + (∑_{i=0}^{n−1} α^{i}β^{n−1−i})zy^{n−1}, and if α ≠ β and α^{n} = β^{n}, then y^{n−1}(αxy + z) = α^{n}xy^{n}.

Proof. The first formula in (1) is easily proved by induction and the second formula follows from the first since ∑_{i=0}^{n−1} α^{i}β^{n−1−i} = 0. (2) is dual to (1). □

Definition 3.5 Define

U^{+} := 𝕜⟨x_{1}, z, x_{2} | x_{2}x_{1} = χ_{2}(g_{1})^{−1}x_{1}x_{2} − χ_{2}(g_{1})^{−1}z,
zx_{1} = χ_{1}(g_{2})x_{1}z,
x_{2}z = χ_{1}(g_{2})zx_{2}⟩

as a Hopf algebra in the category of left Yetter-Drinfel’d modules over 𝕜[Γ] with primitive elements x_{1}, x_{2} and action and coaction of Γ given by

h · x_{i} = χ_{i}(h)x_{i}, for all h ∈ Γ, δ(x_{i}) = g_{i} ⊗ x_{i}, 1 ≤ i ≤ 2.

Let U := U^{+}#𝕜[Γ] be the Radford biproduct. Identifying y#u = yu , y ∈ U^{+}, u ∈ 𝕜[Γ] we have in U

Δ(x_{i}) = g_{i} ⊗ x_{i} + x_{i} ⊗ 1, hx_{i}h^{−1} = χ_{i}(h)x_{i}, for all h ∈ Γ, 1 ≤ i ≤ 2.

In quantum group theory our U is usually called U^{≥0}. Note that U^{+} is a welldefined braided Hopf algebra since by Lemma 3.1 the Serre relations are primitive in the free algebra generated by the primitive elements x_{1}, x_{2}. It follows easily from the diamond lemma that U^{+} has the PBW-basis x_{1}^{i}z^{j}x_{2}^{l}, i, j, l ≥ 0. Then V := 𝕜x_{1} + 𝕜x_{2} ⊂ U^{+} is a Yetter-Drinfel’d module over 𝕜[Γ] of type A_{2}.

Theorem 3.6 Let μ_{1}, μ_{2} ∈ {0, 1} and λ ∈ 𝕜 be scalars such that

μ_{i} = 0, if g_{i}^{p} = 1 or χ_{i}^{p} ≠ ε, 1 ≤ i ≤ 2.  (9)
λ = 0, if g_{1}^{p}g_{2}^{p} = 1 or χ_{1}^{p}χ_{2}^{p} ≠ ε.  (10)

Define u_{i} := μ_{i}(1 − g_{i}^{p}), 1 ≤ i ≤ 2, and u := α(1 − g_{2}^{p}) + λ(1 − g_{1}^{p}g_{2}^{p}) in 𝕜[Γ], where α := μ_{1}μ_{2}(q − 1)^{p}. Then

A := U/(x_{1}^{p} − u_{1}, z^{p} − u, x_{2}^{p} − u_{2})

is a quotient Hopf algebra of U of dimension ord(Γ)p^{3}, 𝔅(V) ≅ U^{+}/(x_{1}^{p}, z^{p}, x_{2}^{p}), and

gr(A) ≅ 𝔅(V)#𝕜[Γ] ≅ U/(x_{1}^{p}, z^{p}, x_{2}^{p}).

Proof. 1) We first show that A is a quotient Hopf algebra of U. By Lemma 3.1 (4), x_{i}^{p} − u_{i} is (g_{i}^{p}, 1)-primitive, since u_{i} is (g_{i}^{p}, 1)-primitive by construction. Let σ := (q − 1)^{p}χ_{1}(g_{2})^{\frac{p(p−1)}{2}} . Note that μ_{1}μ_{2}σ = α since p is odd and χ_{1}^{p} = ε if μ_{1} ≠ 0. By Lemma 3.1 (5) and Lemma 3.3 (1),

Δ(z^{p}) = g_{1}^{p}g_{2}^{p} ⊗ z^{p} + σx_{1}^{p}g_{2}^{p} ⊗ x_{2}^{p} + z^{p} ⊗ 1

and

Δ(u) = g_{1}^{p}g_{2}^{p} ⊗ u + σu_{1}g_{2}^{p} ⊗ u_{2} + u ⊗ 1.

