On finite quantum groups at −1

Nicolás Andruskiewitsch (andrus@mate.uncor.edu)^{∗}

CIEM - CONICET Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba (5000) Ciudad Universitaria, Córdoba, Argentina

www.mate.uncor.edu/andrus
Sorin Dăscălescu (sdascal@al.math.unibuc.ro)

Facultatea de Matematica, University of Bucharest Str. Academiei 14, RO-70109 Bucharest 1, Romania Kuwait University, Faculty of Science, Det. Mathematics & Computer Sc. PO Box 5969, Safat 13060, Kuwait

Abstract. This is a contribution to the classification of finite dimensional pointed Hopf algebras. We are concerned with the case when the group of group-like elements is abelian of exponent 2. We attach to such a pointed Hopf algebra a generalized, simply-laced, Cartan matrix; we conjecture that the Hopf algebra is finite dimensional if and only if the Cartan matrix is of finite type. We prove the conjecture for the types A_{n} and A_{n}^{(1)}. We obtain the classification of all possible Hopf algebras with Cartan matrix A_{n}. We use the lifting method developed by Hans-Jürgen Schneider and the first-named author.

Keywords: Pointed Hopf algebras, finite quantum groups

Mathematics Subject Classification: Primary: 17B37; Secondary: 16W30

1. Introduction

In recent years, there has been intense activity in the classification of finite dimensional Hopf algebras; see [1] for an account of the main known results, and further references. Substantial advances were attained in the subclass of pointed Hopf algebras over an algebraically closed field 𝕜 with characteristic 0, see [1, Section 5], and the detailed survey [5]. In the present paper, we are specifically concerned with pointed Hopf algebras whose group of group-likes is abelian of prime exponent p, with p = 2. The case when p is odd was treated in [4]; we should also mention that the case when the group has order 2 was done in [12, Th. 4.2.1]; the result was rediscovered and proved by a different method in [6]. In this paper, we approach the classification of pointed Hopf algebras following the lifting method [2, 3]. See section 2 for a brief exposition; a more complete reference is [5].

∗ This work was partially supported by CONICET, Agencia Córdoba Ciencia – CONICOR, FOMEC and Secyt (UNC).

\begin{matrix}c \\ ◯ \end{matrix} 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Let A be a pointed Hopf algebra and assume that G(A) is a finite abelian group of exponent 2. We may consider several invariants of A; in particular, the infinitesimal braiding of A is a pair (V, c), where V is a vector space and c : V ⊗ V → V ⊗ V is an invertible solution of the braid equation. If A is finitely generated as an algebra, then V has finite dimension and there exists a basis x_{1}, …, x_{θ} of V and scalars b_{ij} ∈ {±1} such that

c(x_{i} ⊗ x_{j}) = b_{ij}x_{j} ⊗ x_{i},  1 ≤ i, j ≤ θ.  (1.1)

We can assume that b_{ii} ≠ 1, since we are interested in finite dimensional pointed Hopf algebras. We attach a simply-laced generalized Cartan matrix (a_{ij}) of rank θ to A by the rules a_{ii} = 2, for all i, and

b_{ij}b_{ji} = (−1)^{a_{ij}},  1 ≤ i ≠ j ≤ θ.  (1.2)

Out of the infinitesimal braiding (V, c), one constructs a braided Hopf algebra 𝔅(V), the so-called Nichols algebra of V. A fundamental question is when 𝔅(V) is finite dimensional. We conjecture that, in our case, 𝔅(V) is finite dimensional if and only if the generalized Cartan matrix (a_{ij}) is of finite type. An analogous result, when the scalars b_{ij} are roots of 1 of odd order, was proved in [3, Th. 1.1]. Support to this conjecture is given by the following result.

Theorem 1.1. (a). If (V, c) is of type A_{ℓ}, then 𝔅(V) has finite dimension 2^{\frac{ℓ(ℓ+1)}{2}} .

(b). If (V, c) is of type A_{ℓ}^{(1)}, then 𝔅(V) has infinite dimension. (c). If the Dynkin diagram of (V, c) contains a cycle, then 𝔅(V) has

infinite dimension.

The proof of [3, Th. 1.1] was derived via twisting from Lusztig’s results on Frobenius-Lusztig kernels [8, 9]. This technique is not applicable in our case; indeed, the quantum Serre relations have no meaning in the present situation. Our proof of Theorem 1.1 (a) is direct and provides at the same time a presentation of 𝔅(V) by generators and relations. At a key point, we use an inductive argument to show that we have the correct dimension; we believe, this idea will be useful in many other instances. After writing the first version of this paper, the argument was applied in the proof of [5, Th. 6.13]. Also, the proof of Theorem 1.1 (b) relies on it. It is likely that it can be used to prove the conjecture for Cartan matrices of affine type, and hence for indefinite type [7, Chapter 5]. Part (c) of Theorem 1.1 follows immediately from part (b).

The technique used for A_{ℓ} could be also of profit for D_{ℓ}. However, for the exceptional types one needs other tools.

It was conjectured in [3, Conjecture 1.4] that any finite dimensional pointed Hopf algebra (over the algebraically closed field 𝕜 of characteristic 0) is generated by group-like and skew-primitive elements. This is one of the main steps in the Lifting method; a positive partial answer, including the case of prime odd exponent, was given in [4, Th. 7.6].

With similar ideas, we can prove:

Theorem 1.2. Let H be a finite dimensional pointed Hopf algebra. Assume that the infinitesimal braiding (V, c) of H is as in (1.1), and that b_{ij} ∈ {±1}, b_{ii} = −1, for all i, j. Furthermore, assume that the simply-laced Cartan matrix (a_{ij}) is a union of connected components of type A_{ℓ}, for possibly different ℓ’s. Then H is generated by group-like and skew-primitive elements.

Putting together these results, and with more labour, we can compute all pointed Hopf algebras with prescribed infinitesimal braiding of certain type.

Theorem 1.3. Let H be a finite dimensional pointed Hopf algebra. Assume that Γ := G(H) is an abelian group of exponent 2. Furthermore, assume that the simply-laced Cartan matrix (a_{ij}) corresponding to the infinitesimal braiding (V, c) of H, as in (1.2), is a union of connected components of type A_{ℓ}, for possibly different ℓ’s. Then H is isomorphic, as Hopf algebra, to 𝔅(V)#𝕜Γ.

The paper is organized as follows. In Section 2 we briefly recall some known facts and overview the lifting method. In Section 3 we prove part (a) of Theorem 1.1; in Section 4 we prove Theorem 1.2. Theorem 1.3 depends on the previous main results; we conclude its proof in Section 5. Finally, we prove part (b) of Theorem 1.1 in Section 6.

Conventions

We work over an algebraically closed field 𝕜 of characteristic 0. We denote by τ : V ⊗W → W ⊗V the usual transposition, that is τ(v⊗w) = w ⊗ v.

To avoid confusion we use the following variant of Sweedler’s notation for the comultiplication of a braided Hopf algebra R: Δ_{R}(r) = r^{(1)} ⊗ r^{(2)}.

Let H be a Hopf algebra with bijective antipode. The group of grouplikes of H is denoted by G(H). A Yetter-Drinfeld module V over H is simultaneously a left H-module and a left H-comodule, both structures related by the compatibility condition

δ(h.v) = h_{(1)}v_{(−1)}Sh_{(3)} ⊗ h_{(2)}.v_{(0)},  v ∈ V, h ∈ H.  (1.3)

We denote by _{H}^{H}YD the braided category of Yetter-Drinfeld modules over H; the morphisms in this category preserve both the action and the coaction by H. The category _{H}^{H}YD is a braided monoidal category. Hence, we can consider braided Hopf algebras in _{H}^{H}YD. If H = 𝕜Γ, where Γ is a finite abelian group, then an H-comodule V is just a Γgraded vector space: V = ⊕_{g∈Γ}V_{g}, where V_{g} = {v ∈ V : δ(v) = g ⊗ v}. Therefore, if Γ is abelian, a Yetter-Drinfeld module over H = 𝕜Γ is nothing but a Γ-graded, Γ-module. That is, V = ⊕_{χ∈_{Γ}}V^{χ}, where V^{χ} =

{v ∈ V : g.v = χ(g)v}. Here Γ̂ denotes the character group of Γ. Then

V = ⊕_{g∈Γ,χ∈_{Γ}}V_{g}^{χ},  (1.4)

where V_{g}^{χ} = V^{χ}∩V_{g}. Conversely, any vector space with a decomposition (1.4) is a Yetter-Drinfeld module over Γ. The braiding is given by

c(v ⊗ w) = χ(g)w ⊗ v, v ∈ V_{g}, w ∈ V^{χ}.

We shall say Yetter-Drinfeld module over Γ, for short, and denote the corresponding category by _{Γ}^{Γ}YD (since the field is fixed).

2. The lifting method

In this Section, we recall the main steps of the Lifting method, in the setting of our interest. See [5] for more details.

2.1. Nichols algebras

We fix a group Γ = ℤ/(2)^{s}. A finite dimensional Yetter-Drinfeld module V over Γ is described (up to permutation) by sequences g(1), …, g(θ) ∈ Γ, χ(1), …, χ(θ) ∈ Γ̂. Indeed, there exists a basis x_{1}, …, x_{θ} of V such that the action and coaction of Γ are given by

h.x_{j} = χ(j)(h)x_{j},  δ(x_{j}) = g(j) ⊗ x_{j},  j = 1, …, θ. (2.1)

The braiding c_{V,V} of the category _{Γ}^{Γ}YD is given in the basis x_{i} ⊗ x_{j} by the matrix (b_{ij})_{1≤i,j≤θ} = (⟨χ(j), g(i)⟩)_{1≤i,j≤θ}; so that c(x_{i} ⊗ x_{j}) = b_{ij}x_{j} ⊗ x_{i}. By assumption, b_{ij} = ±1.

