ON POINTED HOPF ALGEBRAS OF DIMENSION p^{5}

MATÍAS GRAÑA

Abstract. We describe all possible coradically graded pointed Hopf algebras of dimension p^{5} (p an odd prime number) over an algebraically closed field of characteristic 0.

1. Introduction

The lifting procedure described in [2] is a powerful tool for classifying pointed Hopf algebras. It has been applied successfully to the classification of pointed Hopf algebras of dimension p^{3} in [2] and dimension p^{4} in [4]. It has been used also in the classification of pointed Hopf algebras of dimension 32 in [10]. We describe here all pointed coradically graded Hopf algebras of dimension p^{5} (we assume p is odd since the case p = 2 is treated in [10]). Some of these algebras are known and can be found in the referred articles as well as in [3], [8]. Classification problems of pointed Hopf algebras have been also treated in [6], [9] and [7].

Our main references for Hopf algebras are [13] and [11]. For Nichols algebras we refer to [12] and [1].

The article is organized as follows: in section 2 we give the notation and definitions we use and the first results we need. In section 3 we describe all possible Nichols algebras of dimension p^{5−j} over groups of order p^{j}, j = 1, …, 4. In section 4 we prove necessary auxiliary results; some of them have interest on their own, e.g. 4.4.3. In section 5 we prove that any pointed Hopf algebra of dimension p^{5} over k is generated by group-like and skew-primitive elements. In other words, any coradically graded pointed Hopf algebra of dimension p^{5} can be recovered by bosonization (or biproduct) from one of the Nichols algebras appearing in 3.3.2. Furthermore, this proves also that any pointed Hopf algebra of dimension p^{5} can be recovered by lifting (in the sense of [2]) of one of these bosonizations. The classification of the pointed Hopf algebras of dimension p^{5} could be done in principle using the lifting procedure of [2]. This article contains the first steps in this direction. In section 6 we address the remaining steps and consider some illustrating examples.

This is part of the doctoral thesis of the author, who thanks N. Andruskiewitsch for his guidance.

2. Notation and preliminary results

The letter k shall stand for an algebraically closed field of characteristic 0. All Hopf algebras are k-algebras. For Γ a group and g ∈ Γ we denote by Γ_{g} the isotropy subgroup Γ_{g} = {h ∈ Γ | hg = gh}. Let q ∈ k. For n ≥ m ∈ ℕ, we use the standard notation

(n)_{q} = ∑_{i=0}^{n−1} q^{i}, (0)_{q} = 1; (n)_{q}^{!} = ∏_{i=1}^{n}(i)_{q}; \binom{n}{m}_{q} = \frac{(n)_{q}^{!}}{(m)_{q}^{!}(n − m)_{q}^{!}} .

Date: December, 1998.

This work was partially supported by CONICET, CONICOR and SeCyT-UNC.

1991 Mathematics Subject Classification: 16W30.

For A a Hopf algebra, we say that A is pointed iff the simple subcoalgebras of A are 1-dimensional (iff the irreducible representations of A^{∗} are 1-dimensional).

Let A = ⊕_{i≥0}A(i) be a graded Hopf algebra. We say that A is coradically graded if the graduation corresponds to the coradical filtration of A, i.e. if

A_{r} = ⊕_{i=0}^{r}A(i) ∀r ≥ 0, where A_{0} ⊆ A_{1} ⊆ … stands for the coradical filtration of A.

In particular, A being coradically graded and pointed implies that A(0) ≃ kΓ where Γ is the group of group-likes of A.

Let H be a Hopf algebra. We denote by _{H}^{H}YD the category of (left-left) Yetter–Drinfeld modules over H (see [11]) and by c its braiding. Let A be a coradically graded pointed Hopf algebra and A(0) = kΓ; then

R = A^{coA(0)} = {x ∈ A | (id ⊗π)Δ(x) = x ⊗ 1} = ⊕_{i}R(i), (2.2.1)

(where π : A → A(0) is the canonical projection) is a braided Hopf algebra in the category _{kΓ}^{kΓ}YD. The Hopf algebra A can be recovered by bosonization: A = R#kΓ. Furthermore, R is coradically graded and R(0) = k1. If moreover R is generated as an algebra by R(1) then we say that R is a Nichols algebra.

If R is a Nichols algebra then R is uniquely determined (up to isomorphism) by V = R(1), which coincides with the space of primitive elements P(R). We denote R = 𝔅(V).

We refer to the survey [1] for details on these constructions (Nichols algebras are called TOBAs in that article). Proposition 2.2.2. Let f be any field, and let H be a Hopf algebra over f. Let V be an object in _{H}^{H}YD. Suppose V has a basis {x_{1}, …, x_{θ}} such that c(x_{i} ⊗ x_{j}) = b_{ij}x_{j} ⊗ x_{i} for certain b_{ij} ∈ f (since c is an automorphism, b_{ij} ∈ f^{×}). We take for each i = 1, …, θ

N_{i} = \begin{cases}order of b_{ii} & if b_{ii} ≠ 1 and is a root of unity, \\ ∞ & if b_{ii} is not a root of unity, \\ ∞ & if b_{ii} = 1 and char f = 0, \\ char f & if b_{ii} = 1 and char f > 0. \end{cases}

Then dim 𝔅(V) ≥ ∏_{i} N_{i}. Moreover, if 𝔅(V) is finite dimensional, then the equality holds if and only if b_{ij}b_{ji} = 1 ∀i ≠ j.

Proof. See [2, §3]. □

We recall (see for instance [1]) that if Γ is a finite group, the category _{kΓ}^{kΓ}YD is semisimple. The simple objects are the modules M(g, ρ) defined as follows: let g ∈ Γ, ρ an irreducible representation of the isotropy group Γ_{g}. Let W be the space affording ρ, and take

M(g, ρ) = Ind_{Γ_{g}}^{Γ} W = kΓ ⊗_{kΓ_{g}} W,

with the usual module structure and the comodule structure given by

δ(h ⊗ w) = hgh^{−1} ⊗ (h ⊗ w) ∈ kΓ ⊗ M(g, ρ).

Remark 2.2.3. Since g is central in Γ_{g}, if ρ is an irreducible representation of Γ_{g} then the Schur lemma says that ρ(g) = q id for some q ∈ k^{×}.

