ON NICHOLS ALGEBRAS OF LOW DIMENSION

MATÍAS GRAÑA

Abstract. This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius–Lusztig kernels in order to compute Nichols algebras of diagonal group type. With this, we classify Nichols algebras 𝔅(V) with dimension < 32 or with dimension p^{3}, p a prime number, when V lies in a Yetter–Drinfeld category over a finite group. With the so called Lifting Procedure, this allows to classify pointed Hopf algebras of index < 32 or p^{3}.

1. Introduction, notation, preliminaries

1.1. Introduction. The “Lifting Procedure” (see [AS1]) is a method intended to classify finite dimensional pointed Hopf algebras. Among the achievements of this method, we mention the classification of pointed Hopf algebras of order p^{3}, p^{4} (p a prime number) [AS1, AS3], the classification of pointed Hopf algebras whose group is of exponent p [AS4], the classification of pointed Hopf algebras of order p^{5} [AS4, G2, G1], the classification of pointed Hopf algebras of index p, p^{2} [G3], the answer, by the negative, to the tenth Kaplansky conjecture [AS2] (some of these results have been obtained independently by other methods, see for instance [BDG, Ge, D, CD, CDR, SvO, B]). The first step in this procedure is the classification of Nichols algebras, i.e., graded braided Hopf algebras with the base field in degree 0 and which are coradically graded and generated by its primitive elements (see 1.4.5). Being braided and graded, such an algebra gives a braiding c : V ⊗ V → V ⊗ V, where V is the space of primitive elements (which turns out to coincide with the homogeneous component of degree 1). Furthermore, the pair (V, c) determines the Nichols algebra up to isomorphism. A major goal in the classification program of pointed Hopf algebras is thus to have an algorithm describing (and in particular giving the dimension of) a Nichols algebra starting out with the pair (V, c) as its space of primitive elements. Although any solution to the Braid Equation c : V ⊗ V → V ⊗ V gives rise to a Nichols algebra, the objects V arising in the application of the Lifting Procedure are objects of Yetter–Drinfeld categories over finite groups, and hence we shall restrict ourselves to those pairs (V, c) with this additional property (see 1.4.10).

The theory divides at this moment in two branches: the pair (V, c) is “of diagonal group type” (see 1.4.10) or it is not. In the first case, a substantial progress is made in [AS2] through the notion of “braiding of Cartan type”, using the theory of Frobenius–Lusztig kernels. This notion applies to a large class of pairs (V, c) of diagonal group type. In this article we propose to enlarge the class of Frobenius– Lusztig kernels and we give the examples of type A_{2} and B_{2} (see 2.11 and 2.14). To compute the PBW bases of these generalized FL-kernels we use a generalized form of the quantum Serre relations. Though this generalized form can be found in [R] for the diagonal case, we prefer to derive it from an equality in the group algebra of the braid group, since this equality could provide quantum Serre relations for the general case (see 2.2 and 2.5).

1991 Mathematics Subject Classification. Primary 16W30; Secondary 17B37.

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

In the non-diagonal case very little is known. In fact, up to now there were only two examples known to produce finite dimensional Nichols algebras (and hence, finite dimensional pointed Hopf algebras). The first one is 3-dimensional and it generates a 12-dimensional Nichols algebra (it is given for the first time in [MS] and was reproduced in [AG]). The second one is 6 dimensional and it is given also in [MS]. Another example given there, which is 4 dimensional, is proved in this article to be of diagonal group type (see 5.2.1).^{1} In this article we propose some tools to classify all Yetter–Drinfeld modules of a given dimension over finite groups. We use these tools to parameterize those modules of dimension 3, 4 (sections 4, 5). As a consequence of this, we find a new example of a finite dimensional Nichols algebra with a 4-dimensional space of primitive elements which is of non-diagonal group type; its dimension is 72 (see 5.3.2). We also give some necessary conditions for an irreducible YD-module to be of “finite Nichols rank” (i.e., to give a finite dimensional Nichols algebra). This allows to focus the attention mainly on the specific cases V = M(g, χ) where χ is a character (see 3.1 for the condition, 1.4.7 for the definition of M(g, ρ)).

Finally, as a consequence of these methods together with those of [G3], we classify Nichols algebras over finite groups with dimension < 32. As said above, this is the first step in the classification of pointed Hopf algebras of index < 32. Specifically, we have: Theorem 1.1.1. Let R ∈ _{kΓ}^{kΓ}YD be a Nichols algebra, where Γ is a finite group and dim R < 32. Let V = PR be its space of primitive elements. Then dim V ≤ 4 and either

• R is a Quantum Linear Space (see definition in 2.10), or

• R is of type A_{2} (see remark 2.14), or

• R is of type A_{2} × A_{1}, or

• V = V_{3,−1} and dim R = 12 (see 4.1 for the definition), or

• V = V_{3,−1} ⊕V′, dim 𝔅(V′) = 2, c^{2} : V_{3,−1} ⊗V′ → V_{3,−1} ⊗V′ is the identity, and dim R = 24.

1.2. Acknowledgements. I thank N. Andruskiewitsch for the careful reading of the material and for general and particular advice. I thank J. Fernández Bonder for a hint on 3.3. The referee has suggested the best terminology I have seen for the braided pairs that this article deals with. It has been a pleasure to read his comments. Thanks to A. Schüler, who pointed out a mistake in the published version for the relation z_{2}z_{1} = q_{11}q_{12}q_{21}^{2}q_{22}^{2}z_{1}z_{2} in the proof of 2.11. I thank also M. Lamas for many valuable conversations.

1.3. Notation. For a general reference on Hopf algebras see [M]. We fix an algebraically closed field k of characteristic 0. Most of the results of the article are still true for positive characteristic, as long as it does not divide the dimension of the Nichols algebras under consideration (see [G4]). In this case, the definition of N(1) below must be given as N(1) = char(k).

1After submitting the paper I was informed that a third module given in [MS] is known to produce a finite dimensional Nichols algebra (see [FK]).

All vector spaces, algebras, tensor products, homs are considered over k unless explicitly stated. If Γ is a group and g ∈ Γ, we denote by Γ_{g} the centralizer Γ_{g} = {x ∈ Γ | xg = gx}.

For a root of unity q, we denote by N(q) its order if q ≠ 1. If q = 1 or q is not a root of unity then we define N(q) = ∞. We recall the definition of q-numbers: for q an indeterminate and n, m ∈ ℕ, n ≥ m, define

(n)_{q} = 1 + q + q^{2} + ⋯ + q^{n−1}, (n)_{q}^{!} = ∏_{i=1}^{n}(i)_{q} , ( \begin{matrix}n \\ m \end{matrix} )_{q} = \frac{(n)_{q}^{!}}{(m)_{q}^{!} (n − m)_{q}^{!}} .