Hence

Δ(z^{p}−u) = g_{1}^{p}g_{2}^{p}⊗(z^{p}−u)+σx_{1}^{p}g_{2}^{p}⊗(x_{2}^{p}−u_{2})+σ(x_{1}^{p}−u_{1})g_{2}^{p}⊗u_{2}+(z^{p}−u)⊗1.

Thus (x_{1}^{p} − u_{1}, z^{p} − u, x_{2}^{p} − u_{2}) is a biideal of U, and A is a quotient Hopf algebra of U since U is pointed.

2) Now we show that dim A = ord(Γ)p^{3}. Let a_{1}, c, a_{2} be the canonical images of x_{1}, z, x_{2} in A. Then the elements ha_{1}^{i}c^{j}a_{2}^{l}, h ∈ Γ, i, j, l ≥ 0 form a 𝕜-basis of A. This can be seen using the diamond lemma. As in the proof of [2, Proposition 5.2] we use the ordering h_{1} < … < h_{s} < a_{1} < c < a_{2} where h_{1}, …, h_{s} are generators of the group Γ. If M is a monomial in h_{1}, …, a_{2} we let Φ(M) be its {a_{1}, c, a_{2}}-part. If M, N are monomials we define M < N if Φ(M) < Φ(N) in the standard degree-lexicographic order or Φ(M) = Φ(N) and M is smaller than N in the standard order. The relations of A are the relations of Γ, the relations of U, namely for all h ∈ Γ, 1 ≤ i ≤ 2,

a_{2}a_{1} = χ_{2}(g_{1})^{−1}a_{1}a_{2} − χ_{2}(g_{1})^{−1}c,
ca_{1} = χ_{1}(g_{2})a_{1}c,
a_{2}c = χ_{1}(g_{2})ca_{2},
a_{i}h = χ_{i}(h)^{−1}ha_{i},
ch = (χ_{1}χ_{2})(h)^{−1}hc,

and the additional relations a_{1}^{p} = u_{1}, c^{p} = u, a_{2}^{p} = u_{2}. We only need to consider overlaps involving the additional relations. In the overlapping pair (a_{2}a_{1}^{p}, (a_{2}a_{1})a_{1}^{p−1}) the first element is reduced to a_{2}u_{1}. The second element is reduced to (χ_{2}(g_{1})^{−1}a_{1}a_{2} − χ_{2}(g_{1})^{−1}c)a_{1}^{p−1}. By Lemma 3.4 (1) this expression is equal to χ_{2}(g_{1})^{−p}a_{1}^{p}a_{2} = χ_{2}(g_{1})^{−p}u_{1}a_{2}. To see that we have reached the same normal form we need a_{2}u_{1} = χ_{2}(g_{1})^{−p}u_{1}a_{2}. This equality holds in case u_{1} = 0. If u_{1} ≠ 0, we have χ_{1}^{p} = ε by (10), hence by (9) also χ_{2}(g_{1})^{p} = 1, and the desired equality again follows. In the same way we treat (a_{2}a_{1}^{p}, a_{2}^{p−1}(a_{2}a_{1})) using Lemma 3.4 (2). The remaining overlaps

(a_{i}h, a_{i}^{p−1}(a_{i}h)), (c^{p}h, c^{p−1}ch), (ca_{1}^{p}, (ca_{1})a_{1}^{p−1}), (c^{p}a_{1}, c^{p−1}(ca_{1})),

(a_{2}^{p}c, a_{2}^{p−1}(a_{2}c)), (a_{2}c^{p}, (a_{2}c)c^{p−1}), (a_{i}^{p}a_{i}^{r}, a_{i}^{r}a_{i}^{p}) and (c^{p}c^{r}, c^{r}c^{p})

are easily reduced to the same normal form since respectively

u_{i} = χ_{i}(h)^{−1}u_{i}, u = (χ_{1}χ_{2})(h)^{−p}u, cu_{1} = χ_{1}(g_{2})^{p}u_{1}c, ua_{1} = χ_{1}(g_{2})^{p}a_{1}u,

u_{2}c = χ_{1}(g_{2})^{p}cu_{2}, a_{2}u = χ_{1}(g_{2})ua_{2}, u_{1}a_{i} = a_{i}u_{1}, and uc = cu.