Definition 2.2.4. We say that V ∈ _{H}^{H}YD has a matrix (b_{ij}) if it has a basis {x_{1}, …, x_{θ}} such that c(x_{i} ⊗ x_{j}) = b_{ij}x_{j} ⊗ x_{i}. This happens for instance if Γ is abelian. This happens also under a weaker condition: let V = ⊕_{i}M(g_{i}, ρ_{i}) and suppose that the subgroup Γ′ of Γ generated by the conjugacy classes of all the g_{i} is abelian. Then V comes from the abelian case in the sense of [1, Def. 3.1.8] and consequently has a matrix. In this case V can be considered as a Yetter–Drinfeld module over Γ′ and 𝔅(V)#kΓ can be reconstructed as an extension of Γ/Γ′ by 𝔅(V)#kΓ′. A sufficient condition for Γ′ to be abelian in the case V = M(g, ρ) is that the isotropy subgroup Γ_{g} be invariant in Γ (see [1, Lemma 3.1.9]). Since we are working in characteristic 0, if V has a matrix (b_{ij}) and 𝔅(V) is finite dimensional then by 2.2.2 b_{ii} ≠ 1 ∀i.

If V has a matrix (b_{ij}) with b_{ij}b_{ji} = 1 ∀i ≠ j then it can be shown that 𝔅(V) has a PBW basis of the form

{x_{1}^{n_{1}} ⋯ x_{θ}^{n_{θ}} | 0 ≤ n_{i} < N_{i}}

(N_{i} defined as in 2.2.2) and the relations are given by

x_{i}^{N_{i}} = 0, x_{i}x_{j} = b_{ij}x_{j}x_{i} ∀i > j.

Thus 𝔅(V) is a quantum linear space as an algebra. We notice that the lines kx_{i} (i = 1, …, θ) are not Yetter–Drinfeld submodules in general. In order to agree with the terminology of [2], we shall denote such an algebra by QLS only when the lines kx_{i} are Yetter–Drinfeld modules ∀i. Thus, a QLS in _{kΓ}^{kΓ}YD is given by a module V = ⊕_{i=1}^{θ}M(g_{i}, χ_{i}) where

g_{1}, …, g_{θ} ∈ Γ are central elements,
χ_{1}, …, χ_{θ} ∈ Γ̂ are characters such that
χ_{i}(g_{j})χ_{j}(g_{i}) = 1 ∀i ≠ j.

(2.2.5)

For V ∈ _{kΓ}^{kΓ}YD, we shall say that 𝔅(V) is a QLS over Γ′ ⊂ Γ if V is a Yetter–Drinfeld module in _{kΓ′}^{kΓ′}YD and conditions 2.2.5 hold for Γ′. A 1-dimensional QLS will be called also Quantum Line (or QL), and a 2-dimensional QLS will be called also Quantum Plane (or QP).

According to [3], if V has a matrix (b_{ij}) we say that V is of Cartan type if there exists a (generalized) Cartan matrix (a_{ij}) such that

b_{ij}b_{ji} = b_{ii}^{a_{ij}} ∀i, j = 1, …, θ.

We transfer to V the terminology over the Cartan matrix (a_{ij}). Lemma 2.2.6. Let g be central in Γ and ρ an irreducible representation of Γ. Let V = M(g, ρ). By 2.2.3, g acts by a scalar on V, say q. Let N be the order of q. Then dim 𝔅(V) ≥ N^{deg ρ}.

Proof. Since g is central, V comes from the abelian case, and consequently c has a matrix (b_{ij}). It is straightforward to see that b_{ij} = q ∀i, j. Then 2.2.2 applies and the result follows. □

Lemma 2.2.7. Let g ∈ Γ and ρ an irreducible representation of Γ_{g}. Let V = M(g, ρ). Suppose dim V < p, where p is the smallest prime dividing |Γ|. Then g is central, deg ρ = 1 and thus 𝔅(V) is a QL over Γ with dim 𝔅(V) = N, where N is the order of ρ(g).

Proof. We have dim V = [Γ : Γ_{g}] deg(ρ) < p. Since [Γ : Γ_{g}] and deg(ρ) both divide |Γ|, necessarily [Γ : Γ_{g}] = 1, whence g is central, and deg(ρ) = 1. The result follows by 2.2.2. □

Remark 2.2.8. Since dim 𝔅(V) ≥ 1 + dim V, the hypothesis of the preceding lemma is verified if dim 𝔅(V) ≤ p. Remark 2.2.9. By 2.2.7, we have that if V = ⊕_{i}M(g_{i}, ρ_{i}) is such that dim V < p, then g_{i} is central ∀i and ρ_{i} is a character ∀i, and furthermore dim 𝔅(V) ≥ ∏_{i} N_{i}, where N_{i} is the order of ρ_{i}(g_{i}).

3. Main results

Lemma 3.3.1. Let A be a coradically graded pointed Hopf algebra of dimension p^{5}. Let Γ = G(A) be the group of group-likes of A. Let R = A^{cokΓ} ∈ _{kΓ}^{kΓ}YD be the coinvariants (thus A = R#kΓ) and let V = R(1) be the primitive elements of R. Assume that V generates R as an algebra (i.e. R = 𝔅(V)). Then the following possibilities arise:

(1) If |Γ| = p^{5} then V = 0 and A = kΓ.

(2) If |Γ| = p^{4} then V is 1-dimensional and R is a QLS.

(3) If |Γ| = p^{3} then V may be 1 or 2-dimensional and R is a QLS over some subgroup Γ′ of Γ.

(4) If Γ = C_{p} × C_{p} then V is 2-dimensional (and then R is a twisting of a Nichols algebra of type A_{2}) or V is 3-dimensional (and R is a QLS).

(5) If Γ = C_{p^{2}} then V is 2-dimensional (and then R is a QLS or a twisting of a Nichols algebra of type A_{2}) or V is 3-dimensional (and R is a QLS).

(6) If Γ = C_{p} then either V is 2-dimensional, R is of type B_{2} and necessarily p ≡ 1 mod 4, or V is 3-dimensional, R is of type A_{2} × A_{1} and p = 3.

Proof. We prove that 𝔅(V) is of the claimed form:

(1) Immediate.

(2) By 2.2.9 we have dim V = 1, V = (x) = M(g, χ), g ∈ Z(Γ). Furthermore, by the same remark, χ(g) = q is such that q^{p} = 1 (and q ≠ 1 since A is finite dimensional), from where by 2.2.4 the structure of R is given by

x^{p} = 0, ε(x) = 0,
Δ(x^{r}) = ∑_{i=0}^{r} \binom{r}{i}_{q} x^{i} ⊗ x^{r−i},
δ(x) = g ⊗ x, h ⇀ x = χ(h)x.