All these expressions lie in ℤ[q]. For q ∈ k we evaluate them in q = q to get (n)_{q},

(n)_{q}^{!} and ( \begin{matrix}n \\ m \end{matrix} )_{q}.

Let 𝕊_{n} be the symmetric group, presented with n − 1 generators {τ_{1}, …, τ_{n−1}} and relations τ_{i}τ_{j} = τ_{j}τ_{i} if |i − j| > 1, τ_{i}τ_{i+1}τ_{i} = τ_{i+1}τ_{i}τ_{i+1}, τ_{i}^{2} = 1. We denote by 𝔹_{n} the braid group, with n − 1 generators {σ_{1}, …, σ_{n−1}} and relations

σ_{i}σ_{j} = σ_{j}σ_{i} if |i − j| > 1, σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}.

We consider the sequence of inclusions

𝔹_{1} ↪ 𝔹_{2} ↪ … ↪ 𝔹_{n} ↪ 𝔹_{n+1} ↪ …

where σ_{i} ↦ σ_{i}, and consider 𝔹_{∞} the limit of this sequence. Definition 1.3.2. For m < n, we denote by 𝔹_{m n} the subgroup of 𝔹_{n} generated by σ_{m}, …, σ_{n−1}. We denote by ι_{m n} : 𝔹_{n−m+1} → 𝔹_{n} the inclusion σ_{i} ↦ σ_{i+m−1}. Then 𝔹_{m n} = Im(ι_{m n}). We may view all these groups as subgroups of 𝔹_{∞}. If x_{1}, x_{2} are elements of 𝔹_{j_{1}}, 𝔹_{j_{2}} respectively, then we denote

(x_{1}|x_{2}) = x_{1} · ι_{(j_{1}+1) (j_{1}+j_{2})}(x_{2}) ∈ 𝔹_{j_{1}+j_{2}},

and inductively we define

(x_{1}| ⋯ |x_{s}) = (x_{1}|(x_{2}| ⋯ |x_{s})) ∈ 𝔹_{j_{1}+···+j_{s}}.

For instance, if x_{1} = σ_{2}σ_{1} ∈ 𝔹_{3} and x_{2} = σ_{1} ∈ 𝔹_{2}, we have (x_{1}|x_{2}) = σ_{2}σ_{1}σ_{4} ∈ 𝔹_{5}. We extend (bi)linearly the definition of (|) to the group algebra, i.e.

(|) : k𝔹_{j_{1}} ⊗ k𝔹_{j_{2}} → k𝔹_{j_{1}+j_{2}}.

1.4. Preliminaries. We consider the canonical projection 𝔹_{n} ↠ 𝕊_{n}, σ_{i} ↦ τ_{i}. By [CuRe, 64.20] we have a section (of sets) s : 𝕊_{n} → 𝔹_{n} such that if x = τ_{i_{1}} ⋯ τ_{i_{l}}, where l = ℓ(x) (i.e., x can not be written with fewer than l generators τ_{i}’s), then s(x) = σ_{i_{1}} ⋯ σ_{i_{l}}. Notice that if ℓ(xy) = ℓ(x) + ℓ(y) then s(xy) = s(x)s(y).

We denote by S^{n} the element of the group algebra k𝔹_{n}

(1.4.3)  S^{n} = ∑_{x∈𝕊_{n}} s(x).

Let i_{1} +⋯+i_{r} = n. We denote by Sh_{i_{1},...,i_{r}} ⊆ 𝕊_{n} the subset of (i_{1}, …, i_{r})-shuffles, i.e.

Sh_{i_{1},...,i_{r}} = {x ∈ 𝕊_{n} | x^{−1}(i_{1} + ⋯ + i_{j} + 1) < x^{−1}(i_{1} + ⋯ + i_{j} + 2) < ⋯

⋯ < x^{−1}(i_{1} + ⋯ + i_{j} + i_{j+1}) for j = 0, …, r − 1}.

We consider also the elements

S_{i_{1},...,i_{r}} = ∑_{x∈Sh_{i_{1},...,ir}} s(x) ∈ k𝔹_{i_{1}+···+i_{r}},

T_{i_{1},...,i_{r}} = ∑_{x^{−1}∈Sh_{i_{1},...,ir}} s(x) ∈ k𝔹_{i_{1}+···+i_{r}},
S^{j_{1}|j_{2}|···|j_{s}} = (S^{j_{1}}|S^{j_{2}}| ⋯ |S^{j_{s}}) ∈ k𝔹_{j_{1}+···+j_{s}}.

Notice that

S^{i|j}S_{i,j} = S_{i+j} = T_{i,j}S^{i|j}.

Definition 1.4.4. We call a braided pair (or BP) a pair (V, c), where V is a vector space and c : V ⊗ V → V ⊗ V is a (bijective) solution of the braid equation. We say that (V, c) is a rigid braided pair (or RBP) if it is a finite dimensional BP such that c^{♭} ∈ End(V^{∗} ⊗ V, V ⊗ V^{∗}) is an isomorphism (see for instance [T1] for the definition of c^{♭}).

Let H be a Hopf algebra with bijective antipode. We denote by _{H}^{H}YD the category of (left-left) Yetter–Drinfeld modules over H of arbitrary dimension. If V is an object of _{H}^{H}YD with braidings c_{V,W} : V ⊗ W → W ⊗ V, then (V, c = c_{V,V}) is a braided pair. If V is moreover finite dimensional then (V, c = c_{V,V}) is an RBP.

Let (V, c) be a BP. The group 𝔹_{n} acts on V^{⊗n} by σ_{i} ↦ (id_{V^{⊗i}} ⊗c ⊗ id_{V^{⊗n−i−2}}). We denote respectively by AV, CV the tensor algebra and the tensor coalgebra of V. As vector spaces, they coincide with T V = ⊕_{i}V^{⊗i} = ⊕_{i}A^{i}V = ⊕_{i}C^{i}V. The (i, j)-component of the multiplication of AV is simply the identity map m_{i,j} = id : A^{i}V ⊗ A^{j}V → A^{i+j}V. Dually, the (i, j)-component of the comultiplication of CV are the identity maps Δ_{i,j} = id : C^{i+j}V → C^{i}V ⊗ C^{j}V. Both AV and CV are graded braided Hopf algebras. The comultiplication components of AV are nothing but S_{i,j} : A^{i+j}V → A^{i}V ⊗ A^{j}V. Dually, the multiplication components of CV are T_{i,j} : C^{i}V ⊗ C^{j}V → C^{i+j}V. The antipode in both cases is determined by S(x) = −x ∀x ∈ V.