3) The braided Hopf algebra 𝔅(V) is a quotient of the free algebra T (V) generated by the primitive elements x_{1}, x_{2} with action and coaction of Γ as in V. Define z := x_{1}x_{2} − χ_{2}(g_{1})x_{2}x_{1} in the free algebra. By Lemma 3.1 applied to the biproduct T (V)#𝕜[Γ] it follows that the elements zx_{1} − χ_{1}(g_{2})x_{1}z, x_{2}z = χ_{1}(g_{2})zx_{2} and x_{1}^{p}, x_{2}^{p} are primitive in T (V). Since P (𝔅(V)) = V by definition of 𝔅(V), these elements are mapped onto 0 in 𝔅(V). Then it follows from Lemma 3.1 (5) that also z^{p} is primitive in 𝔅(V) hence 0. Thus 𝔅(V) is a braided Hopf algebra quotient of U^{+}/(x_{1}^{p}, z^{p}, x_{2}^{p}). We know that 𝔅(V) has dimension p^{3} [3]. Hence 𝔅(V) ≅ U^{+}/(x_{1}^{p}, z^{p}, x_{2}^{p}) since the last algebra also has dimension p^{3} for example by the proof in 2). Finally consider the Hopf algebra map U → gr(A) which maps x_{i} onto the residue class ā_{i} of a_{i} in gr(A)(1) = A_{1}/A_{0}, 1 ≤ i ≤ 2, z onto c̄ in A_{2}/A_{1} and h onto h for the elements in the group Γ. This map is surjective since A is generated by a_{1}, a_{2} and Γ, and x_{1}^{p}, x_{2}^{p}, z^{p} are in the kernel since ā_{1}^{p}, ā_{2}^{p}, c̄^{p} are all 0 since by definition a_{1}^{p}, a_{2}^{p}, c^{p} are in the group algebra A_{0} = 𝕜[Γ]. Hence

U/(x_{1}^{p}, x_{2}^{p}, z^{p}) ≅ 𝔅(V)#𝕜[Γ] ≅ gr(A).

□

As an example of dimension p^{5} consider Γ = ⟨g⟩ cyclic of order p^{2}. Let q be a primitive p-th root of unity and b, d integers such that bd ≡ 1, b + d ≡ −1 mod p. Such integers exist if p = 3 or p ≡ 1 mod 3 and −3 is a square mod p by the quadratic reciprocity law. Define g_{1} := g, g_{2} := g^{b}, and χ_{1}(g) := q, χ_{2}(g) := q^{d}. Then we can choose in Theorem 3.6 μ_{1} = 1, μ_{2} = 1 and λ arbitrary in 𝕜.

Theorem 3.7 Let A be a pointed Hopf algebra with

gr(A) ≅ 𝔅(V)#𝕜[Γ]

where V is of type A_{2} as above.

(1) If p > 3 or if Γ is cyclic of order 3, then A is one of the Hopf algebras described in Theorem 3.6.

(2) If Γ has exponent p where p > 3 is a prime, or Γ is cyclic of order 3, then A ≅ 𝔅(V)#𝕜[Γ].

Proof. We show (1); (2) follows immediately. Let A_{0} ⊂ A_{1} ⊂ … be the coradical filtration of A. Then we know from [2, Lemma 5.4] that

A_{1} = 𝕜[Γ] ⊕ ⨁_{g,h,χ} P_{g,h}(A)^{χ}  (11)

where g, h run through all elements of Γ and χ runs through all non-trivial characters of Γ. Since gr(A) ≅ 𝔅(V)#𝕜[Γ] we conclude for all elements g ∈ Γ and non-trivial characters χ of Γ

P_{g,1}(A)^{χ} ≠ 0 if and only if g = g_{i}, χ = χ_{i}, i ∈ {1, 2}.  (12)

Choose a_{i} ∈ P_{g_{i},1}(A)^{χ_{i}} with residue class x_{i} in A_{1}/A_{0}, 1 ≤ i ≤ 2, and define c := a_{1}a_{2} − χ_{2}(g_{1})a_{2}a_{1}. By Lemma 3.1, ca_{1} − χ_{1}(g_{2})a_{1}c ∈ P_{g_{1}^{2}g_{2},1}(A)^{χ_{1}^{2}χ_{2}} and a_{2}c − χ_{1}(g_{2})ca_{2} ∈ P_{g_{1}g_{2}^{2},1}(A)^{χ_{1}χ_{2}^{2}}. If χ_{1}^{2}χ_{2} is non-trivial, then P_{g_{1}^{2}g_{2},1}(A)^{χ_{1}^{2}χ_{2}} = 0 by (12) since g_{1}^{2}g_{2} ≠ g_{i} by Lemma 3.2 (1). In particular, P_{g_{1}^{2}g_{2},1}(A)^{χ_{1}^{2}χ_{2}} = 0 if p > 3 by Lemma 3.2 (2). If Γ is cyclic of order 3, and χ_{1}^{2}χ_{2} = ε, then g_{1}^{2}g_{2} = 1 by Lemma 3.2 (3). Hence P_{g_{1}^{2}g_{2},1}(A) = P_{1,1}(A) = P (A) = 0 since A is finite-dimensional and the characteristic of 𝕜 is 0. Similarly, P_{g_{1}g_{2}^{2},1}(A)^{χ_{1}χ_{2}^{2}} = 0. Thus we have shown the Serre relations