Let a = x#1 ∈ A. Then A is generated by Γ and a, with the structure given by

a^{p} = 0, ε(a) = 0, hah^{−1} = χ(h)a ∀h ∈ Γ,
Δ(a^{r}) = ∑_{i=0}^{r} \binom{r}{i}_{q} (a^{i}g^{r−i}) ⊗ a^{r−i}.

(3) The bound dim V ≤ 2 is a consequence of 4.4.3 below. If V is 1-dimensional then V = M(g, χ) and A is given exactly as in the case |Γ| = p^{4} with the only exception that q has order p^{2} and the relation on a is a^{p^{2}} = 0. If V is 2-dimensional, [1, Prop 3.1.11] applies and V comes from the abelian case, i.e. V has a basis {x_{1}, x_{2}} with (x_{i}) = M(g_{i}, χ_{i}) (g_{i} and χ_{i} are respectively central elements and characters of a certain subgroup Γ′ of Γ). Let N_{i} be the order of χ_{i}(g_{i}). Then by 2.2.2 we have p^{2} ≥ N_{1}N_{2}, from where N_{1} = N_{2} = p (χ_{i}(g_{i}) ≠ 1 since A is finite dimensional), and again by 2.2.2 we have that 𝔅(V) is a QLS over Γ′. Let b_{12} = χ_{2}(g_{1}), and let for each h ∈ Γ the matrix ρ(h)_{ij} be defined by h ⇀ x_{j} = ∑_{i=1}^{2} ρ(h)_{ij}x_{i}. Then A is generated by Γ, a_{1}, a_{2} with structure and relations given by

a_{i}^{p} = 0, ε(a_{i}) = 0, ha_{j}h^{−1} = ∑_{i=1}^{2} ρ(h)_{ij}a_{i} ∀h ∈ Γ,
Δ(a_{i}) = g_{i} ⊗ a_{i} + 1 ⊗ a_{i},
a_{1}a_{2} = b_{12}a_{2}a_{1}.

(4) The bounds 2 ≤ dim V ≤ 3 are immediate consequences of 2.2.2. If dim V = 2 then by 4.4.10 it is a twisting of an algebra of type A_{2}. If dim V = 3 then by 2.2.2 it is a QLS.

(5) As in the case Γ = C_{p} × C_{p}, the bounds 2 ≤ dim V ≤ 3 are consequences of 2.2.2. Suppose dim V = 2, V have basis {x_{1}, x_{2}} and c be given in this basis by the matrix (b_{ij}). If b_{11} (resp. b_{22}) has order p^{2} then by, 2.2.2, b_{22} (resp. b_{11}) has order p and 𝔅(V) is a QLS. If both b_{11} and b_{22} have order p then by 4.4.9 and 4.4.10 𝔅(V) is a twisting of an algebra of type A_{2}. If dim V = 3 then by 2.2.2 it is a QLS.

(6) This is proved in [3, Thm. 1.3].

□

In section 5 we prove that if Γ is a group of order p^{j} and R = ⊕_{i}R(i) ∈ _{kΓ}^{kΓ}YD is a coradically graded braided Hopf algebra of dimension p^{5−j} with R(1) ≃ k, then R is generated by R(1). With this and the previous lemma we can prove

Theorem 3.3.2. Let A = ⊕_{i}A(i) be a coradically graded pointed Hopf algebra of dimension p^{5}. Let Γ = G(A) be the group of group-likes of A. Let R = ⊕_{i}R(i) = A^{coA(0)} ∈ _{kΓ}^{kΓ}YD and let V = R(1). Then R is generated by V (i.e. A = 𝔅(V)#kΓ) and 𝔅(V) is one in the list below. By B(·) we denote the group of order p^{4} in [5, Page 145].

Γ  dim 𝔅(V) Type Conditions
(C_{p})^{4}  1  QLS
(C_{p})^{2} × C_{p^{2}}  1  QLS
C_{p^{2}} × C_{p^{2}}  1  QLS
C_{p} × C_{p^{3}}  1  QLS
C_{p^{4}}  1  QLS
B(vi)  1  QLS
B(vii)  1  QLS
B(viii)  1  QLS
B(ix)  1  QLS
B(x)  1  QLS
B(xiv)  1  QLS
(C_{p})^{3}  2  QLS
C_{p^{2}} × C_{p}  1  QLS
2  QLS
C_{p^{3}}  1  QLS
2  QLS
(C_{p})^{2}  2  A_{2}
3  QLS
C_{p^{2}}  2  A_{2}  p = 3 or p ≡ 1 mod 3
C_{p}  2  B_{2}  p ≡ 1 mod 4
3  A_{2} × A_{1} p = 3

Proof. For the groups of order p^{4}, the only condition for the existence of a QLS is the existence of a central element g ∈ Γ and a character χ ∈ Γ̂ such that χ(g) has order p. This is possible if and only if g ∉ [Γ, Γ] ([Γ, Γ] the commutator subgroup of Γ). It follows by inspection of each case that the groups in the table are those Γ such that Z(Γ) ⊄ [Γ, Γ].

We go now to |Γ| = p^{3}. It is clear that QLS of rank one exist for Γ = C_{p^{2}} × C_{p} and Γ = C_{p^{3}}, and not for Γ = (C_{p})^{3}. The two non abelian groups of order p^{3} verify that their centers are included in their commutators subgroups, whence the 1-dimensional Yetter–Drinfeld modules give infinite dimensional Nichols algebras. We prove now that for the three abelian groups there exist QLS of rank 2: let q_{1}, q_{2}, q_{3} denote respectively (fixed) roots of unity of orders p, p^{2}, p^{3}. We denote the generators of (C_{p})^{3} by \{g_{1}, g_{2}, g_{3}\} and the generators of { \atop (C_{p})^{3}} by \{ĝ_{1}, ĝ_{2}, ĝ_{3}\}, where ĝ_{i}(g_{j}) = q_{1}^{δ_{ij}}. We denote the generators of C_{p^{2}} × C_{p} by \{g_{1}, g_{2}\} and the generators of { \atop C_{p^{2}} × C_{p}} by \{ĝ_{1}, ĝ_{2}\}, where ĝ_{i}(g_{j}) = q_{3−i}^{δ_{ij}}. We denote the generator of C_{p^{3}} by {g} and the generator of Ĉ_{p^{3}} by {ĝ}, where ĝ(g) = q_{3}. It is straightforward that the following Yetter–Drinfeld modules give QLS of dimension p^{2}:

Γ = (C_{p})^{3}, V = M(g_{1}, ĝ_{1}) ⊕ M(g_{2}, ĝ_{2}),
Γ = C_{p^{2}} × C_{p}, V = M(g_{1}^{p}, ĝ_{1}) ⊕ M(g_{1}^{−p}, ĝ_{1}),
Γ = C_{p^{3}}, V = M(g^{p^{2}}, ĝ) ⊕ M(g^{−p^{2}}, ĝ).