We shall use systematically the following result. See e. g. [2] for a proof.

Lemma 2.1. Let R be a finite dimensional braided Hopf algebra in _{Γ}^{Γ}YD and let x ∈ P (R)_{g}^{χ}. If χ(g) = 1 then x = 0.

As we are interested in the finite dimensional case, we assume that b_{ii} = −1 (i. e. b_{ii} ≠ 1) for all i, 1 ≤ i ≤ θ.

Definition 2.2. Let V be a Yetter-Drinfeld module over H. A braided graded Hopf algebra R = ⊕_{n≥0}R(n) in _{H}^{H}YD is called a Nichols algebra of V if 𝕜 ≃ R(0) and V ≃ R(1) in _{H}^{H}YD, and

P (R) = R(1),  (2.2)
R is generated as an algebra by R(1).  (2.3)

It is known that a Nichols algebra of V exists and is unique (up to isomorphisms); it will be denoted by 𝔅(V) = ⊕_{n≥0}𝔅^{n}(V).

We shall identify in what follows V with 𝔅^{1}(V). We shall systematically use (2.2) to obtain relations between the elements of 𝔅(V). Indeed, an expression providing a primitive element, homogeneous of degree greater than one, is necessarily 0.

It is clear that the bosonization assignment R ↦ R#𝕜Γ, where R is a braided Hopf algebra in _{Γ}^{Γ}YD, is functorial. Also, V ↦ 𝔅(V) is functorial. Hence, V ↦ 𝔅(V)#𝕜Γ is functorial.

2.2. The lifting method

Let A be a pointed Hopf algebra. We assume in the rest of this paper that Γ := G(A) is a finite abelian group of exponent 2, so that Γ ≃ (ℤ/2)^{s} for some positive integer s.

Let A_{0} = 𝕜Γ ⊆ A_{1} ⊆ … A_{n} ⊆ A_{n+1} … be the coradical filtration of A; it is an exhaustive coalgebra filtration, and since A is pointed, it is also an algebra filtration. Therefore, gr A = ⊕_{n≥0} gr A(n), the graded coalgebra associated to the coradical filtration of A, is indeed a graded Hopf algebra. Clearly, gr A(0) ≃ 𝕜Γ; and the unique homogeneous projection π : gr A → 𝕜Γ is a retraction of the inclusion. We can then reconstruct gr A by ”bosonization” (or biproduct), cf. [13, 10]; that is,

gr A ≃ R#𝕜Γ,

where R = A^{coπ} is the algebra of coinvariants of π and # stands for ”semidirect product algebra and semidirect coproduct coalgebra”. It is known that R is a braided Hopf algebra in the category _{Γ}^{Γ}YD of Yetter-Drinfeld modules over 𝕜Γ. In the present situation, R inherits the grading from gr A and is then a graded braided Hopf algebra:

R = ⊕_{n≥0} R(n), R(n) := R ∩ gr A(n).

Moreover, R has some special properties:

R(0) = 𝕜1, R(1) = P (R).

Summarizing, we have several invariants of the pointed Hopf algebra A:

− The diagram of A is the graded braided Hopf algebra R.

− The infinitesimal braiding of A is the braided vector space (V, c), where V := R(1) = P (R) and c : V ⊗ V → V ⊗ V is the braiding in _{H}^{H}YD

We know that the subalgebra R′ of R generated by R(1) is isomorphic to the Nichols algebra of V: R′ ≃ 𝔅(V). The dimension of V is finite if A is finitely generated as an algebra.

The lifting method seeks to reconstruct A, or at least to determine its dimension, from the listed invariants. In this case, the braided vector space V is of diagonal type, i. e. the braiding is given by (1.1); and since V ∈ _{Γ}^{Γ}YD, the b_{ij}’s are necessarily ±1. We can, therefore, attach to V a simply-laced (generalized) Dynkin diagram Δ as explained above, cf. (1.2).

Here we sketch the main steps of the Lifting method, in the present setting.

First step We conjecture that 𝔅(V) is finite dimensional if and only if Δ is a finite Dynkin diagram. In Section 3 we give some partial answers to this conjecture, as stated in Th. 1.1.

Second step We then show that R′ = R whenever A is finite dimensional, i. e. that A is generated by group-like and skew-primitive elements, for those V as in the previous step.

Third step For each Γ = ℤ/(2)^{s} one has to determine all possible finite Dynkin diagrams which actually appear from a Yetter-Drinfeld module over Γ. This is a problem of a computational nature.

Fourth step (lifting) Given Γ = ℤ/(2)^{s} and a finite Dynkin diagram appearing from a Yetter-Drinfeld module V over Γ, one has to determine all possible Hopf algebras A such that gr A ≃ 𝔅(V)#𝕜Γ. Our contribution to this step is summarized in Th. 1.3 above.

A Dynkin diagram can be realized in more than one way from a Yetter-Drinfeld module. We fix a standard choice by the following Lemma.

Lemma 2.3. Let V be a Yetter-Drinfeld module over Γ and preserve all the conventions above. Then there exists a finite abelian group Υ of exponent 2 and a Yetter-Drinfeld module Ṽ over Υ with a basis x̃_{1} …, x̃_{θ} such that the action and coaction of Υ are given by (2.1) with respect to some g̃(1), …, g̃(θ) ∈ Υ, χ̃(1), …, χ̃(θ) ∈ Υ̂ and

− dim 𝔅(V) = dim 𝔅(Ṽ),

− the matrix (b̃_{ij})_{1≤i,j≤θ} = (⟨χ̃(j), g̃(i)⟩)_{1≤i,j≤θ} determining the braiding satisfies b̃_{ij} = 1 whenever j < i,

− Ṽ has the same Dynkin diagram as V.

Proof. This is a variation of the proof of [3, Lemma 4.1]. We take Υ :=< y(1) > ⊕ ⋯ ⊕ < y(θ) >, where the order of y(j) is 2. Let γ(i) ∈ Υ̂ be the unique character such that ⟨γ(i), y(j)⟩ = (−1)^{δ_{ij}}, for 1 ≤ i, j ≤ θ, and let h(i) be defined by ⟨γ(j), h(i)⟩ = b_{ij}, 1 ≤ i, j ≤ θ. Then Υ̂ =< γ(1) > ⊕ ⋯ ⊕ < γ(θ) >. On the other hand, write b_{ij} = (−1)^{α_{ij}}, where α_{ij} = 0 or 1.

Given a collection c_{ij} ∈ {0, 1}, 1 ≤ i < j ≤ θ, we can consider the 2-cocycle ω : Υ̂ × Υ̂ → 𝕜^{×},

ω(τ, μ) = (−1)^{∑_{1≤i<j≤θ} c_{ij}τ_{j}μ_{i}},

where τ = ∑_{j} τ_{j}γ(j) and similarly for μ. Then F ∈ 𝕜Υ ⊗ 𝕜Υ given by F = ∑_{τ,ζ∈_{Γ}} ω(τ, ζ)δ_{τ} ⊗ δ_{ζ} is a twisting data for 𝔅(V)#𝕜Υ. By [3, Lemma 2.3], (𝔅(V)#𝕜Υ)_{F} is isomorphic (as a vector space) to 𝔅(Ṽ)#𝕜Υ for some Yetter-Drinfeld module Ṽ. Moreover, the braiding of Ṽ is determined by b̃_{ij} = (−1)^{α_{ji}^{F}} where α_{ij}^{F} := α_{ij} − c_{ji}, if i ≥ j, and α_{ji}^{F} := α_{ji} + c_{ji} if j < i; cf. [3, (2.10)]. It is enough to choose α_{ij} = c_{ji} to conclude the Lemma, cf. [3, Remark 2.4].

An important technical rôle in what follows is played by the braided adjoint action. Recall that the usual adjoint action of a Hopf algebra H on itself is ad (x)(y) = x_{(1)}yS(x_{(2)}). Now, if R is a braided Hopf algebra in _{H}^{H}YD and x, y ∈ R then the braided adjoint action is given by

ad_{c}(x)(y) = μ(μ ⊗ S_{R})(id ⊗c)(Δ_{R} ⊗ id )(x ⊗ y),

a formula that makes sense in any braided category. We observe that ad_{c}(x)(y)#1 = ad (x#1)(y#1) where ad is the usual adjoint action in R#H. Also, ad_{c}(x)(y) = xy − μc(x ⊗ y) whenever x ∈ P (R); we denote in this case [x, y]_{c} := ad_{c}(x)(y). So that in 𝔅(V), we have [x_{i}, x_{j}]_{c} = x_{i}x_{j} − b_{ij}x_{j}x_{i}.

3. Nichols algebras of types A_{ℓ}

We consider in this section a Yetter-Drinfeld module V_{ℓ} over our fixed group Γ = (ℤ/2)^{s}. Let x_{1}, …, x_{ℓ} be a basis of V_{ℓ} with x_{i} ∈ (V_{ℓ})_{g(i)}^{χ(i)}, for

some fixed g(1), …, g(ℓ) ∈ Γ, χ(1), …, χ(ℓ) ∈ Γ̂. Our fundamental assumption in this section is the following. The braiding is given by (1.1) with the matrix:

(b_{ij})_{1≤i,j≤ℓ} = (\begin{matrix}−1 & −1 & 1 & & … & 1 \\ 1 & −1 & −1 & 1 \\ & 1 & −1 & −1 & ⋮ & ⋮ \\ & & & ... & ... \\ ⋮ & ⋮ & & & ... & −1 \\ 1 & & … & & & −1 \end{matrix});

that is, V_{ℓ} is of type A_{ℓ}.