ca_{1} = χ_{1}(g_{2})a_{1}c, a_{2}c = χ_{1}(g_{2})ca_{2}.

By Lemma 3.1 (4), a_{i}^{p} ∈ P_{g_{i}^{p},1}(A)^{χ_{i}^{p}}, 1 ≤ i ≤ 2. Since χ_{i}^{p}(g_{i}^{p}) = 1, 1 ≤ i ≤ 2, g_{i}^{p} and a_{i}^{p} commute. Thus by Lemma 3.1 in [2],

a_{i}^{p} = μ_{i}(1 − g_{i}^{p}) for some μ_{i} ∈ 𝕜, 1 ≤ i ≤ 2.

If μ_{i} ≠ 0, i ∈ {1, 2}, then we can choose μ_{i} = 1 by replacing a_{i} by a scalar multiple of a_{i}. If χ_{i}^{p} ≠ ε, and g_{i}^{p} ≠ 1, then μ_{i} must be zero. From Lemma 3.1 (5) we then obtain

Δ(c^{p}) = g_{1}^{p}g_{2}^{p} ⊗ c^{p} + c^{p} ⊗ 1 + α(1 − g_{1}^{p})g_{2}^{p} ⊗ (1 − g_{2}^{p}),

where α := μ_{1}μ_{2}(q − 1)^{p}χ_{1}(g_{2})^{\frac{p(p−1)}{2}} . Then α = μ_{1}μ_{2}(q − 1)^{p} since p is odd and χ_{1}^{p} = ε if μ_{1} ≠ 0. Hence Δ(c^{p}) ∈ A_{0} ⊗ A + A ⊗ A_{0}, and c^{p} ∈ A_{1}. Since Γ acts on c^{p} via the character χ_{1}^{p}χ_{2}^{p}, we see from (11) that c^{p} ∈ 𝕜[Γ] if χ_{1}^{p}χ_{2}^{p} = ε. Hence if χ_{1}^{p}χ_{2}^{p} = ε, then by Lemma 3.3

c^{p} = α(1 − g_{2}^{p}) + λ(1 − g_{1}^{p}g_{2}^{p})

for some scalar λ. Assume that χ_{1}^{p}χ_{2}^{p} is non-trivial. Then χ_{1}^{p} or χ_{2}^{p} is non-trivial, and a_{1}^{p} = 0 or a_{2}^{p} = 0 by the above argument. Hence μ_{1}μ_{2} = 0. Therefore α = 0, and c^{p} ∈ P_{g_{1}^{p}g_{2}^{p},1}(A)^{χ_{1}^{p}χ_{2}^{p}}. Thus again by Lemma 3.1 in [2], c^{p} is a scalar multiple of 1 − g_{1}^{p}g_{2}^{p}. □

4 Pointed Hopf algebras of dimension p^{4}

Let p be an odd prime and Γ = ⟨g⟩ the cyclic group of order p. We describe Nichols algebras of type A_{2} over Γ. Let q be a primitive p-th root of unity in 𝕜, b, d integers and

g_{1} = g, g_{2} = g^{b}, χ_{1}(g) = q, χ_{2}(g) = q^{d}.