For the two non abelian groups we should have V a Yetter–Drinfeld module of dimension 2. There are three possibilities:

(1) V = M(h_{1}, χ_{1}) ⊕ M(h_{2}, χ_{2}) where h_{i} are central and χ_{i} are characters, but by the same reason as in the rank one case, this would give infinite dimensional Nichols algebras.

(2) V = M(g, χ) where χ is a character and [Γ : Γ_{g}] = 2, but this is impossible since p ≠ 2 (this case arises when p = 2, see [10]).

(3) V = M(g, ρ) where g is central and ρ is an irreducible representation of Γ with deg ρ = 2. Since p ≠ 2, the Frobenius theorem applies to say that this is impossible (this case arises when p = 2, see [10]).

Let now Γ = (C_{p})^{2}. It is immediate that there are no QLS of rank 1 neither 2, since there should be a character with a p^{2}-th root of unity in the image. The existence of a QLS of rank 3 is a consequence of [2, Lemma 4.1]. An explicit construction is as follows: let Γ have generators {g_{1}, g_{2}} and Γ̂ have generators {ĝ_{1}, ĝ_{2}} where ĝ_{i}(g_{j}) = q_{1}^{δ_{ij}} (as before q_{1} is a fixed p-th root of unity). Take then V = M(g_{1}, ĝ_{1}) ⊕ M(g_{1}, ĝ_{1}^{−1}) ⊕ M(g_{2}, ĝ_{2}). It is straightforward to see that V generates a QLS. For a construction of a Nichols algebra of type A_{2}, let r = \frac{1}{2} ∈ ℤ/p (the construction for p = 2 is slightly different, see [10]). Take then V = M(g_{1}, ĝ_{1}ĝ_{2}^{−r}) ⊕ M(g_{2}, ĝ_{1}^{−r}ĝ_{2}). It is clear then that V has the matrix

(b_{ij}) = ( \begin{matrix}q_{1} & q_{1}^{−r} \\ q_{1}^{−r} & q_{1} \end{matrix} ) , whence b_{ij}b_{ji} = b_{ii}^{a_{ij}} with (a_{ij}) = ( \begin{matrix}2 & −1 \\ −1 & 2 \end{matrix} ) .

Let Γ = C_{p^{2}}. The non-existence of a p^{3}-dimensional QLS is a consequence of 4.4.1. Let g, ĝ be respectively generators of Γ and Γ̂, and let q = ĝ(g). Suppose V ∈ _{kΓ}^{kΓ}YD generates an algebra of type A_{2}. Let V = M(g^{e_{1}}, ĝ^{f_{1}}) ⊕ M(g^{e_{2}}, ĝ^{f_{2}}). Since V has a matrix b_{ij} = q^{e_{i}f_{j}} and b_{11}, b_{22} must have order p, then p divides e_{1} and e_{2}, or p divides f_{1} and f_{2}. Then the same arguments as in [3, Thm 1.3] give the condition p = 3 or p ≡ 1 mod 3. Furthermore, let b be such that b^{2} + b + 1 ≡ 0 mod p (the condition on p is equivalent to the existence of such a b) and take e_{1} = p, f_{1} = 1, e_{2} = −p(b + 1), f_{2} = b. It is straightforward to see that this gives a Cartan matrix of type A_{2}.

For Γ = C_{p} it is proved in [3, Thm 1.3] that there exist an algebra of type B_{2} if and only if p ≡ 1 mod 4, and of type A_{2} × A_{1} if and only if p = 3. □

4. Subsidiary results

The following lemma may be considered as an addendum to [2, Lemma 4.2]. Lemma 4.4.1. Let Γ = C_{p^{n}} and V ∈ _{kΓ}^{kΓ}YD generate a finite dimensional QLS. Then V may be 1-dimensional (and hence dim 𝔅(V) = p^{v} with 1 ≤ v ≤ n) or it may be 2-dimensional (and hence dim 𝔅(V) = p^{2v} with 1 ≤ v ≤ n).

Proof. The bound dim V ≤ 2 is the content of [2, Lemma 4.2]. Let Γ have a generator g and Γ̂ have a generator ĝ. Let q = ĝ(g), which is a primitive p^{n}-th root of unity. If V is 1-dimensional, the result is an easy consequence of 2.2.2.

Suppose V = M(g^{e′_{1}}, ĝ^{f′_{1}}) ⊕ M(g^{e′_{2}}, ĝ^{f′_{2}}), and let e′_{i} = p^{r}e_{i} such that e_{1}, e_{2} are not both divisible by p, f′_{i} = p^{s}f_{i} such that f_{1}, f_{2} are not both divisible by p. Then V has a matrix given by b_{ij} = q^{e′_{i}f′_{j}} = q^{e_{i}f_{j}p^{r+s}}. Since 𝔅(V) is finite dimensional, r + s < n (for if not b_{11} = b_{22} = 1). Let u = n − r − s. Suppose p |̸e_{1} (if p |̸e_{2} it is analogous). Suppose first that p |̸f_{2} then b_{12} has order p^{u}. Since V generates a QLS, b_{21}b_{12}^{−1} also has order p^{u} and thus p |̸e_{2}, p |̸f_{1}. This implies the result with v = u.