We shall consider a braided Hopf algebra R provided with an inclusion of braided vector spaces V ↪ P (R). We identify the elements x_{1}, …, x_{ℓ} with their images in R; in case of confusion, ad-hoc notation will be introduced. Distinguished examples of such R are the tensor algebra T (V) and the Nichols algebra 𝔅(V). Convenient hypotheses on R will be stated accordingly.

We introduce the family (e_{ij})_{1≤i<j≤ℓ+1} of elements of R as follows:

e_{i,i+1} := x_{i};  (3.1)
e_{i,j} := [e_{i,j−1}, e_{j−1,j}]_{c}, 1 ≤ i < j ≤ ℓ + 1, j − i ≥ 2.  (3.2)

We shall need the braiding between elements of this family. Since, clearly,

e_{i,j} ∈ (V_{ℓ})_{g(i)...g(j−1)}^{χ(i)...χ(j−1)},

we have

c(e_{i,j} ⊗ e_{p,q}) = B_{p,q}^{i,j}e_{p,q} ⊗ e_{i,j}, 1 ≤ i < j ≤ ℓ + 1, 1 ≤ p < q ≤ ℓ + 1, (3.3)

where

B_{p,q}^{i,j} = ∏_{i≤h≤j−1, p≤s≤q−1} χ(s)(g(h)) = ∏_{i≤h≤j−1, p≤s≤q−1} b_{h,s} = (−1)^{α_{p,q}^{i,j}};

here

α_{p,q}^{i,j} = card {s ∈ ℕ : i ≤ s ≤ j − 1, p ≤ s ≤ q − 1}

+ card {s ∈ ℕ : i ≤ s ≤ j − 1, p − 1 ≤ s ≤ q − 2}.

In particular,

c(e_{i,j} ⊗ e_{j,q}) = −e_{j,q} ⊗ e_{i,j}, i < j < q;  (3.4)
c(e_{i,j} ⊗ e_{p,i}) = e_{p,i} ⊗ e_{i,j}, p < i < j;  (3.5)
c(e_{i,p} ⊗ e_{j,q}) = e_{j,q} ⊗ e_{i,p}, p < j;  (3.6)
c(e_{j,q} ⊗ e_{i,p}) = e_{i,p} ⊗ e_{j,q}, p < j;  (3.7)
c(e_{i,i+1} ⊗ e_{i,i+2}) = e_{i,i+2} ⊗ e_{i,i+1};  (3.8)
c(e_{i,j} ⊗ e_{i,p}) = −e_{i,p} ⊗ e_{i,j}, i < p < j;  (3.9)
c(e_{i,i+2} ⊗ e_{i+1,i+2}) = e_{i+1,i+2} ⊗ e_{i,i+2};  (3.10)
c(e_{i,j+1} ⊗ e_{j−1,j}) = e_{j−1,j} ⊗ e_{i,j+1}, i < j − 1;  (3.11)
c(e_{j−1,j} ⊗ e_{i,j+1}) = e_{i,j+1} ⊗ e_{j−1,j}, i < j − 1;  (3.12)
c(e_{p,j} ⊗ e_{i,j}) = −e_{i,j} ⊗ e_{p,j}, i < p < j;  (3.13)
c(e_{p,j} ⊗ e_{i,s}) = −e_{i,s} ⊗ e_{p,j}, i < p < s < j.  (3.14)

Thus,

e_{i,j} = e_{i,j−1}e_{j−1,j} + e_{j−1,j}e_{i,j−1}.  (3.15)

We now establish relations that hold in the braided Hopf algebra R, either if R is finite dimensional or else if R ≃ 𝔅(V). We shall need both situations later.

Lemma 3.1. Assume that R is finite dimensional or R ≃ 𝔅(V). (a). If 1 ≤ i ≤ ℓ, then

x_{i}^{2} = e_{i,i+1}^{2} = 0.  (3.16)

(b). If 1 ≤ i < p ≤ ℓ, and p − i ≥ 2, then [e_{i,i+1}, e_{p,p+1}]_{c} = 0; that is,

e_{i,i+1}e_{p,p+1} = e_{p,p+1}e_{i,i+1}.  (3.17)

Proof. It is easy to see that e_{i,i+1}^{2} and [e_{i,i+1}, e_{p,p+1}]_{c} are primitive elements of R. If R is finite dimensional, the claim follows from Lemma 2.1. If R ≃ 𝔅(V), the claim follows since e_{i,i+1}^{2} and [e_{i,i+1}, e_{p,p+1}]_{c} are homogeneous of degree 2.

Lemma 3.2. Assume that (3.17) holds in R.

(a). If 1 ≤ i < j < p < r ≤ n + 1, then [e_{i,j}, e_{p,r}]_{c} = 0; that is,

e_{i,j}e_{p,r} = e_{p,r}e_{i,j}.  (3.18)

(b). If 1 ≤ i < p < j ≤ n + 1, then [e_{i,p}, e_{p,j}]_{c} = e_{i,j}; that is,

e_{i,j} = e_{i,p}e_{p,j} + e_{p,j}e_{i,p}.  (3.19)

Proof. (a). For j = i+1 and r = p+1, this is (3.17). In the general case, e_{i,j} (resp., e_{p,r}) lies in the subalgebra of R generated by x_{i}, …, x_{j−1} (resp., by x_{p}, …, x_{q−1}) and the result follows from the case already considered.

(b). We show by induction on j − p that the formula holds for any i < p < j. If j − p = 1 then (3.19) is just the definition (3.15). For the recursion step, let p < j; then

e_{i,j+1} = e_{i,j}e_{j,j+1} + e_{j,j+1}e_{i,j}
= (e_{i,p}e_{p,j} + e_{p,j}e_{i,p}) e_{j,j+1} + e_{j,j+1} (e_{i,p}e_{p,j} + e_{p,j}e_{i,p})

= e_{i,p}e_{p,j}e_{j,j+1} + e_{p,j}e_{i,p}e_{j,j+1} + e_{j,j+1}e_{i,p}e_{p,j} + e_{j,j+1}e_{p,j}e_{i,p}

= e_{i,p}e_{p,j+1} + e_{p,j+1}e_{i,p},

where we used (3.15), the induction hypothesis and (3.18).

Lemma 3.3. Assume that (3.17), and a fortiori (3.18), hold in R. For any 1 ≤ i < j ≤ n + 1 we have

Δ(e_{i,j}) = e_{i,j} ⊗ 1 + 1 ⊗ e_{i,j} + 2 ∑_{i<p<j} e_{i,p} ⊗ e_{p,j}.  (3.20)

Proof. We proceed by induction on j − i. If j − i = 1, the formula holds since the x_{i}’s are primitive. For the inductive step, we compute Δ(e_{i,j}e_{j,j+1}) =

= (e_{i,j} ⊗ 1 + 1 ⊗ e_{i,j} + 2 ∑_{i<p<j} e_{i,p} ⊗ e_{p,j}) (e_{j,j+1} ⊗ 1 + 1 ⊗ e_{j,j+1})

= e_{i,j}e_{j,j+1} ⊗ 1 − e_{j,j+1} ⊗ e_{i,j} − 2 ∑_{i<p<j} e_{i,p}e_{j,j+1} ⊗ e_{p,j}

+ e_{i,j} ⊗ e_{j,j+1} + 1 ⊗ e_{i,j}e_{j,j+1} + 2 ∑_{i<p<j} e_{i,p} ⊗ e_{p,j}e_{j,j+1}.

and also Δ(e_{j,j+1}e_{i,j}) =

= (e_{j,j+1} ⊗ 1 + 1 ⊗ e_{j,j+1}) (e_{i,j} ⊗ 1 + 1 ⊗ e_{i,j} + 2 ∑_{i<p<j} e_{i,p} ⊗ e_{p,j})

= e_{j,j+1}e_{i,j} ⊗ 1 + e_{j,j+1} ⊗ e_{i,j} + 2 ∑_{i<p<j} e_{j,j+1}e_{i,p} ⊗ e_{p,j}

+ e_{i,j} ⊗ e_{j,j+1} + 1 ⊗ e_{j,j+1}e_{i,j} + 2 ∑_{i<p<j} e_{i,p} ⊗ e_{j,j+1}e_{p,j};

Summing up the two equations, using the defining formula (3.15) and the fact that e_{j,j+1} and e_{i,p} commute (3.18) we find the required formula (3.20) for Δ(e_{i,j+1}).

Lemma 3.4. Assume that R is finite dimensional or R ≃ 𝔅(V). For any 1 ≤ i ≤ p < q ≤ j ≤ ℓ + 1 we have

e_{i,j}e_{p,q} = e_{p,q}e_{i,j}.  (3.21)

Proof. Since e_{p,q} lies in the subalgebra generated by x_{p}, …, x_{q−1}, it is enough to show that e_{i,j} commutes with each of e_{i,i+1}, …, e_{j−1,j}. We prove this by induction on j. If j = 2, then i = 1 = p, q = 2 and the claim is evident.

We assume this is true for j and show it for j + 1. Since both e_{i,j}, e_{j,j+1} commute with each of e_{i,i+1}, …, e_{j−2,j−1} and e_{i,j+1} = e_{i,j}e_{j,j+1} + e_{j,j+1}e_{i,j}, we see that e_{i,j+1} commutes with all these, too. On the other hand, we see using (3.16) that

e_{j,j+1}e_{i,j+1} = e_{j,j+1}e_{i,j}e_{j,j+1} = e_{i,j+1}e_{j,j+1}.