There exists a unique (graded) map of braided Hopf algebras which reduces to the identity id : V → V in degree 1; we denote this map by S. Its graded components are the maps S^{n} : A^{n}V → C^{n}V. Definition 1.4.5. Let (V, c) be an RBP. We say that the image of

S : AV → CV

is the Nichols algebra generated by V and we denote it by 𝔅(V) ^{2}. We say that dim(𝔅(V)) is the Nichols rank of V. Remark 1.4.6. It can be proved that a finite dimensional braided pair (V, c) belongs to a braided category, and that an RBP (V, c) belongs to a rigid braided category. We warn that in [AS2] the term “Nichols algebra” is used for the case in which the rigid braided category is clearly defined. For the purpose of this article, it is more convenient not to emphasize the rigid braided category in which (V, c) lies. Furthermore, we shall often consider (V, c) as an object of different categories.

2If (V, c) is just a BP, the object 𝔅(V) can be constructed in the same manner. In this general case, the result has been called a Quantum Symmetric Algebra.

We refer to [AG] for first properties of Nichols algebras. We refer to [G3] for further results, in particular the freeness result of Nichols algebras over certain Nichols subalgebras which we state in 2.7.

Definition 1.4.7. Let Γ be a finite group, g ∈ Γ, and ρ ∈ Γ̂_{g} an irreducible representation of Γ_{g}. We denote by M(g, ρ) ∈ _{kΓ}^{kΓ}YD the object

M(g, ρ) = Ind_{Γ_{g}}^{Γ} ρ = kΓ ⊗_{kΓ_{g}} ρ, h ⇀ (t ⊗ x) = ht ⊗ x, δ(t ⊗ x) = tgt^{−1} ⊗ (t ⊗ x).

It has been proved in [DPR, W] and then more generally in [CiRo] in the language of Hopf bimodules (see also [AG]) that if one takes one element g in each conjugacy class of Γ and if ρ runs over all irreducible non-isomorphic representations of Γ_{g}, the objects M(g, ρ) are a full set of irreducible non-isomorphic objects of _{kΓ}^{kΓ}YD, which turns out to be a semisimple category.

Definition 1.4.8. Let V = ⊕_{i=1}^{θ}M(g_{i}, ρ_{i}), where ρ_{i} : Γ_{g_{i}} → Aut(W_{i}) is an irreducible representation of Γ_{g_{i}}. Let {x_{1}^{i}, …, x_{r_{i}}^{i}} be a basis of W_{i} and let {h_{1}^{i}, …, h_{s_{i}}^{i}} be a set of representatives of left cosets Γ/Γ_{g_{i}}. We define t_{j}^{i} = h_{j}^{i}g_{i}(h_{j}^{i})^{−1} and z_{jl}^{i} = h_{j}^{i} ⊗ x_{l}^{i} ∈ Ind_{Γ_{i}}^{Γ} W_{i} = M(g_{i}, ρ_{i}). Then {z_{jl}^{i}}_{ijl} is a basis of V and the braiding is given by

c(z_{jl}^{i} ⊗ z_{vw}^{u}) = (t_{j}^{i}h_{v}^{u} ⊗ x_{w}^{u}) ⊗ z_{jl}^{i} = t_{j}^{i} ⇀ z_{vw}^{u} ⊗ z_{jl}^{i}.

Definition 1.4.9. Let (V, c) be a BP, V = ⊕_{i}V_{i}. We say that this direct sum is a decomposition of (V, c) (and that (V, c) is decomposable) if

c|_{V_{i}⊗V_{j}} : V_{i} ⊗ V_{j} → V_{j} ⊗ V_{i}.

In this case we denote by c_{ij} the restriction c_{ij} = c|_{V_{i}⊗V_{j}}. It is straightforward to see that if (V, c) is an RBP and V = ⊕_{i}V_{i} is a decomposition of V as a BP, then this decomposition induces a decomposition of V^{∗} = ⊕_{i}V_{i}^{∗} and taking the corresponding restrictions c_{ii} of c then each (V_{i}, c_{ii}) becomes an RBP.

If Γ is a finite group and V ∈ _{kΓ}^{kΓ}YD is finite dimensional, then as noticed in 1.4.8 V can be decomposed as V = ⊕_{i}M(g_{i}, ρ_{i}). However, sometimes it is possible to further decompose some of the summands, or to give a different decomposition. These decompositions usually fail to lie in _{kΓ}^{kΓ}YD, but since (V, c) is an RBP any decomposition gives a direct sum of RBP’s.

Definition 1.4.10. Let (V, c) be a BP. We say that V is of group type (or of GT) if V has a basis {x_{1}, …, x_{n}} and there exist g_{i} ∈ GL(V) (i = 1, …, n) such that

(1.4.11)  c(x_{i} ⊗ x_{j}) = g_{i}(x_{j}) ⊗ x_{i}.

The terminology, introduced by the referee, comes from the fact that for V a finite dimensional object of _{kΓ}^{kΓ}YD, one can find such a basis and maps in GL(V) (in fact, 1.4.8 tells explicitly how to do this when Γ is finite), and conversely, a BP which such a property is automatically an object of _{kΓ}^{kΓ}YD, where Γ ⊂ GL(V) is the group generated by {g_{1}, …, g_{n}}. We further say that (V, c) is of finite group type (or finite GT) if the group Γ generated by {g_{1}, …, g_{n}} is finite. We say that (V, c) is of abelian group type (or abelian GT) if Γ is abelian, and we finally say that (V, c) is of diagonal group type (or diagonal GT) if Γ acts in a diagonal way on V. It is clear that “diagonal” implies “abelian” and that the latter implies the former when k is algebraically closed of characteristic 0 (as we supposed it to be). It is clear also that “diagonal” means that there exist a basis {x_{1}, …, x_{n}} and a matrix (q_{ij}) ∈ (k^{×})^{n×n} such that

(1.4.12)  c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i}.

This condition is called “to come from the abelian case” in [AG], since for V in _{kΓ}^{kΓ}YD, Γ finite abelian, it is fulfilled. The basis {x_{1}, …, x_{n}} and the matrix (q_{ij}) verifying 1.4.12 will be called the basis and the matrix of (V, c).

In general, for a BP of group type (V, c), we say that {x_{1}, …, x_{n}} is a braided basis if it verifies 1.4.11. The maps {g_{1}, …, g_{n}} are called the group-likes associated to the basis. It can be seen that

(1.4.13)  g_{i}(x_{j}) ∈ V^{g_{i}g_{j}g_{i}^{−1}},

where we denote V^{g} := span{x_{t} | g_{t} = g}.