Suppose then p|f_{2}. Then p |̸f_{1}, and let f_{2} = p^{t}b, e_{2} = p^{y}a with p |̸b, p |̸a. We prove that t = y: we have t < u since if not b_{22} = 1. Now, b_{12} has order p^{u−t}, whence b_{21} has order p^{u−t}. Since p |̸f_{1} then p^{t}|e_{2}, whence y ≥ t. By similar considerations, y ≤ t. This implies the result with v = u − t. □

We shall make use of the following important tool for Nichols algebras. Definition 4.4.2. Let V ∈ _{H}^{H}YD and c = c_{V,V}. For i + j = n, we denote by Δ_{i,j} : 𝔅^{n}(V) → 𝔅^{i}(V) ⊗ 𝔅^{j}(V) the (i, j)-component of the comultiplication of 𝔅(V). It is proved in [14] (or see [1, Def. 3.2.10]) that Δ_{i,j} is injective ∀i, j. Let {x_{1}, …, x_{θ}} be a basis of V and let {x_{1}^{∗}, …, x_{θ}^{∗}} be its dual basis. We denote by ∂_{x_{i}} the differential operator on 𝔅(V) given by

∂_{x_{i}}(z) = (id ⊗x_{i}^{∗})Δ_{n−1,1}(z), if z ∈ 𝔅^{n}(V), n > 0, and ∂_{x_{i}}(1) = 0.

By the injectivity of Δ_{i,j} it is immediate that for z ∈ 𝔅^{n}(V) (n > 0) we have z = 0 if and only if ∂_{x_{i}}(z) = 0 for all i = 1, …, θ. Suppose now that V ∈ _{kΓ}^{kΓ}YD and ∂_{x_{i}} verifies that there exists g ∈ Γ such that ∂_{x_{i}}(v) = 0 if δ(v) = h ⊗ v and h ≠ g (this happens for instance if δ(x_{j}) = g_{j} ⊗ x_{j}, j = 1, …, θ and g = g_{i}). Then it is easy to see that ∂_{x_{i}} verifies the Leibniz rule

∂_{x_{i}}(z_{1}z_{2}) = ∂_{x_{i}}(z_{1})(g ⇀ z_{2}) + z_{1}∂_{x_{i}}(z_{2}).

The following theorem is proved in [2, Thm 0.2] when Γ is an abelian group. Theorem 4.4.3. Let Γ be a finite group. Let V ∈ _{kΓ}^{kΓ}YD be such that dim 𝔅(V) = p^{2}, where p is the smallest prime number dividing |Γ|. Then dim V ≤ 2 and 𝔅(V) is a QLS over some subgroup Γ′ ⊂ Γ. Furthermore, if p > 2 then V = M(g, χ) with g central, χ is a character such that χ(g) has order p^{2} and hence 𝔅(V) is a QL over Γ, or V = M(g_{1}, χ_{1}) ⊕ M(g_{2}, χ_{2}) where g_{i} is central, χ_{i} is a character (i = 1, 2) such that χ_{i}(g_{i}) has order p and hence 𝔅(V) is a QP over Γ.

Proof. Let V = ⊕_{i=1}^{θ}M(g_{i}, ρ_{i}). It can be shown that dim 𝔅(V) ≥ dim 𝔅(M(g_{i}, ρ_{i})) ∀i. Let I = [Γ : Γ_{g_{1}}] and d = deg(ρ_{1}). We have dim M(g_{1}, ρ_{1}) = dI. The numbers d, I can be equal to 1 or ≥ p. Since dim 𝔅(V) ≥ 1 + dim V we have dim V < p^{2}.

Suppose first that d ≥ p. This implies that I = 1, from where g_{1} is central in Γ. By 2.2.6 we have p^{2} = dim 𝔅(V) ≥ dim 𝔅(M(g_{1}, ρ_{1})) ≥ N^{d} with N the order of q, where q id = ρ_{1}(g_{1}). Since 𝔅(V) is finite dimensional, we have q ≠ 1 and hence N ≥ p. If p > 2 we have a contradiction. If p = 2 we must have θ = 1, d = 2, N = 2. The condition d = 2 implies that V comes from the abelian case, as explained in 2.2.4. The condition on N tells that q = −1. Furthermore, 2.2.2 says

that 𝔅(V) is a QLS, and it is shown in [1] that the matrix of c is ( \begin{matrix}−1 & −1 \\ −1 & −1 \end{matrix} ).

Suppose then that I ≥ p. This implies that d = 1, from where ρ_{1} is a character of Γ_{g_{1}}. Let q = ρ_{1}(g_{1}) and let N be the order of q. Let x be a generator of the space affording ρ_{1}, and let {h_{1} = 1, h_{2}, …, h_{I}} be a set of representatives of the cosets of Γ/Γ_{g_{1}}. Then M(g_{1}, ρ_{1}) has as basis the elements {h_{1} ⇀ x, …, h_{I} ⇀ x} and we have

c(h_{i} ⇀ x ⊗ h_{i} ⇀ x) = h_{i}g_{1}h_{i}^{−1}h_{i} ⇀ x ⊗ h_{i} ⇀ x = h_{i}g_{1} ⇀ x ⊗ h_{i} ⇀ x = q(h_{i} ⇀ x ⊗ h_{i} ⇀ x). (4.4.4)

It is straightforward to see using derivations that the elements

{1, (h_{i} ⇀ x)^{r} | 1 < r < N, i = 1, …, I}

are linearly independent, from where

p^{2} = dim 𝔅(V) ≥ 1 + I(N − 1). (4.4.5)

Thus, N ≤ p. On the other hand, if q ≠ 1 for if not it would be easy to see using derivations that the elements {x^{r} | r ≥ 0} would be linearly independent and 𝔅(V) would be infinite dimensional (note that we had not proved at the moment that kx is a sub-YD-module neither that M(g_{1}, ρ_{1}) comes from the abelian case, and hence 2.2.2 can not be used.) We have then proved that N = p. Suppose for a moment that I > p. It is clear that if p > 2 then I ≥ p + 2, but then (4.4.5) tells that this is a contradiction. If p = 2 then I = 3, but then [1, Prop. 3.2.2] tells that dim 𝔅(M(g_{1}, ρ_{1})) ≥ 5, also a contradiction. Hence, we have that I = p and then Γ_{g}, having index the smallest prime dividing |Γ|, is invariant in Γ. As said in 2.2.4, this implies that 𝔅(M(g_{1}, ρ_{1})) comes from the abelian case, but then 2.2.2 applies and 4.4.4 tells that dim 𝔅(M(g_{1}, ρ_{1})) ≥ p^{p}. This is a contradiction if p > 2. If p = 2 then θ = 1, q = −1 and it is proved in [1] that the matrix of c is

( \begin{matrix}−1 & 1 \\ 1 & −1 \end{matrix} ) or ( \begin{matrix}−1 & −1 \\ −1 & −1 \end{matrix} ) .