It remains to show that e_{i,j+1} commutes with e_{j−1,j}. Using again (3.16), we see that this is evident for i = j − 1. We prove: if e_{t,j+1} commutes with e_{j−1,j} for any t, i < t ≤ j − 1, then e_{i,j+1} commutes with e_{j−1,j}. For, we compute

Δ(e_{i,j+1}e_{j−1,j}) = (e_{i,j+1} ⊗ 1 + 1 ⊗ e_{i,j+1} + 2 ∑_{i<p<j+1} e_{i,p} ⊗ e_{p,j+1}) ×

× (e_{j−1,j} ⊗ 1 + 1 ⊗ e_{j−1,j})

= e_{i,j+1}e_{j−1,j} ⊗ 1 + e_{j−1,j} ⊗ e_{i,j+1}

+ 2 ∑_{i<p<j−1} e_{i,p}e_{j−1,j} ⊗ e_{p,j+1} − 2e_{i,j−1}e_{j−1,j} ⊗ e_{j−1,j+1}

+ 2e_{i,j}e_{j−1,j} ⊗ e_{j,j+1} + e_{i,j+1} ⊗ e_{j−1,j} + 1 ⊗ e_{i,j+1}e_{j−1,j}

+ 2 ∑_{i<p<j+1} e_{i,p} ⊗ e_{p,j+1}e_{j−1,j},

where we have used (3.11), (3.9), (3.5), in this order. Also, we have

Δ(e_{j−1,j}e_{i,j+1}) = (e_{j−1,j} ⊗ 1 + 1 ⊗ e_{j−1,j}) ×

× (e_{i,j+1} ⊗ 1 + 1 ⊗ e_{i,j+1} + 2 ∑_{i<p<j+1} e_{i,p} ⊗ e_{p,j+1})

= e_{j−1,j}e_{i,j+1} ⊗ 1 + e_{j−1,j} ⊗ e_{i,j+1}

+ 2 ∑_{i<p<j+1} e_{j−1,j}e_{i,p} ⊗ e_{p,j+1}

+ e_{i,j+1} ⊗ e_{j−1,j} + 1 ⊗ e_{j−1,j}e_{i,j+1}

+ 2 ∑_{i<p<j−1} e_{i,p} ⊗ e_{j−1,j}e_{p,j+1}

+ 2e_{i,j−1} ⊗ e_{j−1,j}e_{j−1,j+1} − 2e_{i,j} ⊗ e_{j−1,j}e_{j,j+1},

where we have used (3.12), (3.7), (3.13), in this order. Let

z := e_{i,j+1}e_{j−1,j} − e_{j−1,j}e_{i,j+1}.

Using the induction hypotheses on j and i, and Lemma 3.2, we conclude that

Δ(z) = z ⊗ 1 + 1 ⊗ z − 2(e_{j−1,j}e_{i,j−1} + e_{i,j−1}e_{j−1,j}) ⊗ e_{j−1,j+1}
+ 2e_{i,j} ⊗ (e_{j,j+1}e_{j−1,j} + e_{j−1,j}e_{j,j+1}) = z ⊗ 1 + 1 ⊗ z.

If R is finite dimensional, the claim follows from Lemma 2.1. If R ≃ 𝔅(V), the claim follows since z is homogeneous of degree 2.

Lemma 3.5. (a). Assume that (3.17), (3.21) hold in R. If 1 ≤ i < j ≤ ℓ + 1, then

Δ(e_{i,j}^{2}) = e_{i,j}^{2} ⊗ 1 + 1 ⊗ e_{i,j}^{2} + 4 ∑_{i+1≤p≤j−1} e_{i,p}^{2} ⊗ e_{p,j}^{2}.  (3.22)

(b). Assume that R is finite dimensional or R = 𝔅(V). If 1 ≤ i < j ≤ ℓ + 1, then

e_{i,j}^{2} = 0.  (3.23)

Proof. (a). We prove that (3.22) holds for i, j = i + t. By (3.20), we have

Δ(e_{i,i+t})^{2} = (e_{i,i+t} ⊗ 1 + 1 ⊗ e_{i,i+t} + 2 ∑_{i+1≤p≤i+t−1} e_{i,p} ⊗ e_{p,i+t})^{2}

= e_{i,i+t}^{2} ⊗ 1 + e_{i,i+t} ⊗ e_{i,i+t} + 2 ∑_{i+1≤p≤i+t−1} e_{i,i+t}e_{i,p} ⊗ e_{p,i+t}

− e_{i,i+t} ⊗ e_{i,i+t} + 1 ⊗ e_{i,i+t}^{2} − 2 ∑_{i+1≤p≤i+t−1} e_{i,p} ⊗ e_{i,i+t}e_{p,i+t}

− 2 ∑_{i+1≤p≤i+t−1} e_{i,p}e_{i,i+t} ⊗ e_{p,i+t}

+ 2 ∑_{i+1≤p≤i+t−1} e_{i,p} ⊗ e_{p,i+t}e_{i,i+t}

+ 4 ∑_{i+1≤p<s≤i+t−1} e_{i,p}e_{i,s} ⊗ e_{p,i+t}e_{s,i+t}

− 4 ∑_{i+1≤p<s≤i+t−1} e_{i,s}e_{i,p} ⊗ e_{s,i+t}e_{p,i+t}

+ 4 ∑_{i+1≤p≤i+t−1} e_{i,p}^{2} ⊗ e_{p,i+t}^{2}

= e_{i,i+t}^{2} ⊗ 1 + 1 ⊗ e_{i,i+t}^{2} + 4 ∑_{i+1≤p≤i+t−1} +e_{i,p}^{2} ⊗ e_{p,i+t}^{2};

here we have used (3.9), (3.13), (3.5), (3.7), (3.14) and the commutation relations (3.21).

(b). We prove by induction on t that e_{i,i+t}^{2} = 0. For t = 1 this is just (3.16). For t > 1, we see by (3.22) and the induction hypothesis that e_{i,i+t}^{2} is primitive. Again, if R is finite dimensional, the claim follows from Lemma 2.1. And if R ≃ 𝔅(V), the claim follows since e_{i,i+t}^{2} is homogeneous of degree ≥ 2.

Lemma 3.6. Assume that (3.17) holds in R. Then, for any 1 ≤ i < r < p < j ≤ ℓ + 1, we have

e_{i,p}e_{r,j} + e_{r,j}e_{i,p} = 2e_{i,j}e_{r,p}.  (3.24)

Proof. We know that e_{i,j} = e_{i,p}e_{p,j} + e_{p,j}e_{i,p} (3.19) and that e_{r,p} commutes with each of e_{i,p}, e_{i,j}, e_{r,j} (3.18); cf. Lemma 3.2. We compute

2e_{i,j}e_{r,p} = e_{r,p}e_{i,j} + e_{i,j}e_{r,p}
= e_{r,p}e_{i,p}e_{p,j} + e_{r,p}e_{p,j}e_{i,p} + e_{i,p}e_{p,j}e_{r,p} + e_{p,j}e_{i,p}e_{r,p}
= e_{i,p}e_{r,p}e_{p,j} + e_{r,p}e_{p,j}e_{i,p} + e_{i,p}e_{p,j}e_{r,p} + e_{p,j}e_{r,p}e_{i,p}
= e_{i,p}(e_{r,p}e_{p,j} + e_{p,j}e_{r,p}) + (e_{r,p}e_{p,j} + e_{p,j}e_{r,p})e_{i,p}
= e_{i,p}e_{r,j} + e_{r,j}e_{i,p}.

The method of proof of the following statement is well-known.

Lemma 3.7. Assume that (3.17), (3.21), (3.23), hold in R. Assume, furthermore, that R is generated as an algebra by the elements x_{1}, …, x_{ℓ}. Then the algebra R is spanned as a vector space by the elements

e_{1,2}^{ϵ_{1,2}}e_{1,3}^{ϵ_{1,3}} … e_{1,ℓ+1}^{ϵ_{1,ℓ+1}}e_{2,3}^{ϵ_{2,3}} … e_{2,ℓ+1}^{ϵ_{2,ℓ+1}} … e_{ℓ,ℓ+1}^{ϵ_{ℓ,ℓ+1}}, with ϵ_{i,j} ∈ {0, 1}. (3.25)

Proof. Let us order the the monomials (3.25) lexicographically with respect to the order of the family (e_{ij}) given by

e_{1,2} ≺ e_{1,3} ≺ … e_{1,ℓ+1} ≺ e_{2,3} ≺ … e_{2,ℓ+1} ≺ … e_{ℓ,ℓ+1}.

We define the symbol of an ordered monomial M by σ(M) := e_{r,s} where e_{r,s} is the first element appearing in M. Let B be the subspace generated by the monomials in (3.25). We claim that, for any ordered monomial M and for any i, e_{i,i+1}M ∈ B and it is 0 or a combination of monomials N with σ(N) ⪰ min{e_{i,i+1}, σ(M)}, length of N ≤ length of M + 1. The claim implies that B = R since it is a left ideal containing 1.

It remains to prove the claim. We do this by induction on the length, the case when the length of M is 0 being obvious. We assume then that the length is positive. Write M = e_{p,q}M′ where e_{p,q} ≺ M′. We consider several cases:

− If i < p or i = p and i + 1 < q, e_{i,i+1} ≺ e_{p,q}.

− If i = p and i + 1 = q, e_{i,i+1}M = 0 by (3.23).