Type A_{2} means b + d ≡ −1 ≡ −bd mod p. Therefore b^{2} + b + 1 ≡ 0 mod p. Hence p = 3 and b ≡ 1 mod p, or b ≢ 1 mod p is a cubic root of 1. In the last case, b exists if and only if p ≡ 1 mod 3. The corresponding Radford biproduct U^{+}#𝕜[Γ] is the Hopf algebra

U(q, b) := 𝕜⟨g, x_{1}, z, x_{2} | gx_{1}g^{−1} = qx_{1}, gx_{2}g^{−1} = q^{−1−b}x_{2},

x_{1}x_{2} − q^{−1−b}x_{2}x_{1} = z, zx_{1} = q^{b}x_{1}z, x_{2}z = q^{b}zx_{2},

g^{p} = 1, x_{1}^{p} = 0, z^{p} = 0, x_{2}^{p} = 0⟩

with g group-like, x_{1} (g, 1)-primitive and x_{2} (g^{b}, 1)-primitive.

Theorem 4.1 Let p be an odd prime. Let A be a pointed non-semisimple Hopf algebra of dimension p^{4} with coradical 𝕜[Γ]. Then A is isomorphic to one of the Hopf algebras in the list of Section 3 if Γ has order p^{3} or p^{2}, or to one of the Hopf algebras U(q, b) if Γ has order p, where p = 3 or p ≡ 1 mod 3.

Proof. This follows from Theorem 1.1, Section 2 and Theorem 3.7. □

In Section 2, the Hopf algebras are classified up to isomorphism. We know from [3, Lemma 6.4] that two Hopf algebras U(q, b) and U(q̃, b̃) are isomorphic if and only if q̃ = q, and b̃ ≡ b mod p or b̃ ≡ −1 − b mod p. Hence there are exactly 2(p − 1) non-isomorphic Hopf algebras of the form U(q, b) if p ≡ 1 mod 3, and 2 of dimension 81 if p = 3.

5 Appendix. Quasi-isomorphism types.

We now prove that there are only a finite number of quasi-isomorphism types of pointed Hopf algebras of order p^{4}.

Twisting the multiplication by a cocycle. Let A be a Hopf algebra and let σ : A ⊗ A → k be a normalized cocycle; that is, σ is a convolution invertible linear map such that σ(x, 1) = ε(x) = σ(1, x) and

σ(x_{(1)}, y_{(1)})σ(x_{(2)}y_{(2)}, z) = σ(y_{(1)}, z_{(1)})σ(x, y_{(2)}z_{(2)})

for all x, y, z ∈ A. Let ._{σ} : A ⊗ A → A be the linear map given by

x._{σ}y := σ(x_{(1)}, y_{(1)})x_{(2)}y_{(2)}σ^{−1}(x_{(3)}, y_{(3)})  (13)

Let A_{σ} be the vector space underlying A with the comultiplication of A and the multiplication ._{σ}; it is again a Hopf algebra.

Definition 5.1 We shall say that two Hopf algebras A and B are quasi-isomorphic if there exists an isomorphism ϕ : A_{σ} → B for some normalized cocycle σ.

Remarks 5.2 (i). Dually, it is possible to twist the comultiplication via F ∈ A ⊗ A; we obtain a new Hopf algebra A^{F} and a corresponding notion of coquasi-isomorphic. (ii). If C is a coalgebra, σ̃ : C ⊗ C → k is a normalized cocycle and ψ : A → C is a coalgebra map, then evidently σ : A ⊗ A → k, σ = σ̃(ψ ⊗ ψ) is a normalized cocycle. If A is pointed with coradical 𝕜[Γ] then there exists always coalgebra maps A → 𝕜[Γ]; hence, we get normalized cocycles on A from normalized group cocycles on Γ. (iii). It is known that two finite dimensional Hopf algebras are quasi-isomorphic if and only if their categories of comodules are equivalent as braided categories.

Lifting of quantum linear spaces. Let Γ be a finite abelian group. We fix a decomposition Γ = ⟨y_{1}⟩ ⊕ … ⊕ ⟨y_{ν}⟩ and we denote by M_{ℓ} the order of y_{ℓ}, 1 ≤ ℓ ≤ ν.

Let g_{1}, …, g_{θ} ∈ Γ, χ_{1}, …, χ_{θ} ∈ Γ̂ be a datum for quantum linear space; i. e., the following conditions hold

q_{i} := χ_{i}(g_{i}) ≠ 1, for all i .We denote by N_{i} the order of q_{i}.

χ_{j}(g_{i})χ_{i}(g_{j}) = 1, for all i ≠ j (14)

We abbreviate R := R(g_{1}, …, g_{θ}; χ_{1}, …, χ_{θ}) for the quantum linear space as in [2, Section 3]. The integer θ is called the rank of R.