Suppose finally that I = d = 1. Then g_{1} is central and ρ_{1} is a character. Let q = ρ_{1}(g_{1}) and let N be its order. Then dim 𝔅(V) ≥ dim 𝔅(M(g_{1}, ρ_{1})) = N implies that N ≤ p^{2}. If N = p^{2} then θ = 1 and the result follows at once. If N < p^{2} then N is a prime number ≥ p. Since dim 𝔅(M(g_{1}, ρ_{1})) = N, we have θ > 1. Since dim 𝔅(M(g_{2}, ρ_{2})) ≤ p^{2} − 1 (because if x is a generator of M(g_{1}, ρ_{1}) then x does not belong to 𝔅(M(g_{2}, ρ_{2}))) by the same arguments as above applied to M(g_{2}, ρ_{2}) we have necessarily that g_{2} is central and ρ_{2} is a character. Let N_{2} be the order of ρ_{2}(g_{2}). Thus N_{2} < p^{2}, and since g_{1}, g_{2} are both central then M(g_{1}, ρ_{1}) ⊕ M(g_{2}, ρ_{2}) comes from the abelian case, whence by 2.2.2, 𝔅(M(g_{1}, ρ_{1}) ⊕ M(g_{2}, ρ_{2})) has dimension ≥ NN_{2}. This implies N = N_{2} = p, θ = 2 and 𝔅(V) is a QLS over Γ. □

Remark 4.4.6. It is proved in [8] in a different way that if dim 𝔅(V) = p, p the smallest prime number dividing Γ, then dim V = 1 and 𝔅(V) is a QLS. It is proved also, with the same ideas as here, in [3, Prop 7.5]. Remark 4.4.7. We note that the proof of 4.4.3 above says that there are no V in _{kΓ}^{kΓ}YD such that dim 𝔅(V) = π^{2} if π is a prime number smaller than every prime dividing |Γ|.

The previous theorem implies the following result:

Corollary 4.4.8. Let A be a pointed Hopf algebra of dimension m whose coradical has dimension m/p^{2}, p the smallest prime number dividing m. Then p^{3} divides m and dim A_{1} = (r + 1)m/p^{2}, where r = 1 or 2.

Proof. Consider the coradical filtration of A and let H = ⊕_{i}H(i) be the associated graded algebra. Then H is pointed and H(0) ≃ kΓ, Γ the group of group-likes of A, which has order m/p^{2}. Let R = H^{coH(0)} ∈ _{kΓ}^{kΓ}YD and let R′ ⊂ R be the algebra generated by R(1) (R′ = R iff R is a Nichols algebra). Thus dim R = p^{2} and, thanks to the Nichols–Zoeller theorem, dim R′ = p^{j} with 0 ≤ j ≤ 2. The case j = 0 would imply dim R(1) = 0, which is impossible. The case j = 1 is also impossible, for in that case 4.4.6 says that dim R(1) = dim R′(1) = 1, and [2, Thm 3.2] says that R is a Nichols algebra. Then R′ = R. Remark 4.4.7 says that p divides |Γ| (whence p^{3} divides m) and 4.4.3 says that r = dim R(1) may be 1 or 2, whence dim H(1) = r|Γ| and dim A_{1} = dim H(0) + dim H(1) = (r + 1)|Γ| = (r + 1)m/p^{2}. □

Lemma 4.4.9. Let Γ be a p-group and V a 2-dimensional module in _{kΓ}^{kΓ}YD such that dim 𝔅(V) = p^{3}. Recall that under this assumption c has a matrix (b_{ij}) in some basis {x, y}. Let q be a primitive p^{2}-th root of unity, and suppose b_{ij} = q^{c_{ij}}. If p divides c_{11} and c_{22} then p divides c_{12} + c_{21}.

Proof. We have x^{p} = y^{p} = 0. Let z = Ad_{x}(y) = xy − b_{12}yx. Let σ = 1 − b_{12}b_{21}. We compute

∂_{x}(z) = b_{12}y − b_{12}y = 0, ∂_{y}(z) = x − b_{12}b_{21}x = σx ⇒ ∂_{x}∂_{y}(z) = σ.

Furthermore, we have

∂_{x}(x^{r}y^{s}z^{t}) = (r)_{b_{11}}b_{11}^{t}b_{12}^{s+t}x^{r−1}y^{s}z^{t},
∂_{y}(y^{s}z^{t}) = (s)_{b_{22}}b_{21}^{t}b_{22}^{t}y^{s−1}z^{t} + ∑_{i=0}^{t−1} σb_{21}^{i}b_{22}^{i}y^{s}z^{t−1−i}xz^{i},
∂_{x}∂_{y}(y^{s}z^{t}) = σ(t)_{(b_{11}b_{12}b_{21}b_{22})}y^{s}z^{t−1}.

Thus, if p |̸c_{12} + c_{21}, the order of (b_{11}b_{12}b_{21}b_{22}) is p^{2}, from where the set {z^{t} | 0 ≤ t < p^{2}} is linearly independent. This implies inductively that the set {y^{s}z^{t} | 0 ≤ s < p, 0 ≤ t < p^{2}} is linearly independent, and then that the set {x^{r}y^{s}z^{t} | 0 ≤ r, s < p, 0 ≤ t < p^{2}} is linearly independent, whence dim 𝔅(V) ≥ p^{4}. □

The following result is a consequence of [3, Cor. 1.2]. We give a direct proof here. Lemma 4.4.10. Let V = (x, y) be a 2-dimensional module in _{kΓ}^{kΓ}YD such that dim 𝔅(V) = p^{3}. Let V have a matrix (b_{ij}) and suppose that b_{ij}^{p} = 1 for all i, j. Then b_{ij} is a Cartan matrix of type A_{2}.

Proof. Let q = b_{11} and c_{ij} be given by b_{ij} = q^{c_{ij}}. We may suppose as said above that b_{12} = b_{21}. Let b_{12} = q^{a}, b_{22} = q^{c}. Take z = Ad_{x}(y) = xy − q^{a}yx, and let σ = 1 − q^{2a}; thus σ ≠ 0 since otherwise 𝔅(V) would be a QLS and dim 𝔅(V) would be p^{2}. We compute as before

∂_{x}(z) = 0, ∂_{y}(z) = σx ⇒ ∂_{x}∂_{y}(z) = σ,

from where

∂_{x}(x^{r}y^{s}z^{t}) = (r)_{q}q^{t+a(s+t)}x^{r−1}y^{s}z^{t},
∂_{y}(y^{s}z^{t}) = (s)_{q^{c}}q^{as+ct}y^{s−1}z^{t} + ∑_{i=0}^{t−1} σq^{(a+c)i}y^{s}z^{t−1−i}xz^{i},
∂_{x}∂_{y}(y^{s}z^{t}) = σ(t)_{q^{1+2a+c}}y^{s}z^{t−1}.