− If i > q > p then e_{i,i+1}e_{p,q} = e_{p,q}e_{i,i+1} by (3.18); hence e_{i,i+1}M = e_{i,i+1}e_{p,q}M′ = e_{p,q}e_{i,i+1}M′; we apply the inductive hypothesis and the fact that e_{p,q} ≺ min{e_{i,i+1}, σ(M′)} to get the claim.

− If i = q > p then e_{i,i+1}e_{p,i} = e_{p,i+1} − e_{p,i}e_{i,i+1} by (3.19); again, the inductive hypothesis and e_{p,q} ≺ min{e_{i,i+1}, σ(M′)} imply the claim.

− If q > i > p then e_{i,i+1}e_{p,q} = e_{p,q}e_{i,i+1} by (3.21); the argument is as in the two preceding cases.

In the proof of the following result, we use an inductive argument on the rank of the Nichols algebra; since the first version of this paper appeared, the argument was applied again in [5, Th. 6.13]. Notice also that a similar argument also appears in [14].

Theorem 3.8. The elements

e_{1,2}, e_{1,3}, …, e_{1,ℓ+1}, e_{2,3}, …, e_{2,ℓ+1}, …, e_{ℓ,ℓ+1},

in this order, determine a PBW basis for 𝔅(V_{ℓ}). In particular,

dim 𝔅(V_{ℓ}) = 2^{\frac{ℓ(ℓ+1)}{2}} .

Proof. We argue by induction on ℓ, the case ℓ = 1 being clear. Let us assume the statement for ℓ−1. Let Z_{ℓ} = 𝔅(V_{ℓ})#𝕜Γ. Let i_{ℓ} : V_{ℓ−1} → V_{ℓ} be given by x_{i} ↦ x_{i} and p_{ℓ} : V_{ℓ} → V_{ℓ−1} by x_{i} ↦ x_{i}, 1 ≤ i ≤ ℓ − 1 and x_{ℓ} ↦ 0. Let ι_{ℓ} : Z_{ℓ−1} → Z_{ℓ} and π_{ℓ} : Z_{ℓ} → Z_{ℓ−1} be respectively induced by i_{ℓ}, p_{ℓ}. Since id_{V_{ℓ−1}} = p_{ℓ}i_{ℓ}, we have a splitting of Hopf algebras id_{Z_{ℓ−1}} = π_{ℓ}ι_{ℓ}. We apply Radford’s theorem; if

R_{ℓ} = Z_{ℓ}^{co π_{ℓ}} = {z ∈ Z_{ℓ} : (id ⊗π_{ℓ})Δ(z) = z ⊗ 1},

then Z_{ℓ} ≃ R_{ℓ}#Z_{ℓ−1}. Thus dim Z_{ℓ} = dim R_{ℓ} dim Z_{ℓ−1}. Since dim Z_{ℓ} ≤ 2^{\frac{ℓ(ℓ+1)}{2}} by Lemma 3.7 and dim Z_{ℓ−1} = 2^{\frac{ℓ(ℓ−1)}{2}} by the induction hypothesis, we have dim R_{ℓ} ≤ 2^{ℓ}.

Let us denote h_{i} = e_{i,ℓ+1}, 1 ≤ i ≤ ℓ. We next claim that h_{1}, …, h_{ℓ} are commuting linearly independent primitive elements of the braided Hopf algebra R_{ℓ}; c_{R}(h_{i} ⊗ h_{j}) = −h_{j} ⊗ h_{i}, for any i > j; and the dimension of the subalgebra of R_{ℓ} spanned by h_{1}, …, h_{ℓ} is 2^{ℓ}. Here c_{R_{ℓ}} means the braiding of R_{ℓ} in the category _{Z_{ℓ−1}}^{Z_{ℓ−1}}YD. Below, we shall always refer to the structure of R_{ℓ} as braided Hopf algebra in this category.

We prove now the claim. By (3.15), π_{ℓ}(h_{i}) = 0; this implies that h_{i} ∈ R_{ℓ}, by (3.20). We shall prove now that h_{i} is a primitive element of R_{ℓ}. It is now convenient to proceed by induction on j = ℓ + 1 − i; again, the case j = 1 is clear. Assume the statement for j. Now

x_{i−1} ⇀ h_{i} = g(i − 1)h_{i}S(x_{i−1}) + x_{i−1}h_{i}
= −g(i − 1)h_{i}g(i − 1)x_{i−1} + x_{i−1}h_{i}
= h_{i}x_{i−1} + x_{i−1}h_{i} = h_{i−1}.

So

Δ_{R_{ℓ}}(h_{i−1}) = Δ_{R_{ℓ}}(x_{i−1} ⇀ h_{i}) = x_{i−1} ⇀ Δ(h_{i})
= g(i − 1) ⇀ 1 ⊗ x_{i−1} ⇀ h_{i} + x_{i−1} ⇀ h_{i} ⊗ 1
= h_{i−1} ⊗ 1 + 1 ⊗ h_{i−1}.

Again by induction on j = ℓ + 1 − i, it is not difficult to show that h_{i} ≠ 0; (3.20) and the induction hypothesis on Z_{ℓ−1} are used in this step. But h_{i} is homogeneous of degree i (with respect to the grading of Z_{ℓ}); therefore, we can conclude that h_{1}, …, h_{ℓ} are linearly independent.

We already know that h_{i} and h_{j} commute for i > j, see (3.21). On the other hand, it follows at once from (3.20) and the reconstruction formulas for the bosonization that

Δ_{Z_{ℓ}}(h_{i}) = e_{i,ℓ+1} ⊗ 1 + ( ∏_{i≤h≤ℓ} g(h)) ⊗ e_{i,ℓ+1}

+ 2 ∑_{i<p<ℓ+1} ( ∏_{p≤h≤ℓ} g(h)) e_{i,p} ⊗ e_{p,ℓ+1};

then the coaction of Z_{ℓ−1} on R_{ℓ} satisfies

δ(h_{i}) = ( ∏_{i≤h≤ℓ} g(h)) ⊗ e_{i,ℓ+1} + 2 ∑_{i<p<ℓ+1} ( ∏_{p≤h≤ℓ} g(h)) e_{i,p} ⊗ e_{p,ℓ+1}.

If i > j, we check easily that

e_{i,p} ⇀ h_{j} = (e_{i,p})_{(1)}h_{j}S((e_{i,p})_{(2)}) = h_{j}(e_{p,i+1})_{(1)}S((e_{p,i+1})_{(2)}) = 0,

by (3.21) and the identity g_{s,p}h_{j} = h_{j}g_{s,p}, i ≤ s < p.

Thus

c_{R_{ℓ}}(h_{i} ⊗ h_{j}) = ( ∏_{i≤h≤ℓ} g(h)) ⇀ h_{j} ⊗ h_{i} = −h_{j} ⊗ h_{i}.

We then argue as in the proof of [2, Lemma 3.3] to conclude that the dimension of the subalgebra of R_{ℓ} spanned by h_{1}, …, h_{ℓ} is ≥ 2^{ℓ}; but then this dimension is exactly 2^{ℓ} by the observations before. This proves the claim. It implies, in turn, that dim Z_{ℓ} = 2^{\frac{ℓ(ℓ+1)}{2}} ; in presence of Lemma 3.7, this implies in turn the Theorem.

Theorem 3.9. The algebra 𝔅(V_{ℓ}) is presented by generators e_{i,j}, 1 ≤ i < j ≤ ℓ + 1, and relations (3.15), (3.17), (3.21), (3.23).

Proof. Let 𝔅′ be the algebra presented by generators e_{i,j}, 1 ≤ i < j ≤ ℓ, and relations (3.15), (3.17), (3.21), (3.23). We claim that 𝔅′ is generated as an algebra by the elements x_{i} = e_{i,i+1}, 1 ≤ i ≤ ℓ and that 𝔅′ is a braided Hopf algebra with the e_{i,i+1}’s primitive. The first claim follows clearly from (3.15). The second claim is also not difficult; relations (3.23) are derived from Lemma 3.5.

By Lemma 3.7, we see that the monomials (3.25) span 𝔅′ as vector space, thus dim 𝔅′ ≤ 2^{\frac{ℓ(ℓ+1)}{2}} . By Lemmas 3.1, 3.4 and 3.5, there is a surjective algebra map ψ : 𝔅′ → 𝔅(V_{ℓ}); by Theorem 3.8, ψ is an isomorphism.

In presence of Lemma 2.3, we obtain

Corollary 3.10. If V is a Yetter-Drinfeld module of type A_{ℓ} over a finite abelian group of exponent 2, then the algebra 𝔅(V) is presented by generators e_{i,j}, 1 ≤ i < j ≤ ℓ + 1, and relations (3.15), (3.23),

[e_{i,i+1}, e_{p,p+1}]_{c} = 0, 1 ≤ i < p ≤ ℓ,  (3.26)
[e_{i,j}, e_{p,q}]_{c} = 0, 1 ≤ i ≤ p < q ≤ j ≤ ℓ + 1.  (3.27)

In particular, dim 𝔅(V) = 2^{\frac{ℓ(ℓ+1)}{2}} .

Proof. The presentation by generators and relations follows from Theorem 3.9 and [4, Lemma 3.15]. The claim on the dimension follows at once from Lemma 2.3 and Theorem 3.8.

Remark 3.11. In 𝔅(V), the following relation holds: [e_{i,p}, e_{p,j}]_{c} = e_{i,j}.

4. Generation in degree one

The key for all the results in this Section is the following Theorem; the proof is similar to the proof of [4, Th. 7.6] and uses Theorem 3.9.