It is not hard to prove that a 2-dimensional BP of group type is always of abelian GT, and hence if it is moreover of finite GT, then it is of diagonal GT.

2. On the diagonal case

We prove an equality in the group algebra of the braid group 𝔹_{n}. This equality will give information on the adjoint action. In particular, the quantum Serre relations will turn out to be a direct consequence of this equality. We denote by e the identity element in 𝔹_{n} and consider in k𝔹_{n} the following elements for 1 ≤ i ≤ j ≤ n.

U_{i}^{j} = σ_{j}σ_{j−1} ⋯ σ_{i},
D_{i}^{j} = σ_{i}σ_{i+1} ⋯ σ_{j},
R_{i}^{j} = (e − σ_{j}σ_{j})(e − D_{j−1}^{j}σ_{j})(e − D_{j−2}^{j}σ_{j}) ⋯ (e − D_{i}^{j}σ_{j}).

Lemma 2.1. We have the following identities

(1) Let y ∈ 𝔹_{n} ↪ 𝔹_{n+1} and let x = ι_{2 (n+1)}(y). Then U_{1}^{n}x = yU_{1}^{n} and

xD_{1}^{n} = D_{1}^{n}y.
(2) Let x ∈ 𝔹_{2 (n+1)}. Then xD_{1}^{n}U_{1}^{n} = D_{1}^{n}U_{1}^{n}x.
(3) U_{1}^{j}D_{2}^{n}σ_{n} = D_{1}^{n}σ_{n}U_{1}^{j−1} for j < n.

(4) (e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n−1} − D_{n}^{n}U_{1}^{n} − D_{n−1}^{n}U_{1}^{n} − ⋯ − D_{1}^{n}U_{1}^{n})R_{2}^{n}

= R_{1}^{n}(e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n−1}) for n ≥ 2.

Proof. (1) Since ι_{2 n} is a morphism, we prove it for the generators. Let y = σ_{j−1}, 2 ≤ j ≤ n. Then x = σ_{j}, and we have

U_{1}^{n}x = σ_{n} ⋯ σ_{1}σ_{j} = (σ_{n} ⋯ σ_{j+1})σ_{j}σ_{j−1}σ_{j}(σ_{j−2} ⋯ σ_{1})
= (σ_{n} ⋯ σ_{j+1})σ_{j−1}σ_{j}σ_{j−1}(σ_{j−2} ⋯ σ_{1}) = σ_{j−1}σ_{n} ⋯ σ_{1}
= yU_{1}^{n},

and analogously for D_{1}^{n}.

(2) As the previous item, we can prove it for generators σ_{j}, 2 ≤ j ≤ n. We

have

σ_{j}D_{1}^{n}U_{1}^{n} = D_{1}^{n}σ_{j−1}U_{1}^{n} = D_{1}^{n}U_{1}^{n}σ_{j}.

(3) Simply

U_{1}^{j}D_{2}^{n}σ_{n} = U_{2}^{j}D_{1}^{n}σ_{n} = σ_{j} ⋯ σ_{2}D_{1}^{n}σ_{n} = σ_{j} ⋯ σ_{3}D_{1}^{n}σ_{1}σ_{n} = …
… = D_{1}^{n}σ_{j−1} ⋯ σ_{1}σ_{n} = D_{1}^{n}σ_{n}σ_{j−1} ⋯ σ_{1} = D_{1}^{n}σ_{n}U_{1}^{j−1}.

ON NICHOLS ALGEBRAS OF LOW DIMENSION 7

We remark that these equalities are easily understood with drawings.

(4) By induction. Notice first that

σ_{i}σ_{i+1}σ_{i+1}σ_{i}σ_{i+1}σ_{i+1} = σ_{i}σ_{i+1}σ_{i}σ_{i+1}σ_{i}σ_{i+1} = σ_{i+1}σ_{i}σ_{i+1}σ_{i}σ_{i+1}σ_{i}
= σ_{i+1}σ_{i+1}σ_{i}σ_{i+1}σ_{i+1}σ_{i}.

For n = 2 we have

(e + σ_{1} − σ_{2}σ_{2}σ_{1} − σ_{1}σ_{2}σ_{2}σ_{1})(e − σ_{2}σ_{2})

= e − σ_{2}σ_{2} + σ_{1} − σ_{1}σ_{2}σ_{2} − σ_{2}σ_{2}σ_{1} + σ_{2}σ_{2}σ_{1}σ_{2}σ_{2} − σ_{1}σ_{2}σ_{2}σ_{1} + σ_{1}σ_{2}σ_{2}σ_{1}σ_{2}σ_{2} = e − σ_{2}σ_{2} + σ_{1} − σ_{1}σ_{2}σ_{2} − σ_{2}σ_{2}σ_{1} + σ_{2}σ_{2}σ_{1}σ_{2}σ_{2} − σ_{1}σ_{2}σ_{2}σ_{1} + σ_{2}σ_{2}σ_{1}σ_{2}σ_{2}σ_{1} = (e − σ_{2}σ_{2} − σ_{1}σ_{2}σ_{2} + σ_{2}σ_{2}σ_{1}σ_{2}σ_{2})(e + σ_{1}) = (e − σ_{2}σ_{2})(e − σ_{1}σ_{2}σ_{2})(e + σ_{1})

Suppose the equality is true for n. Then

(e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n} − D_{n+1}^{n+1}U_{1}^{n+1} − D_{n}^{n+1}U_{1}^{n+1} − ⋯ − D_{1}^{n+1}U_{1}^{n+1})R_{2}^{n+1}

= R_{2}^{n+1} + (U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n} − D_{n+1}^{n+1}U_{1}^{n+1} − ⋯ − D_{2}^{n+1}U_{1}^{n+1})R_{2}^{n+1}

− D_{1}^{n+1}U_{1}^{n+1}R_{2}^{n+1} = R_{2}^{n+1} + (e + U_{2}^{2} + ⋯ + U_{2}^{n} − D_{n+1}^{n+1}U_{2}^{n+1} − ⋯ − D_{2}^{n+1}U_{2}^{n+1})σ_{1}R_{2}^{n+1}

− D_{1}^{n+1}U_{1}^{n+1}R_{2}^{n+1} = R_{2}^{n+1} + (e + U_{2}^{2} + ⋯ + U_{2}^{n} − D_{n+1}^{n+1}U_{2}^{n+1} − ⋯ − D_{2}^{n+1}U_{2}^{n+1})σ_{1}R_{3}^{n+1}