As before, the set {x^{r}y^{s}z^{t} | 0 ≤ r, s, t < p} is linearly independent (as a remark, note that we must have 1 + 2a + c ≢ 0 (mod p) since if not 𝔅(V) would be infinite dimensional). Let now w = Ad_{x}(z) = xz − q^{1+a}zx, and we have

∂_{x}(w) = 0, ∂_{y}(w) = σx^{2} − σq^{1+a}q^{a}x^{2} = σ(1 − q^{2a+1})x^{2}.

The (x, y)-bidegree of w is (2, 1), from where the set {x^{2}y, xz, w} must be linearly dependent in order for 𝔅(V) to be p^{3}-dimensional, but this implies 2a + 1 = 0, which means that b_{12}b_{21} = b_{11}^{−1}.

With the same reasoning, we must have b_{12}b_{21} = b_{22}^{−1}, and thus b_{ij}b_{ji} = b_{ii}^{c_{ij}} with

c_{ij} ≡ ( \begin{matrix}2 & −1 \\ −1 & 2 \end{matrix} ) mod p,

and b_{ij} is a Cartan matrix of type A_{2}. As a remark, note that b_{11} = b_{22}. □

5. The classification is complete

We have to prove that those of 3.3.2 are all the coradically graded pointed Hopf algebras of dimension p^{5}. This amounts to be the same as to prove that a coradically graded pointed Hopf algebra is generated by its homogeneous component of degree 1, which in turn is equivalent to prove that if A = ⊕_{i}A(i) is a coradically graded pointed Hopf algebra and R = A^{coA_{0}} is its algebra of coinvariants then R is a Nichols algebra. As in [3, §8], let S = R^{∗} be its dual. Then S is a graded braided Hopf algebra in _{kΓ}^{kΓ}YD, S = ⊕_{i}S(i) and is generated by S(0) ⊕ S(1). Furthermore, we have a surjection S ↠ S′, S′ = 𝔅(S(1)). We have to prove that S is coradically graded, i.e. that P(S) = S(1). This is the same as to say that S′ = S. Now, [2, Thm 3.2] plus 4.4.6 solve the problem for the cases in which Γ has order p^{4} or p^{3}, and [3, Thm 8.2] and [3, Lemma 8.5] solve the problem for the case in which Γ has order p or when Γ = C_{p} × C_{p}. The following theorem solves the pending case.

Theorem 5.5.1. Let Γ be a finite group, p the smallest prime number dividing |Γ|. Let S = ⊕_{i}S(i) in _{kΓ}^{kΓ}YD be a graded braided Hopf algebra of dimension p^{3} such that S(0) = k and S is generated by S(1). Suppose that S(1) comes from the abelian case, i.e. there exist an abelian subgroup Γ′ ⊂ Γ such that S(1) is a YD-module over Γ′. Then S is a Nichols algebra.

Proof. We prove the statement for p > 2, the case p = 2 being treated in [10]. Let S′ = 𝔅(S(1)), and consider the canonical projection S ↠ S′. We must prove that this is an isomorphism. If dim S(1) = 3 then by 2.2.2 we have dim S′ ≥ p^{3}, but this implies that S′ = S and S is a Nichols algebra. If dim S(1) = 1 then [2, Thm 3.2] shows that S is a Nichols algebra. We are led to consider hence the case dim S(1) = 2. We have dim S′ ≤ p^{3}, and we suppose dim S′ < p^{3}. Then by 2.2.2 we have dim S′ ≥ p^{2}, whence dim S′ = p^{2}. Now theorem 4.4.3 says that S′ is a QLS over Γ′, S′(1) has a basis {x, y} and the braiding c has a matrix (b_{ij}) in this basis, where b_{ii} are primitive p-th roots of unity and b_{12}b_{21} = 1. Furthermore, the linear spans kx and ky are sub-YD-modules over Γ′. Let z = x_{1}x_{2} − b_{12}x_{2}x_{1} ∈ S. If we prove that z = 0 in S then dim S = p^{2}, but this would be a contradiction and we would be done.

Suppose z ≠ 0. Now, it is immediate to see that z is primitive in S. Consider the coradical filtration of S and let T = ⊕_{i}T (i) be the associated graded algebra. We have x, y, z ∈ S_{1}, and consider x̄, ȳ, z̄ ∈ T (1). It is easy to see that these elements are linearly independent. We compute the matrix of c for {x̄, ȳ, z̄}. It is given by

(b′_{ij}) = ( \begin{matrix}b_{11} & b_{12} & b_{11}b_{12} \\ b_{21} & b_{22} & b_{21}b_{22} \\ b_{11}b_{21} & b_{12}b_{22} & b_{11}b_{12}b_{21}b_{22} \end{matrix} ) = ( \begin{matrix}b_{11} & b_{12} & b_{11}b_{12} \\ b_{21} & b_{22} & b_{21}b_{22} \\ b_{11}b_{21} & b_{12}b_{22} & b_{11}b_{22} \end{matrix} ) .

Consider now the canonical projection T ↠ T′ = 𝔅(T (1)). Since kx, ky and kz are sub-YD-modules of S over Γ′, then kx̄, kȳ and kz̄ are sub-YD-modules of T over Γ′. Thus, if W = (x̄, ȳ, z̄) we have dim 𝔅(W) ≤ dim T′ ≤ dim T = p^{3}. Now 2.2.2 applies (notice that b′_{ii} has order p ∀i, since b_{11}b_{22} = 1 would imply dim 𝔅(W) = ∞), and 𝔅(W) is a QLS, but this implies that 1 = b_{11}^{2}b_{12}b_{21} = b_{11}^{2}, which is impossible since p ≠ 2. Hence, z = 0 in S and the theorem is proved. □

6. Final remarks

In order to give a complete classification of the pointed Hopf algebras of dimension p^{5}, the following steps should be accomplished:

(1) For each Nichols algebra R in 3.3.2, give all the modules M in _{kΓ}^{kΓ}YD such that 𝔅(M) ≃ R.

(2) Classify the isomorphism classes of the bosonizations of the Nichols algebras in the previous step (note that there exist non isomorphic Nichols algebras which give isomorphic algebras after bosonization).

(3) For each coradically graded p^{5}-dimensional Hopf algebra in the previous step, classify all the liftings.