A compatible datum D for Γ and R consists of

1) a scalar μ_{i} ∈ {0, 1} for each i, 1 ≤ i ≤ θ; it is arbitrary if g_{i}^{N_{i}} ≠ 1 and χ_{i}^{N_{i}} = 1, but 0 otherwise;

2) a scalar λ_{ij} ∈ k for each i, j, 1 ≤ i < j ≤ θ; it is is arbitrary if g_{i}g_{j} ≠ 1 and χ_{i}χ_{j} = 1, but 0 otherwise.

Remark 5.3 If λ_{ij} ≠ 0 and λ_{ih} ≠ 0, then χ_{i}χ_{j} = 1 and χ_{i}χ_{h} = 1; hence χ_{j} = χ_{h}. If in addition the order N_{i} of q_{i} := χ_{i}(g_{i}) is odd, then j = h [2].

Definition 5.4 Let Γ be a finite abelian group, R a quantum linear space and D a compatible datum. Keep the notation above. Let A(Γ, R, D) be the algebra presented by generators h_{ℓ}, 1 ≤ ℓ ≤ ν, and a_{i}, 1 ≤ i ≤ θ with defining relations

h_{ℓ}^{M_{ℓ}} = 1, 1 ≤ ℓ ≤ ν;  (15)
h_{ℓ}h_{t} = h_{t}h_{ℓ}, 1 ≤ t < ℓ ≤ ν;  (16)
a_{i}h_{ℓ} = χ_{i}^{−1}(y_{ℓ})h_{ℓ}a_{i}, 1 ≤ ℓ ≤ ν, 1 ≤ i ≤ θ;  (17)
a_{i}^{N_{i}} = μ_{i} (1 − g_{i}^{N_{i}}) , 1 ≤ i ≤ θ;  (18)
a_{j}a_{i} = χ_{i}(g_{j})a_{i}a_{j} + λ_{ij} (1 − g_{i}g_{j}) , 1 ≤ i < j ≤ θ.  (19)

There exists a unique Hopf algebra structure on A(Γ, R, D) such that

Δ(h_{ℓ}) = h_{ℓ} ⊗h_{ℓ},  Δ(a_{i}) = a_{i} ⊗1+g_{i} ⊗a_{i}, 1 ≤ ℓ ≤ ν, 1 ≤ i ≤ θ. (20)

The set of monomials h_{1}^{r_{1}} … h_{ν}^{r_{ν}}a_{1}^{s_{1}} … a_{θ}^{r_{θ}}, 0 ≤ r_{ℓ} < M_{ℓ}, 0 ≤ s_{i} < N_{i}, 1 ≤ ℓ ≤ ν, 1 ≤ i ≤ θ, is a basis of A(Γ, R, D). See [2].

Let σ̃ : Γ × Γ → 𝕜^{×} be a normalized group cocycle and let ψ : A(Γ, R, D) → 𝕜[Γ] be the coalgebra map given by

ψ(h_{1}^{r_{1}} … h_{ν}^{r_{ν}}a_{1}^{s_{1}} … a_{θ}^{r_{θ}}) = h_{1}^{r_{1}} … h_{ν}^{r_{ν}}δ_{s_{1},0} … δ_{r_{θ},0};

then σ : A(Γ, R, D) ⊗ A(Γ, R, D) → k, σ = σ̃(ψ ⊗ ψ) is a normalized cocycle. Here we extended linearly σ̃ to 𝕜[Γ] × 𝕜[Γ] → 𝕜. It is harmless to write σ instead of σ̃ in what follows.

Let Ψ_{i,n−1} = ∏_{1≤h≤n} σ(g_{i}, g_{i}^{h}). Let ω_{i}^{−1} be a fixed N_{i}-th root of ψ_{i,N_{i}}.

Lemma 5.5 If σ is symmetric, then A(Γ, R, D)_{σ} is isomorphic to A(Γ, R, D_{σ}), where D_{σ} = (μ_{i}, ω_{i}ω_{j}σ(g_{i}, g_{j})λ_{ij}).