× (e − D_{2}^{n+1}σ_{n+1}) − R_{2}^{n+1}D_{1}^{n+1}U_{1}^{n+1} (by 2) = R_{2}^{n+1} + (e + U_{2}^{2} + ⋯ + U_{2}^{n} − D_{n+1}^{n+1}U_{2}^{n+1} − ⋯ − D_{2}^{n+1}U_{2}^{n+1})R_{3}^{n+1}σ_{1}

× (e − D_{2}^{n+1}σ_{n+1}) − R_{2}^{n+1}D_{1}^{n+1}U_{1}^{n+1} (R_{3}^{n+1} ∈ k𝔹_{3 n+2}) = R_{2}^{n+1} + ι_{2 (n+2)}((e + U_{1}^{1} + ⋯ + U_{1}^{n−1} − D_{n}^{n}U_{1}^{n} − ⋯ − D_{1}^{n}U_{1}^{n})R_{2}^{n})σ_{1}

× (e − D_{2}^{n+1}σ_{n+1}) − R_{2}^{n+1}D_{1}^{n+1}U_{1}^{n+1} = R_{2}^{n+1} + ι_{2 (n+2)}(R_{1}^{n}(e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n−1}))σ_{1}(e − D_{2}^{n+1}σ_{n+1})

− R_{2}^{n+1}D_{1}^{n+1}U_{1}^{n+1} (by IH) = R_{2}^{n+1} + R_{2}^{n+1}(e + U_{2}^{2} + U_{2}^{3} + ⋯ + U_{2}^{n})σ_{1}(e − D_{2}^{n+1}σ_{n+1}) − R_{2}^{n+1}D_{1}^{n+1}U_{1}^{n+1} = R_{2}^{n+1}[e + σ_{1} + U_{2}^{2}σ_{1} + ⋯ + U_{2}^{n}σ_{1} − σ_{1}D_{2}^{n+1}σ_{n+1}

− U_{2}^{2}σ_{1}D_{2}^{n+1}σ_{n+1} − ⋯ − U_{2}^{n}σ_{1}D_{2}^{n+1}σ_{n+1} − D_{1}^{n+1}U_{1}^{n+1}] = R_{2}^{n+1}(e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n} − D_{1}^{n+1}σ_{n+1} − U_{1}^{2}D_{2}^{n+1}σ_{n+1} − ⋯

− U_{1}^{n}D_{2}^{n+1}σ_{n+1} − D_{1}^{n+1}σ_{n+1}U_{1}^{n}) = R_{2}^{n+1}(e + U_{1}^{1} + ⋯ + U_{1}^{n} − D_{1}^{n+1}σ_{n+1} − D_{1}^{n+1}σ_{n+1}U_{1}^{1} − ⋯ − D_{1}^{n+1}σ_{n+1}U_{1}^{n−1}

− D_{1}^{n+1}σ_{n+1}U_{1}^{n}) (by 3) = R_{2}^{n+1}(e − D_{1}^{n+1}σ_{n+1})(e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n}) = R_{1}^{n+1}(e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n}).

□

Proposition 2.2. S^{n+1}(e − U_{1}^{n})(e − U_{2}^{n}) ⋯ (e − U_{n}^{n}) = R_{1}^{n}S^{n|1} for n ≥ 1.

Proof. By induction, for n = 1 being

S^{2}(e − σ_{1}) = (e + σ_{1})(e − σ_{1}) = e − σ_{1}σ_{1}.

If the equality holds for n, we have

S^{n+2}(e − U_{1}^{n+1})(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1})

= S^{n+2}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1}) − S^{n+2}U_{1}^{n+1}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1})

= T_{1,n+1}S^{1|n+1}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1})
− T_{n+1,1}S^{n+1|1}U_{1}^{n+1}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1})
= T_{1,n+1}S^{1|n+1}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1})
− T_{n+1,1}U_{1}^{n+1}S^{1|n+1}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1}) (by 2.1,1)
= (T_{1,n+1} − T_{n+1,1}U_{1}^{n+1})S^{1|n+1}(e − U_{2}^{n+1}) ⋯ (e − U_{n+1}^{n+1})
= (T_{1,n+1} − T_{n+1,1}U_{1}^{n+1})ι_{2 n+2}(S^{n+1}(e − U_{1}^{n}) ⋯ (e − U_{n}^{n}))
= (T_{1,n+1} − T_{n+1,1}U_{1}^{n+1})ι_{2 n+2}(R_{1}^{n}S^{n|1}) (by IH)
= (T_{1,n+1} − T_{n+1,1}U_{1}^{n+1})(R_{2}^{n+1}S^{1|n|1})

= (e + U_{1}^{1} + U_{1}^{2} + ⋯ + U_{1}^{n+1} − U_{1}^{n+1} − D_{n+1}^{n+1}U_{1}^{n+1} − ⋯ − D_{1}^{n+1}U_{1}^{n+1})

× R_{2}^{n+1}S^{1|n|1}
= (e + U_{1}^{1} + ⋯ + U_{1}^{n} − D_{n+1}^{n+1}U_{1}^{n+1} − ⋯ − D_{1}^{n+1}U_{1}^{n+1})R_{2}^{n+1}S^{1|n|1}
= R_{1}^{n+1}(e + U_{1}^{1} + ⋯ + U_{1}^{n})S^{1|n|1} (by 2.1,4)
= R_{1}^{n+1}(T_{1,n}|e_{1})(S^{1|n}|e_{1}) (e_{1} the unique element of 𝔹_{1})
= R_{1}^{n+1}(S^{n+1}|e_{1})
= R_{1}^{n+1}S^{(n+1)|1}.
□

Definition 2.3. Let X be a vector space, 0 ≠ x ∈ X and T ∈ End(X) such that T^{n}(x) = 0 for some n. The nilpotency order of T in x is the number r + 1 such that T^{r}(x) ≠ 0 and T^{r+1}(x) = 0. Definition 2.4. Let R be a braided Hopf algebra and x ∈ R. We denote by Ad_{x} ∈ End(R) the (braided) adjoint of x, i.e.,

Ad_{x}(y) = m(x_{(1)} ⊗ m(c(S(x_{(2)}) ⊗ y)),

where c is the braiding of R and m the multiplication map. Thus, if x is a primitive element, we have

Ad_{x}(y) = xy − m(c(x ⊗ y)).

The following result can be found in [R], but, as said in the introduction, we can obtain it directly from 2.2. Theorem 2.5. Let x_{1}, x_{2} ∈ V be such that for some matrix (q_{ij}) the braiding verifies c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i} for i, j = 1, 2. Let N = N(q_{11}) (see the beginning of 1.3) and let t be the least non-negative integer (if it exists) such that q_{11}^{t}q_{12}q_{21} = 1. Let r = min{t, N − 1}. Then the nilpotency order of Ad_{x_{1}} on x_{2} is exactly r + 1.