Theorem 4.1. (i). Let R = ⊕_{n≥0}R(n) be a graded braided Hopf algebra with R(0) = 𝕜1, R(1) = P (R) ≃ V_{ℓ} as braided vector spaces. If R is finite dimensional then R ≃ 𝔅(V_{ℓ}); that is, R is generated in degree one.

(ii). Let S = ⊕_{n≥0}S(n) be a graded braided Hopf algebra generated in degree one, with S(0) = 𝕜1, S(1) ≃ V_{ℓ} as braided vector spaces. If S is finite dimensional then S ≃ 𝔅(V_{ℓ}); that is, S(1) = P (S).

Proof. We prove (ii); it is known that (i) and (ii) are equivalent, by taking R = S^{∗}; see for instance [3, Lemma 5.5].

Assume that S(1) = V_{ℓ} and define elements e_{ij} ∈ S by (3.1), (3.15). By Lemmas 3.1, 3.4 and 3.5, (3.17), (3.21), (3.23) hold in S. By Theorem 3.9, we have an epimorphism of braided graded Hopf algebras ϕ : 𝔅(V) → S. But the restriction of ϕ to the first term of the coradical filtration of 𝔅(V), namely 𝕜1 ⊕ V, is injective; see [11, Chapter 5], ϕ is injective.

Corollary 4.2. Let R = ⊕_{n≥0}R(n) be a graded braided Hopf algebra with R(0) = 𝕜1. Assume that the generalized Cartan matrix associated to R(1) = P (R) is a union of connected components of type A_{ℓ_{j}}, j = 1, …, M. If R is finite dimensional then R is generated in degree one.

Proof. If M = 1, the claim follows from Lemma 2.3 using [3, Lemma 2.3]. The general case follows from the case M = 1 using [3, Lemma 4.2].

To state the next Corollary, we need some notation. Let H be a finite dimensional Hopf algebra whose coradical H_{0} is a Hopf subalgebra. Let V be its infinitesimal braiding, as in subsection 2.2. Assume that there exist g(1), …, g(θ) ∈ G(H), central in H_{0}, and ”multiplicative” characters χ(1), …, χ(θ) (i. e. one-dimensional representations) of H; and a basis x_{1} …, x_{θ} of V such that x_{i} ∈ V_{χ(i)}^{g(i)} for all i. Furthermore, assume that

χ(i)(g(j)) ∈ {±1}, for all i, j.

Since H is finite dimensional, we have then χ(i)(g(i)) = −1 for all i. We can attach to H a generalized Cartan matrix A by (1.2).

Corollary 4.3. If A is a union of connected components of type A_{ℓ_{j}}, j = 1, …, M then H is generated as an algebra by H_{1}. In particular, if H is pointed then H is generated as an algebra by group-like and skew-primitive elements.

Note that Theorem 1.2 is a particular case of this Corollary.

5. Liftings of Nichols algebras of type A_{ℓ}

The aim of this section is to prove the following.

Theorem 5.1. Let H be a finite dimensional pointed Hopf algebra. Assume that G(H) =: Γ is an abelian group of exponent 2. Furthermore, assume that the simply-laced Cartan matrix (a_{ij}) corresponding to the infinitesimal braiding (V, c) of H, as in (1.2), is a union of connected components of type A_{ℓ}, for possibly different ℓ’s.

Then H is isomorphic, as Hopf algebra, to 𝔅(V)#𝕜Γ.

Let us consider first a finite dimensional pointed Hopf algebra H such that G(H) = Γ and the infinitesimal braiding V of H is the direct sum of a finite family (V_{ℓ_{α}})_{α} of Yetter-Drinfeld modules, such that V_{ℓ_{α}} has type A_{ℓ_{α}}, and the braiding matrix is in the standard form, i.e. it has only 1 below the main diagonal. By Section 4, the diagram of H is the Nichols algebra 𝔅(V), thus gr(H) ≃ 𝔅(V)#𝕜Γ. We keep the notation of Section 3 for the basis x_{1}, …, x_{ℓ} of such a Yetter-Drinfeld module V_{ℓ}, with x_{i} ∈ (V_{ℓ})_{g(i)}^{χ(i)}, for the braiding matrix and the generators (e_{i,j})_{1≤i<j≤ℓ} of 𝔅(V_{ℓ}). As in Lemma 3.1(b), elements of the bases of different V_{ℓ_{α}}’s commute, so 𝔅(V) is isomorphic as an algebra to the tensor product of all 𝔅(V_{ℓ_{α}}). If H_{0} ⊆ H_{1} ⊆ … is the coradical filtration of H, we have by [2, Lemma 5.4] that

H_{1} = 𝕜Γ ⊕ (⊕_{g,h∈Γ,χ∈_{Γ},χ≠ϵ}P_{g,h}(H)^{χ}).

Moreover, P_{g,1}(H)^{ϵ} = 𝕜(g − 1) and for any χ ≠ ϵ we have that P_{g,1}(H)^{χ} ≠ 0 if and only if there exists i such that (g, χ) = (g(i), χ(i)) (where g(i) and χ(i) are associated to the basis element x_{i} in one of the V_{ℓ_{α}}’s).

Fix some V_{ℓ} in the family (V_{ℓ_{α}})_{α}. Choose a_{1}, …, a_{ℓ} ∈ H such that a_{i} ∈ P_{g(i),1}(H)^{χ(i)} and the class of a_{i} in gr (H) is x_{i}. We define the elements (E_{i,j})_{1≤i<j≤ℓ+1} starting with a_{1}, …, a_{ℓ} as we defined (e_{i,j})_{1≤i<j≤ℓ+1} in the Nichols algebra 𝔅(V_{ℓ}) starting with x_{1}, …, x_{ℓ}. Thus E_{i,i+1} = a_{i} for any 1 ≤ i ≤ ℓ, and E_{i,j} = E_{i,j−1}E_{j−1,j} + E_{j−1,j}E_{i,j−1} for any 1 ≤ i < j ≤ ℓ + 1, j − i ≥ 2.

Lemma 5.2. For any i, j with i + 1 < j we have that a_{i}a_{j} = a_{j}a_{i}.

Proof. We have that

Δ(a_{i}a_{j}) = (g(i) ⊗ a_{i} + a_{i} ⊗ 1)(g(j) ⊗ a_{j} + a_{j} ⊗ 1)

= g(i)g(j) ⊗ a_{i}a_{j} + g(j)a_{i} ⊗ a_{j} + g(i)a_{j} ⊗ a_{i} + a_{i}a_{j} ⊗ 1 and similarly Δ(a_{j}a_{i}) = g(j)g(i) ⊗ a_{j}a_{i} + g(i)a_{j} ⊗ a_{i} + g(j)a_{i} ⊗ a_{j} + a_{j}a_{i} ⊗ 1. We obtain that a_{i}a_{j} − a_{j}a_{i} ∈ P_{g(i)g(j),1}(A)^{χ(i)χ(j)}. We have that χ(i)χ(j) ≠ ϵ. Indeed, otherwise we would have that 1 = b_{ji} = χ(i)(g(j)) = χ(j)(g(j)) = −1, a contradiction.

Also, (χ(i)χ(j))(g(i)g(j)) = 1, so (g(i)g(j), χ(i)χ(j)) ≠ (g(t), χ(t)) for any t. We conclude that a_{i}a_{j} − a_{j}a_{i} = 0.

With the same proof as for Lemma 3.2 we have the following.

(a). If 1 ≤ i < j < p < r ≤ ℓ + 1, then E_{i,j}E_{p,r} = E_{p,r}E_{i,j}.

(b). If 1 ≤ i < p < j ≤ ℓ + 1, then E_{i,j} = E_{i,p}E_{p,j} + E_{p,j}E_{i,p}.

Using an inductive argument similar to the one in Lemma 3.3 we have that

Δ(E_{i,j}) = g(i) … g(j − 1) ⊗ E_{i,j} + E_{i,j} ⊗ 1

+ 2 ∑_{i<p<j} g(i) … g(p − 1)E_{p,j} ⊗ E_{i,p}.

Lemma 5.3. If 1 ≤ i ≤ p < q ≤ j ≤ ℓ + 1, then E_{i,j}E_{p,q} = E_{p,q}E_{i,j}.