Proof. Let ĥ_{ℓ}, â_{i} be the elements h_{ℓ}, a_{i} when regarded in A(Γ, R, D)_{σ}. We claim that

ĥ_{ℓ}._{σ}ĥ_{t} = h_{ℓ}h_{t};  (21)
â_{i}._{σ}ĥ_{t} = σ(g_{i}, h_{t})a_{i}h_{t}, ĥ_{t}._{σ}â_{i} = σ(h_{t}, g_{i})a_{i}h_{t};  (22)
â_{i}._{σ}â_{j} = σ(g_{i}, g_{j})a_{i}a_{j};  (23)
â_{i}^{n} = Ψ_{i,n}a_{i}^{n}.  (24)

Indeed, (21), (22) and (23) follow easily from the definition (13) of the twisted multiplication using that Δ^{2}(h) = h ⊗ h ⊗ h, Δ^{2}(a_{i}) = a_{i} ⊗ 1 ⊗ 1 + g_{i} ⊗ a_{i} ⊗ 1 + g_{i} ⊗ g_{i} ⊗ a_{i}. In turn, (24) follows from (13) by induction; the first step is (23) and the inductive step uses the expression

Δ^{2}(a_{i}^{n}) = ∑_{0≤h≤n} ∑_{0≤s≤h} \binom{n}{h}_{q_{i}} \binom{h}{s}_{q_{i}}a_{i}^{s}g_{i}^{n−s} ⊗ a_{i}^{h−s}g_{i}^{n−h} ⊗ a_{i}^{n−h}g_{i}^{n−h},

which follows applying twice the quantum binomial formula. Let now h̃_{t}, ã_{i} be the generators of A(Γ, R, D)s satisfying the relations ˜(15), ⋯, ˜(19) analogous to (15), …, (19) but with respect to the data D_{σ}. We claim that there exists a Hopf algebra isomorphism F : A(Γ, R, D)s → A(Γ, R, D)_{σ} such that

F(h̃_{t}) = ĥ_{t}, F(ã_{i}) = ω_{i}â_{i}, 1 ≤ t ≤ ν, 1 ≤ i ≤ θ.

We need only to verify that relations ˜(15), ⋯, ˜(19) hold. Relations ˜(15), ˜(16), follow immediately from (21); ˜(17) from (22) and ˜(19) from (23) using the symmetry of σ; ˜(18) from (24) by the choice of ω_{i}. □

Lemma 5.6 Let D = (μ_{i}, λ_{ij}) be a compatible datum of rank θ = 2. Let E be the compatible datum (μ_{i}, η_{ij}) where η_{ij} = 1 if λ_{ij} ≠ 0 and 0 otherwise. Then A(Γ, R, D) is quasi-isomorphic to A(Γ, R, E) .

Proof. By Lemma 5.5, it is enough to find a symmetric cocycle σ : Γ × Γ → 𝕜^{×} such that ω_{1}ω_{2}σ(g_{1}, g_{2})λ_{12} = 1 whenever λ_{12} ≠ 0. Assume that σ is the coboundary of a map f : Γ → 𝕜^{×} satisfying f(1) = 1. Then σ(g_{1}, g_{2}) = f(g_{1})f(g_{2})f(g_{1}g_{2})^{−1} and

Ψ_{1,n} = σ(g_{1}, g_{1})σ(g_{1}, g_{1}^{2}) … σ(g_{1}, g_{1}^{n−1})
= f(g_{1})f(g_{1})f(g_{1}^{2})^{−1}f(g_{1})f(g_{1}^{2})f(g_{1}^{3})^{−1} … f(g_{1})f(g_{1}^{n−1})f(g_{1}^{n})^{−1}
= f(g_{1})^{n}f(g_{1}^{n})^{−1}.

We can choose ω_{1} = f(g_{1})^{−1} ^{N_{1}} \sqrt{f(g_{1}^{N_{1}})} for some N_{1}-th root of f(g_{1}^{N_{1}}) and similarly for ω_{2}. So that we are reduced to find a function f such that f(1) = 1 and

λ_{12}^{−1} = ^{N_{1}} \sqrt{f(g_{1}^{N_{1}})} ^{N_{2}} \sqrt{f(g_{2}^{N_{2}})}f(g_{1}g_{2})^{−1}.

Since λ_{12} ≠ 0, g_{1}g_{2} ≠ 1 and such f exists. □

Proposition 5.7 There are only a finite number of quasi-isomorphism types of pointed Hopf algebras of order p^{4}.

Proof. This follows at once from Lemma 5.6 and the classification results in the previous Sections of the paper. □

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FAMAF. UNC. (5000), C. Universitaria, Córdoba, Argentina, e-mail andrus@mate.uncor.edu

Mathematisches Seminar der Universität München, Theresienstrasse 39, D-8033 München, Germany, e-mail hansschrz.mathematik.uni-muenchen.de