Proof. Let n ∈ ℕ. We consider Ad_{x_{1}}^{n}(x_{2}) = S^{n+1}(Ãd_{x_{1}}^{n}(x_{2})) ∈ 𝔅(V), where Ãd is the adjoint in AV. It is easy to see by induction that this element is the action of the left hand side of 2.2 on x_{1}^{n}x_{2}. Now, the right hand side acts on x_{1}^{n}x_{2} as a scalar, namely

(1 − q_{12}q_{21})(1 − q_{11}q_{12}q_{21})(1 − q_{11}^{2}q_{12}q_{21}) ⋯ (1 − q_{11}^{n−1}q_{12}q_{21})(n)_{q_{11}}^{!} .

The assertion follows at once. □

Definition 2.6. Let (V, c) be an RBP. For y ∈ V^{∗}, we denote by ∂_{y} the operator

∂_{y} = (id ⊗y) ∘ Δ^{i−1,1} : 𝔅^{i}(V) → 𝔅^{i−1}(V).

If B = {x_{1}, …, x_{n}} is a basis of V, let {∂_{1}, …, ∂_{n}} be the basis of V^{∗} dual to B. We denote also by ∂_{j}, j = 1, …, n the operator ∂_{∂_{j}}. We warn that for the case V ∈ _{H}^{H}YD, these are usually not morphisms in _{H}^{H}YD, but only linear maps.

The following facts, which we shall use, are consequences of results in [G3].

Theorem 2.7. Let V = ⊕_{i=1}^{n}M(g_{i}, ρ_{i}) = ⊕_{i}V_{i}, where ρ_{i} ∈ Γ̂_{g_{i}}. Let q_{i} = ρ_{i}(g_{i}) (since g_{i} lies in the center of Γ_{g_{i}} and ρ_{i} is irreducible, ρ_{i}(g_{i}) is a scalar) and let N_{i} = N(q_{i}).

(1) Take a basis {z_{jl}^{i}} of V as in 1.4.8 and {∂_{jl}^{i}} its dual basis. Let i, j, l be fixed, K = ker(∂_{jl}^{i}) ⊂ 𝔅(V), and let W = span(x_{jl}^{i}). Then for those i such that N_{i} < ∞ the multiplication map K ⊗ 𝔅(W) → 𝔅(V) is an isomorphism. In particular, if {y_{1}, …, y_{k}} is a basis of K, then

{y_{r}(z_{jl}^{i})^{s} | 1 ≤ r ≤ k, 0 ≤ s < N_{i}}

is a basis of 𝔅(V).

(2) If V is of finite Nichols rank, then

N_{i} | dim 𝔅(V_{i}) | ∏_{i} dim 𝔅(V_{i}) | dim 𝔅(V).

Furthermore, ∏_{i} dim 𝔅(V_{i}) = dim 𝔅(V) iff c_{ij} = id_{V_{i}⊗V_{j}} ∀i ≠ j, where c_{ij} = c|_{V_{i}⊗V_{j}} : V_{i} ⊗ V_{j} → V_{j} ⊗ V_{i}.

(3) N_{i}^{dim V_{i}} ≤ dim 𝔅(V_{i}) ∀i. Furthermore, if {z_{1}^{i}, …, z_{d_{i}}^{i}} is a basis of V_{i}, then the set

{(z_{1}^{1})^{i_{1}^{1}} ⋯ (z_{d_{1}}^{1})^{i_{d_{1}}^{1}} (z_{1}^{2})^{i_{1}^{2}} ⋯ (z_{d_{n}}^{n})^{i_{dn}^{n}} | 0 ≤ i_{k}^{j} < N_{j}}

is linearly independent in 𝔅(V). This allows to give a relation between the dimension and the Nichols rank of a

BP of finite group type: Definition 2.8. Let n = ∏_{i=1}^{R} p_{i}^{v_{i}}, where p_{i} are distinct primes, v_{i} > 0 and p_{i} > p_{1} for i > 1. We define r(n) = log_{p_{1}}(n). Remark 2.9. Applying 2.7 part 1 together with part 3, we get that if dim 𝔅(V) = n, then d = dim V ≤ r(n). Applying 2.7 part 2, we get that if V = ⊕_{i=1}^{θ}V_{i} = ⊕_{i}M(g_{i}, ρ_{i}) ∈ _{kΓ}^{kΓ}YD and n = dim(𝔅(V)), n = ∏_{i=1}^{R} p_{i}^{v_{i}}, then θ ≤ ∑_{i} v_{i}. Definition 2.10. If (V, c) is of diagonal GT with basis {x_{1}, …, x_{n}} and matrix (q_{ij}), we say, following [AS1], that 𝔅(V) is a Quantum Linear Space (or QLS) if q_{ij}q_{ji} = 1 for i ≠ j. It is proved in [AS1] (and it is also a consequence of 2.7 part 2), that in this case dim 𝔅(V) = ∏ N(q_{ii}).

In [AS3] the authors associate to some Nichols algebras of diagonal GT a Frobenius– Lusztig kernel with the same dimension. This association can be done whenever q_{ij}q_{ji} = q_{ii}^{a_{ij}} ∀i, j for a matrix (a_{ij}) with entries in ℤ, where (q_{ij}) is the matrix of the braiding. The following proposition deals with general rank-2 Nichols algebras. Proposition 2.11. Let (V, c) be a 2 dimensional BP of finite GT (and thus of diagonal GT), with basis {x_{1}, x_{2}} and matrix (q_{ij}), and let N_{j} = N(q_{jj}) for j = 1, 2. As in 2.5, let r + 1 be nilpotency order of Ad_{x_{2}} on x_{1}. For 1 ≤ i ≤ r let M_{i} = N(q_{11}(q_{12}q_{21})^{i}q_{22}^{i^{2}}). Then

dim 𝔅(V) ≥ N_{1}N_{2} ∏_{1≤i≤r} M_{i}.

Furthermore, suppose that the nilpotency order of Ad_{x_{1}} on x_{2} is 2. We have: (1) If r = 1 then the equality holds.

(2) If r = 2, and N(q_{11}) ≠ 2 or N(q_{22}) ≠ 3 then the equality holds.

(3) If r = 2, N(q_{11}) = 2 and N(q_{22}) = 3 then the equality holds if and only if

q_{12}q_{21} = −1, or q_{12}q_{21} = q_{22}, or q_{12}q_{21} = −q_{22}.