Proof. We argue as in the proof of Lemma 3.4. The difference appears when we compute Δ(E_{j−1,j}E_{i,j+1}−E_{i,j+1}E_{j−1,j}). For this we have that

Δ(E_{j−1,j}E_{i,j+1}) = (g(j − 1) ⊗ E_{j−1,j} + E_{j−1,j} ⊗ 1)(E_{i,j+1} ⊗ 1 + g(i) … g(j) ⊗ E_{i,j+1}

+ 2 ∑_{i<p<j+1} g(p) … g(j)E_{i,p} ⊗ E_{p,j+1})

= g(j − 1)E_{i,j+1} ⊗ E_{j−1,j} + g(i) … g(j − 2)g(j) ⊗ E_{j−1,j}E_{i,j+1}

+ 2 ∑_{i<p<j+1} g(p) … g(j)g(j − 1)E_{i,p} ⊗ E_{j−1,j}E_{p,j+1}

+ 2 ∑_{i<p<j+1} b_{p,j−1} … b_{j,j−1}g(p) … g(j)E_{j−1,j}E_{i,p} ⊗ E_{p,j+1}

+ E_{j−1,j}E_{i,j+1} ⊗ 1 + b_{i,j−1} … b_{j,j−1}g(i) … g(j)E_{j−1,j} ⊗ E_{i,j+1} = g(i) … g(j − 2)g(j) ⊗ E_{j−1,j}E_{i,j+1} + g(j − 1)E_{i,j+1} ⊗ E_{j−1,j}

+ 2 ∑_{i<p<j−2} g(p) … g(j)E_{j−1,j}E_{i,p} ⊗ E_{p,j+1} + E_{j−1,j}E_{i,j+1} ⊗ 1

+ 2 ∑_{i<p<j} g(p) … g(j − 2)g(j)E_{i,p} ⊗ E_{j−1,j}E_{p,j+1}

+ 2g(j − 1)g(j)E_{i,j} ⊗ E_{j−1,j}E_{j,j+1} + g(i) … g(j)E_{j−1,j} ⊗ E_{i,j+1} − 2g(j − 1)g(j)E_{j−1,j}E_{i,j−1} ⊗ E_{j−1,j+1} + 2g(j − 1)g(j)E_{j−1,j}E_{i,j} ⊗ E_{j,j+1}

On finite quantum groups at −1 21

Similarly we obtain that Δ(E_{i,j+1}E_{j−1,j}) = g(i) … g(j − 2)g(j) ⊗ E_{i,j+1}E_{j−1,j} + g(j − 1)E_{i,j+1} ⊗ E_{j−1,j}

+ 2 ∑_{i<p<j} g(p) … g(j − 2)g(j)E_{i,p} ⊗ E_{p,j+1}E_{j−1,j}

− 2g(j − 1)g(j)E_{i,j} ⊗ E_{j,j+1}E_{j−1,j} + E_{i,j+1}E_{j−1,j} ⊗ 1 + g(i) … g(j)E_{j−1,j} ⊗ E_{i,j+1}

+ 2 ∑_{i<p<j+1} g(p) … g(j)E_{i,p}E_{j−1,j} ⊗ E_{p,j+1}

Using these relations we obtain that Δ(E_{j−1,j}E_{i,j+1} − E_{i,j+1}E_{j−1,j})

= g(i) … g(j − 2)g(j) ⊗ (E_{j−1,j}E_{i,j+1} − E_{i,j+1}E_{j−1,j})
+ (E_{j−1,j}E_{i,j+1} − E_{i,j+1}E_{j−1,j}) ⊗ 1

so E_{j−1,j}E_{i,j+1} − E_{i,j+1}E_{j−1,j} ∈ P_{(g(1)...g(j−2)g(j),1)}(A)^{χ(1)...χ(j−2)χ(j−1)}, Since (χ(i) … χ(j − 2)χ(j − 1))(g(j)) = −1 ≠ ϵ(g(j)), we have that χ(i) … χ(j − 2)χ(j − 1) ≠ ϵ. Then

(χ(i) … χ(j − 2)χ(j − 1))(g(i) … g(j − 2)g(j)) = 1 ≠ χ(t)(g(t)) for any t, therefore E_{j−1,j}E_{i,j+1} − E_{i,j+1}E_{j−1,j} = 0.

Lemma 5.4. For any 1 ≤ i < j ≤ ℓ + 1 we have E_{i,j}^{2} = 0. Proof. We proceed by induction as in Lemma 3.5. The only difference is at the following computation.

Δ(E_{i,i+t}^{2}) = (E_{i,i+t} ⊗ 1 + g(i) … g(i + t − 1) ⊗ E_{i,i+t}

+ 2 ∑_{i<p<i+t} g(i) … g(p − 1)E_{p,i+t} ⊗ E_{i,p})^{2}

= E_{i,i+t}^{2} ⊗ 1 + g(i) … g(i + t − 1)E_{i,i+t} ⊗ E_{i,i+t}

+ 2 ∑_{i<p<i+t} g(i) … g(p − 1)E_{i,p} ⊗ E_{i,i+t}E_{p,i+t}

− g(i) … g(i + t − 1)E_{i,i+t} ⊗ E_{i,i+t} + 1 ⊗ E_{i,i+t}^{2}

− 2 ∑_{i<p<i+t} g(p) … g(i + t − 1)E_{i,i+t}E_{i,p} ⊗ E_{p,i+t}

− 2 ∑_{i<p<i+t} g(i) … g(p − 1)E_{i,p} ⊗ E_{p,i+t}E_{i,i+t}

+ 2 ∑_{i<p<i+t} g(p) … g(i + t − 1)E_{i,p}E_{i,i+t} ⊗ E_{p,i+t}

+ 4 ∑_{i<p<s<i+t} g(p) … g(s − 1)E_{i,p}E_{i,s} ⊗ E_{p,i+t}E_{s,i+t}

− 4 ∑_{i<s<p<i+t} g(s) … g(p − 1)E_{i,p}E_{i,s} ⊗ E_{p,i+t}E_{s,i+t}

+ 4 ∑_{i<p<i+t} E_{i,p}^{2} ⊗ E_{p,i+t}^{2}

= E_{i,i+t}^{2} ⊗ 1 + 1 ⊗ E_{i,i+t}^{2}

showing that E_{i,i+t}^{2} is a primitive element, so then it is zero.

Proof of Theorem 5.1. Let H be as in the statement of the Theorem. If F ∈ 𝕜Γ⊗𝕜Γ defines a twisting H_{F} of H, then the coradical filtration of H is a coalgebra filtration of H_{F}, therefore it contains the coradical filtration of H_{F}. Since (H_{F})_{F^{−1}} = H, the coradical filtrations of H and H_{F} must be equal. In particular H_{F} is pointed and moreover gr H_{F} = (gr H)_{F}. By Lemma 2.3 there exists some twisting H_{F} of H, with F ∈ 𝕜Γ ⊗ 𝕜Γ, such that the associated infinitesimal braiding V_{F} has standard form. We know that gr (H_{F}) ≃ 𝔅(V_{F})#𝕜Γ, and this is generated as an algebra by the elements of the group Γ (where g ∈ Γ is identified with 1#g) and all the families (e_{i,j})_{1≤i<j≤l+1} associated to the Yetter-Drinfeld modules V_{ℓ_{α}} (the direct summands in V_{F}), subject to the relations (3.15), (3.17), (3.21), (3.23), ge_{i,i+1} = χ(i)(g)e_{i,i+1}g for any i and g ∈ Γ, and elements of the bases of different V_{ℓ_{α}}’s commute. By [2, Lemma 2.2], H_{F} is generated as an algebra by Γ and all the families of elements a_{1}, …, a_{ℓ} coming from the V_{ℓ_{α}}’s, thus H_{F} is also generated by Γ and all the families (E_{i,j})_{1≤i<j≤ℓ+1} coming from the V_{ℓ_{α}}’s. Clearly gE_{i,i+1} = χ(i)(g)E_{i,i+1}g and (3.15), (3.17), (3.21), (3.23) are satisfied (with E instead of e) and as in Lemma 5.2 elements of the form a_{i} coming from different V_{ℓ_{α}}’s commute. We obtain that H_{F} is a factor algebra of 𝔅(V_{F})#𝕜Γ, and by taking into account the dimension we must have H_{F} ≃ 𝔅(V_{F})#𝕜Γ ≃ gr H_{F} as algebras. It is clear that this is also an isomorphism of coalgebras. Then twisting by F^{−1} we obtain H ≃ (gr H_{F})_{F^{−1}} ≃ gr H ≃ 𝔅(V)#𝕜Γ, which ends the proof.

6. Nichols algebras of types A_{ℓ}^{(1)}

We consider in this section a Yetter-Drinfeld module V_{ℓ}^{(1)} over our fixed group Γ = (ℤ/2)^{s}. Let y_{1}, …, y_{ℓ+1} be a basis of V_{ℓ}^{(1)} with y_{i} ∈ (V_{ℓ}^{(1)})_{g(i)}^{χ(i)}, for some fixed g(1), …, g(ℓ + 1) ∈ Γ, χ(1), …, χ(ℓ + 1) ∈

Γ̂. Our fundamental assumption in this section is the following. The braiding is given by (1.1) with the matrix:

(b_{ij})_{1≤i,j≤ℓ+1} = (\begin{matrix}−1 & −1 & 1 & & … & −1 \\ 1 & −1 & −1 & 1 \\ & 1 & −1 & −1 & ⋮ & ⋮ \\ & & & ... & ... \\ ⋮ & ⋮ & & & ... & −1 \\ 1 & & … & & & −1 \end{matrix});

that is, V_{ℓ}^{(1)} is of type A_{ℓ}^{(1)}, so that the Dynkin diagram of V_{ℓ}^{(1)} is a cycle with ℓ + 1 vertices.

We keep the notation introduced in Section 3 for V_{ℓ}.

Theorem 6.1. The dimension of 𝔅(V_{ℓ}^{(1)}) is infinite.

Proof. We consider three ℓ-dimensional subspaces of V_{ℓ}^{(1)}, each with a fixed ordered basis:

W_{1}, with basis y_{1}, …, y_{ℓ};
W_{2}, with basis y_{2}, …, y_{ℓ+1};
W_{3}, with basis y_{ℓ+1}, y_{1}, …, y_{ℓ−1}.

We have two isomorphisms of braided vector spaces:

j_{1} :V_{ℓ} → W_{1}, j_{1}(x_{i}) = y_{i},  1 ≤ i ≤ ℓ;
j_{2} :V_{ℓ} → W_{2}, j_{2}(x_{i}) = y_{i+1}, 1 ≤ i ≤ ℓ.

We denote by the same name the corresponding morphisms of braided Hopf algebras 𝔅(V_{ℓ}) → 𝔅(V_{ℓ}^{(1)}). We also have a morphism of braided Hopf algebras j_{3} : 𝔅(W_{3}) → 𝔅(V_{ℓ}^{(1)}).