These steps are highly non trivial. For instance, let Γ = C_{p^{n}} where n > 0 and p ≠ 2 and let 0 ≤ s ≤ n. The number of QLS of rank 1 over Γ with dimension p^{s} is given by

∑_{\begin{matrix}i+j−n=s \\ i,j≤n \end{matrix}} ϕ(i, j),

while the number of isomorphisms classes of these QLS after bosonization is given by

∑_{\begin{matrix}i+j−n=s \\ i,j≤n \end{matrix}} \frac{ϕ(i, j)}{I(i, j)},

where

ϕ(i) = p^{i−1}(p − 1),
ϕ(i, j) = ϕ(i)ϕ(j),
I(k_{1}, …, k_{r}) = ϕ(max{k_{1}, …, k_{r}}).

Furthermore, the number of QLS of rank 2 over Γ with dimension p^{s} (s even, by 4.4.1) is given by

\frac{1}{2} ∑_{i_{1},j_{1},i_{2},j_{2}} \frac{ϕ(i_{1}, j_{1})ϕ(i_{2}, j_{2})}{L_{n}(i_{1}, j_{2})},

while the number of isomorphism classes of these QLS after bosonization is given by

\frac{1}{2} ∑_{i_{1},j_{1},i_{2},j_{2}} \frac{ϕ(i_{1}, j_{1})ϕ(i_{2}, j_{2})}{L_{n}(i_{1}, j_{2})I(i_{1}, i_{2}, j_{1}, j_{2})},

where

L_{n}(i, j) = ϕ(i + j − n),

and the sum is over the tuples such that

i_{1} + j_{1} − n = s_{1} ≥ 1, i_{2} + j_{2} − n = s_{2} ≥ 1, s_{1} + s_{2} = s, i_{1}, j_{1}, i_{2}, j_{2} ≤ n, i_{1} + j_{2} = i_{2} + j_{1}.

As a result of this, the number of coradically graded non isomorphic Hopf algebras of dimension p^{5} with coradical C_{p^{4}}, C_{p^{3}} is respectively 2(p^{2} − 1) and p(p − 1)[2 + \frac{p(p−1)(p+2)}{2}].

See also the discussion in [3, §9] for the first step, [3, §6] for the second. In particular, a necessary and sufficient condition for two YD-modules to give isomorphic algebras after bosonization is given in [3, Prop. 6.3].

As an example on the last step, let A be a pointed Hopf algebra of dimension p^{5} with coradical Γ of order p^{4}. Let H be the associated graded Hopf algebra and R its invariants as in 2.2.1. Then H = R#kΓ, where R = 𝔅(V). We have then V = M(h, χ) where h is central and χ is a character such that χ(h) has order p. Let x be a generator of V. Then, by [10, Prop. 2.0.17], x can be lift to a ∈ A such that Δ(a) = h ⊗ a + a ⊗ 1 and gag^{−1} = χ(g)a ∀g ∈ Γ. Since the elements {x^{i}#g | 0 ≤ i < p, g ∈ Γ} are a basis of H, then the elements {a^{i}g | 0 ≤ i < p, g ∈ Γ} are a basis of A. The lifting A is then determined by the element a^{p}, the case a^{p} = 0 being the bosonization A = R#kΓ. It is easy to see that a^{p} is a skew-primitive, Δ(a^{p}) = h^{p} ⊗ a^{p} + a^{p} ⊗ 1. Looking at the space of skew-primitives, this implies that

a^{p} = λ(h^{p} − 1), λ ∈ k.

Taking a suitable scalar multiple of a we may suppose that λ ∈ {0, 1}, from where there are no more that 2 liftings. In some of the cases we must have a^{p} = 0. These cases are given by the diamond lemma:

ga^{p} = gλ(h^{p} − 1) = λ(h^{p} − 1)g
ga^{p} = χ(g)aga^{p−1} = … = χ^{p}(g)a^{p}g = χ^{p}(g)λ(h^{p} − 1)g,

from where λ(χ^{p} − 1) = 0 for A to be p^{5}-dimensional. This tells that over the group B(vi) any pointed Hopf algebra of dimension p^{5} is coradically graded.

On the other hand, it is clear that if h^{p} = 1 then a^{p} = 0. This tells that over the groups B(viii), B(ix), B(x) and B(xiv) any pointed Hopf algebra of dimension p^{5} is coradically graded.

As a corollary: a pointed Hopf algebra of dimension p^{5} and non abelian coradical is coradically graded, unless its coradical is isomorphic to kB(vii). We classify all the liftings in this case: B(vii) can be presented with generators X, Y, Z and relations

X^{p^{2}} = Y^{p} = Z^{p} = 1, [Z, Y] = X^{p}, [X, Y] = [X, Z] = 1.

Hence Z(B(vii)) = (X) while [B(vii), B(vii)] = (X^{p}). Let q be a (fixed) p-th root of unity. The Yetter–Drinfeld modules generating Nichols algebras of dimension p are then

V = M(X^{i}, χ) such that p |̸i, χ(X) = q^{a} (p |̸a), χ(Y) = q^{b}, χ(Z) = q^{c}.

However, it can be shown that most of them give isomorphic algebras after bosonization. We are led to consider two modules:

V_{i} = M(X, χ_{i}) (i = 1, 2), χ_{1}(X) = q, χ_{1}(Y) = χ_{1}(Z) = 1, χ_{2}(X) = χ_{2}(Y) = q, χ_{2}(Z) = 1.

We have hence 2 pointed Hopf algebras of dimension p^{5} with non abelian coradical and which are not coradically graded:

A^{(1)} generated by a, X, Y, Z and the relations of B(vii) plus
Xa = qaX, Y a = aY, Za = aZ, a^{p} = X^{p} − 1, Δ(a) = X ⊗ a + a ⊗ 1,
A^{(2)} generated by a, X, Y, Z and the relations of B(vii) plus
Xa = qaX, Y a = qaY, Za = aZ, a^{p} = X^{p} − 1, Δ(a) = X ⊗ a + a ⊗ 1.

A description of the liftings of QLS (resp. of the algebras of type A_{2} over groups of exponent p) is made in [2] (resp. [4]).

References

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[14] P. Schauenburg, A Characterization of the Borel-like Subalgebras of Quantum Enveloping Algebras, Comm. in Alg. 24 (1996), 2811–2823.

Matías Graña, FCEyN – Buenos Aires

 E-mail address: matiasg@dm.uba.ar