Proof. Let z_{1} = Ad_{x_{2}}(x_{1}) = x_{2}x_{1} − q_{21}x_{1}x_{2}, and define inductively z_{i+1} = Ad_{x_{2}}(z_{i}) = x_{2}z_{i} − q_{21}q_{22}^{i}z_{i}x_{2}. Taking on 𝔅(V) the bi-degree given by deg(x_{1}) = (1, 0), deg(x_{2}) = (0, 1), we have z_{i} ∈ 𝔅^{1,i}(V). Furthermore, it is proved in 2.5 that z_{r} ≠ 0 and z_{r+1} = 0. It is immediate to see by induction (it follows also from 2.2) that ∂_{x_{2}}(z_{i}) = 0, and ∂_{x_{1}}(z_{i}) = d_{12}^{1}d_{12}^{2} ⋯ d_{12}^{i}x_{2}^{i}, where d_{12}^{j} = (1 − q_{12}q_{21}q_{22}^{j−1}). Thus, ∂_{x_{2}}^{i}∂_{x_{1}}(z_{i}) ≠ 0 if i ≤ r. We shall prove that the set {x_{1}^{n_{1}}z_{1}^{m_{1}} ⋯ z_{r}^{m_{r}} | 0 ≤ n_{1} < N_{1}, 0 ≤ m_{i} < M_{i}} is linearly independent. To see this, we compute

∂_{x_{2}}^{i}∂_{x_{1}}(z_{i}^{m}) = ∑_{j=0}^{m−1} q_{12}^{ji}q_{11}^{j}d_{12}^{1} ⋯ d_{12}^{j}∂_{x_{2}}^{i}(z_{i}^{m−1−j}x_{2}^{i}z_{i}^{j})

= d_{12}^{1} ⋯ d_{12}^{j} ∑_{j=0}^{m−1} q_{12}^{ji}q_{11}^{j}q_{21}^{ji}q_{22}^{ji^{2}}(i)_{q_{22}}^{!} z_{i}^{m−1}

= d_{12}^{1} ⋯ d_{12}^{j}(m)_{q_{11}q_{12}^{i}q_{21}^{i}q_{22}^{i^{2}}} z_{i}^{m−1},

from where z_{i}^{m} ≠ 0 if m < N(q_{11}q_{12}^{i}q_{21}^{i}q_{22}^{i^{2}}) = M_{i}. It is clear also that if j > i then ∂_{x_{2}}^{j}∂_{x_{1}}(z_{i}^{m}) = 0. Suppose inductively that for s < r, the set

{x_{1}^{n_{1}}z_{1}^{m_{1}} ⋯ z_{s}^{m_{s}} | 0 ≤ n_{1} < N_{1}, 0 ≤ m_{i} < M_{i}}

is linearly independent. Then we have that if there were a linear combination

∑ α_{n_{1},m_{1},...,m_{s+1}}x_{1}^{n_{1}}z_{1}^{m_{1}} ⋯ z_{s+1}^{m_{s+1}} = 0,

we would get, applying (∂_{x_{2}}^{s+1}∂_{x_{1}})^{M_{s+1}−1}, a linear combination

∑ α̃_{n_{1},m_{1},...,m_{s},M_{s+1}−1}x_{1}^{n_{1}}z_{1}^{m_{1}} ⋯ z_{s}^{m_{s}} = 0,

where the α̃’s are the α’s multiplied by non-zero factors. By the inductive hypothesis, we have that α_{n_{1},m_{1},...,m_{s},M_{s+1}−1} = 0. We apply now [∂_{x_{2}}^{s+1}∂_{x_{1}}]^{M_{s+1}−2} and get that α_{n_{1},m_{1},...,m_{s},M_{s+1}−2} = 0. Continuing in this way, we get that α_{n_{1},m_{1},...,m_{s},m_{s+1}} = 0 for all m_{s+1}, proving the inductive thesis. Now the assertion follows at once noting that this set lies in ker(∂_{x_{2}}), and then by 2.7 part 1 the set

(2.12)  {x_{1}^{n_{1}}z_{1}^{m_{1}} ⋯ z_{r}^{m_{r}}x_{2}^{n_{2}} | 0 ≤ n_{i} < N_{i}, 0 ≤ m_{i} < M_{i}}

is linearly independent in 𝔅(V). Notice that we have not proved that z_{i}^{M_{i}} = 0.

Now, for the nilpotency order of Ad_{x_{1}} on x_{2} to be 2 it can happen that q_{11} = −1 or that q_{11}q_{12}q_{21} = 1. In the first case we have x_{1}^{2} = 0, and then

z_{1}x_{1} = −q_{21}x_{1}x_{2}x_{1} = q_{11}q_{21}x_{1}z_{1}.

In the second case it is easy to see using derivations that we also have

z_{1}x_{1} = q_{11}q_{21}x_{1}z_{1},

for ∂_{x_{2}} annihilates and ∂_{x_{1}} gives the same element in 𝔅(V) when applied to both sides of the equality.

We consider now the three cases in the statement.

(1) We have 0 = Ad_{x_{2}}(z_{1}) = x_{2}z_{1} − q_{21}q_{22}z_{1}x_{2}, whence x_{2}z_{1} = q_{21}q_{22}z_{1}x_{2}. These equalities show how to write any monomial in 𝔅(V) as a combination of the elements of the set {x_{1}^{n_{1}}z_{1}^{m_{1}}x_{2}^{n_{2}}}, from where we shall have dim 𝔅(V) ≤ N_{1}N_{2}M_{1} once we prove that z_{1}^{M_{1}} = 0, which we will do at the end of the proof.

(2) We have z_{2} = Ad_{x_{2}}(z_{1}) = x_{2}z_{1} − q_{21}q_{22}z_{1}x_{2}, and hence

0 = Ad_{x_{2}}(z_{2}) = x_{2}z_{2} − q_{21}q_{22}^{2}z_{2}x_{2}.

It can be seen, using derivations, that z_{2}x_{1} = q_{21}^{2}q_{11}x_{1}z_{2} + (q_{11} − q_{22})q_{21}z_{1}^{2}. Furthermore, it can also be seen that z_{2}z_{1} = q_{11}q_{12}q_{21}^{2}q_{22}^{2}z_{1}z_{2}. We conclude now as in the previous case, since these equalities show how to write any monomial as a combination of those of (2.12).