Let Z_{ℓ}^{(1)} = 𝔅(V_{ℓ}^{(1)})#𝕜Γ. We have a splitting of Yetter-Drinfeld modules id_{V_{ℓ}} = qj_{1}, where q : V_{ℓ}^{(1)} → V_{ℓ} is given by

q(y_{i}) = x_{i}, 1 ≤ i ≤ ℓ, q(y_{ℓ+1}) = 0.

This splitting gives rise to a splitting of Hopf algebras id_{Z_{ℓ}} = QJ, where J : Z_{ℓ} → Z_{ℓ}^{(1)} and Q : Z_{ℓ}^{(1)} → Z_{ℓ} are respectively induced by j_{1}, q. Let

R = (Z_{ℓ}^{(1)})^{co Q} = {z ∈ Z_{ℓ}^{(1)} : (id ⊗Q)Δ(z) = z ⊗ 1}.

Then Z_{ℓ}^{(1)} ≃ R#Z_{ℓ}. Our aim is to show that dim R is infinite. We first introduce the family (f_{ij}) of elements of 𝔅(V_{ℓ}^{(1)}), for 1 ≤ i < j ≤ ℓ + 1 or 2 ≤ i < j ≤ ℓ + 2, as follows:

f_{ij} := j_{1}(e_{ij}), 1 ≤ i < j ≤ ℓ + 1;  (6.1)
f_{i+1,j+1} := j_{2}(e_{ij}), 1 ≤ i < j ≤ ℓ + 1.  (6.2)

Clearly, there is no contradiction between (6.1) and (6.2). As in the proof of Theorem 3.8, we see that f_{i+1,ℓ+2} ∈ R; they are even primitive elements of this braided Hopf algebra. We now introduce

f_{1,ℓ+2} := [f_{1,ℓ+1}, f_{ℓ+1,ℓ+2}]_{c} = f_{1,ℓ+1}y_{ℓ+1} − y_{ℓ+1}f_{1,ℓ+1} ∈ 𝔅(V_{ℓ}^{(1)}); (6.3)

thus, f_{1,ℓ+2} ∈ Ker Q. We claim that

Δ(f_{1,ℓ+2}) = f_{1,ℓ+2} ⊗ 1 + 1 ⊗ f_{1,ℓ+2}

+2 ∑_{1<p<ℓ+2} f_{1,p}⊗(2f_{p,ℓ+1}f_{ℓ+1,ℓ+2}−f_{p,ℓ+2})−2 ∑_{1<p<ℓ+2} j_{3}(e_{1,p})⊗f_{p,ℓ+1}. (6.4)

Here Δ means the comultiplication in 𝔅(V_{ℓ}^{(1)}); and below, c is the braiding of this algebra. Let us first state that

c(f_{1,ℓ+1} ⊗ y_{ℓ+1}) = y_{ℓ+1} ⊗ f_{1,ℓ+1};  (6.5)
c(f_{p,ℓ+1} ⊗ y_{ℓ+1}) = −y_{ℓ+1} ⊗ f_{p,ℓ+1}; 1 < p < ℓ + 1;  (6.6)
c(y_{ℓ+1} ⊗ f_{r,s}) = f_{r,s} ⊗ y_{ℓ+1}, 1 ≤ r < s ≤ ℓ + 1.  (6.7)

We shall also need

\begin{aligned}f_{1,p}y_{ℓ+1} + y_{ℓ+1}f_{1,p} & = j_{3}(e_{2,p+1}e_{1,2} + e_{1,2}f_{2,p+1}) \\ & = j_{3}(e_{2,p+1}e_{1,2} + e_{1,2}e_{2,p+1}) = j_{3}(e_{1,p}). \end{aligned}

On finite quantum groups at −1 25

We now compute

Δ(f_{1,ℓ+1}y_{ℓ+1}) = (f_{1,ℓ+1} ⊗ 1 + 1 ⊗ f_{1,ℓ+1} + 2 ∑_{1<p<ℓ+1} f_{1,p} ⊗ f_{p,ℓ+1})

× (y_{ℓ+1} ⊗ 1 + 1 ⊗ y_{ℓ+1}) = f_{1,ℓ+1}y_{ℓ+1} ⊗ 1 + f_{1,ℓ+1} ⊗ y_{ℓ+1}

+ y_{ℓ+1} ⊗ f_{1,ℓ+1} + 1 ⊗ f_{1,ℓ+1}f_{j,j+1}

− 2 ∑_{1<p<ℓ+1} f_{1,p}y_{ℓ+1} ⊗ f_{p,ℓ+1}

+ 2 ∑_{1<p<ℓ+1} f_{1,p} ⊗ f_{p,ℓ+1}y_{ℓ+1};

and also

Δ(y_{ℓ+1}f_{1,ℓ+1}) = (y_{ℓ+1} ⊗ 1 + 1 ⊗ y_{ℓ+1})

× (f_{1,ℓ+1} ⊗ 1 + 1 ⊗ f_{1,ℓ+1} + 2 ∑_{1<p<ℓ+1} f_{1,p} ⊗ f_{p,ℓ+1})

= y_{ℓ+1}f_{1,ℓ+1} ⊗ 1 + y_{ℓ+1} ⊗ f_{1,ℓ+1} + f_{1,ℓ+1} ⊗ y_{ℓ+1} + 1 ⊗ y_{ℓ+1}f_{1,ℓ+1}

+ 2 ∑_{1<p<ℓ+1} y_{ℓ+1}f_{1,p} ⊗ f_{p,ℓ+1}

+ 2 ∑_{1<p<ℓ+1} f_{1,p} ⊗ y_{ℓ+1}f_{p,ℓ+1}.

Hence

Δ(f_{1,ℓ+2}) = f_{1,ℓ+2} ⊗ 1 + 1 ⊗ f_{1,ℓ+2}

− 2 ∑_{1<p<ℓ+1} (f_{1,p}y_{ℓ+1} + y_{ℓ+1}f_{1,p}) ⊗ f_{p,ℓ+1}

+ 2 ∑_{1<p<ℓ+1} f_{1,p} ⊗ (f_{p,ℓ+1}y_{ℓ+1} − y_{ℓ+1}f_{p,ℓ+1})

= f_{1,ℓ+2} ⊗ 1 + 1 ⊗ f_{1,ℓ+2}

− 2 ∑_{1<p<ℓ+1} j_{3}(e_{1,p}) ⊗ f_{p,ℓ+1}

+ 2 ∑_{1<p<ℓ+1} f_{1,p} ⊗ (2f_{p,ℓ+1}f_{ℓ+1,ℓ+2} − f_{p,ℓ+2}) ,

as desired. Now, equation (6.4) implies at once that f_{1,ℓ+2} ∈ R; and using Theorem 3.8 we conclude that f_{1,ℓ+2} ≠ 0.

Let now c_{R} mean the braiding of R in the category _{Z_{ℓ}}^{Z_{ℓ}}YD. We next claim that

f_{1,ℓ+2} is primitive in R,  (6.8)
c_{R}(f_{1,ℓ+2} ⊗ f_{1,ℓ+2}) = f_{1,ℓ+2} ⊗ f_{1,ℓ+2}.  (6.9)

As in Theorem 3.8, we compute

y_{ℓ+1} ⇀ f_{1,ℓ+1} = g(ℓ + 1)f_{1,ℓ+1}S(y_{ℓ+1}) + y_{ℓ+1}f_{1,ℓ+1}
= −g(ℓ + 1)f_{1,ℓ+1}g(ℓ + 1)y_{ℓ+1} + y_{ℓ+1}f_{1,ℓ+1}
= −f_{1,ℓ+1}y_{ℓ+1} + y_{ℓ+1}f_{1,ℓ+1} = −f_{1,ℓ+2}.

Thus,

Δ_{R}(f_{1,ℓ+2}) = −Δ_{R}(y_{ℓ+1} ⇀ f_{1,ℓ+1}) = −y_{ℓ+1} ⇀ Δ_{R}(f_{1,ℓ+1})
= − (g(ℓ + 1) ⇀ 1 ⊗ y_{ℓ+1} ⇀ f_{1,ℓ+1} + y_{ℓ+1} ⇀ f_{1,ℓ+1} ⊗ 1)
= f_{1,ℓ+2} ⊗ 1 + 1 ⊗ f_{1,ℓ+2}.

This is (6.8). For (6.9), it is enough to show that

f_{1,p} ⇀ f_{1,ℓ+2} = 0, 1 < p < ℓ + 1,
j_{3}(e_{1,p}) ⇀ f_{1,ℓ+2} = 0, 1 < p < ℓ + 1,
f_{1,ℓ+2} ⇀ f_{1,ℓ+2} = 0,

which is not difficult.

We conclude from (6.8) and (6.9) that R has infinite dimension. For,

Δ_{R}(f_{1,ℓ+2}^{N}) = ∑_{0≤i≤N} \binom{N}{i}f_{1,ℓ+2}^{i} ⊗ f_{1,ℓ+2}^{N−i};

by induction, we see that f_{1,ℓ+2}^{N} ≠ 0 for all N, and a fortiori the family of homogeneous elements f_{1,ℓ+2}^{N}, N ≥ 0, is linearly independent.

Remark 6.2. The element f_{1,ℓ+2} is analogous to the imaginary root vector of the affine Lie algebra with Dynkin diagram A_{ℓ}^{(1)}, see [7, Ch. 5]. It is likely that by the same method, one would be able to show that Nichols algebras of types D_{ℓ}^{(1)}, E_{6}^{(1)}, E_{7}^{(1)}, E_{8}^{(1)} are also infinitedimensional.

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