(3) The same computations as in the previous case hold here, except for the relation between z_{2}z_{1} and z_{1}z_{2}. We are looking for the condition on the matrix for (2.12) to be a basis of 𝔅(V). For this to happen, we must be able to write z_{2}z_{1} as a linear combination of those elements; but now, the bi-degree of z_{2}z_{1} is (3, 2) and, moreover, it lies in the kernel of ∂_{x_{2}}. By 2.7 part 1, we have to write z_{2}z_{1} as a combination of the elements in (2.12) of bi-degree (3, 2) which also lie in ker ∂_{x_{2}}, but there is only one such element: z_{1}z_{2}. Hence we should have z_{2}z_{1} = λz_{1}z_{2}. Taking ∂_{x_{1}} in both sides, we are led to the equations

− λ(1 − t)tq_{12}q_{22}^{2} = (1 − t)(1 + t − q_{22}t^{2}),
λ(1 − t)(1 − qt) = −(1 − t)(1 − qt)tq_{21}q_{22}^{2},

for t = q_{12}q_{21}. Since (1 − t) ≠ 0 and (1 − qt) ≠ 0, these equations are equivalent to

0 = 1 + t − q_{22}t^{2} − q_{22}t^{3} = q_{22}(1 + t)(q_{22} − t)(q_{22} + t),

from where the conditions we stated are clear. We shall prove now that in these cases z_{i}^{M_{i}} = 0 for i = 1, 2. For this, we shall

use that for n ≥ k ≥ 0 we have

(2.13)  ∑_{i=k}^{n} q^{i} ( \begin{matrix}i \\ k \end{matrix} )_{q} = q^{k} ( \begin{matrix}n + 1 \\ k + 1 \end{matrix} )_{q} ,

which can easily be proved using that q^{n−i} ( \begin{matrix}n \\ i \end{matrix} )_{q} + ( \begin{matrix}n \\ i + 1 \end{matrix} )_{q} = ( \begin{matrix}n + 1 \\ i + 1 \end{matrix} )_{q} .

We know that ∂_{2}(z_{i}^{n}) = 0 ∀n > 0, and then we must prove that ∂_{1}(z_{i}^{M_{i}}) = 0. Let Q_{i} = q_{11}(q_{12}q_{21})^{i}q_{22}^{i^{2}} (then M_{i} = N(Q_{i})). Suppose r = 2. We have, using the Leibniz rule for ∂_{1} and the identities proved above,

∂_{1}(z_{1}^{M_{1}}) = d_{12}^{1} ∑_{i=0}^{M_{1}−1}(q_{11}q_{12})^{i}z_{1}^{M_{1}−i−1}x_{2}z_{1}^{i}

= d_{12}^{1} ∑_{i=0}^{M_{1}−1}(q_{11}q_{12})^{i} [(q_{21}q_{22})^{i}z_{1}^{M_{1}−1}x_{2} + ∑_{j=0}^{i−1}(q_{21}q_{22})^{j}z_{1}^{M_{1}−i+j−1}z_{2}z_{1}^{i−j−1}]

= d_{12}^{1} ∑_{i=0}^{M_{1}−1} Q_{1}^{i}z_{1}^{M_{1}−1}x_{1} + d_{12}^{1} ∑_{0≤j<i≤M_{1}−1} Q_{1}^{2i−j−1}(q_{21}q_{22})^{−1}z_{1}^{M_{1}−2}z_{2}

= d_{12}^{1}(M_{1})_{Q_{1}} z_{1}^{M_{1}−1}x_{1} + d_{12}^{1}(q_{21}q_{22})^{−1}( ∑_{i=0}^{M_{1}−1} Q_{1}^{i}(i)_{Q_{1}} )z_{1}^{M_{1}−2}z_{2}

= 0 + d_{12}^{1}(q_{21}q_{22})^{−1}Q_{1} ( \begin{matrix}M_{1} \\ 2 \end{matrix} )_{Q_{1}} z_{1}^{M_{1}−2}z_{2} = 0 (by (2.13)),

since it is easy to see that if M_{1} = 2 then z_{2} = 0. If r = 1 then z_{2} = 0 and the same computation works. For z_{2}^{M_{2}}, we have

∂_{1}(z_{2}^{M_{2}}) = d_{12}^{1}d_{12}^{2} ∑_{i=0}^{M_{2}−1}(q_{11}q_{12}^{2})^{i}z_{2}^{M_{2}−i−1}x_{2}^{2}z_{2}^{i} = d_{12}^{1}d_{12}^{2} ∑_{i=0}^{M_{2}−1} Q_{2}^{i}z_{2}^{M_{2}−1}x_{2}^{2}
= d_{12}^{1}d_{12}^{2}(M_{2})_{Q_{2}} z_{2}^{M_{2}−1}x_{2}^{2} = 0.
□

Remark 2.14. For (V, c) a BP of diagonal GT, let the matrix a_{ij} be defined by a_{ii} = 2 and a_{ij} = 1 − d_{ij} for i ≠ j, with d_{ij} the nilpotency order of Ad_{x_{i}} on x_{j}. This is a generalized Cartan matrix. The proposition shows that if (a_{ij}) is of type A_{2} then a PBW basis can be constructed for the Nichols algebra 𝔅(V) in the same way as for Frobenius–Lusztig kernels of type A_{2}, though these matrices are not of Cartan type in general. It is natural to ask to what extent this method allows to compute the Nichols algebras as generalizations of Frobenius–Lusztig kernels. In part 3 we show that this does not give the general answer, and we find the conditions for this method to work in the case of type B_{2}.

Concrete examples of these algebras over specific groups can be found in [N, §3]. There, the author presents algebras over abelian groups, all of which can be computed with 2.11. For instance, if Γ = C_{4} the cyclic group of order 4 with generator σ, take V = M(σ, χ) ⊕ M(σ^{2}, χ), where i = χ(σ) is a primitive

fourth root of unity. Then the matrix of c is ( \begin{matrix}−1 & i \\ −1 & i \end{matrix} ). Here N_{1} = 2, N_{2} = 4,

r = 1 and the nilpotency order of Ad_{x_{1}} on x_{2} is 2. Then 𝔅(V) is of type A_{2}, and M_{1} = N(−1) = 2, whence dim 𝔅(V) = 16. The pointed Hopf algebra 𝔅(V)#kC_{4} is hence 64-dimensional. As another example, borrowed from [N], take Γ = C_{6} with generator σ, and take V = M(σ, χ^{3}) ⊕ (σ, χ^{2}), where N(χ(σ)) = 6, i.e. χ is a generator of Γ̂. Then the matrix of c is ( \begin{matrix}−1 & ω \\ −1 & ω \end{matrix} ), where ω = χ^{2}(σ) is a third

root of unity. This braiding is of type B_{2} and fits in the part 3 of the proposition, with q_{12}q_{21} = −q_{22}. We have N_{1} = 2, N_{2} = 3, M_{1} = 3, M_{2} = 2, and hence 𝔅(V) is 36 dimensional. More examples can be found in [G1].

3. On the non-diagonal case

We give some necessary conditions a Yetter–Drinfeld module over a finite group algebra must verify to produce a finite dimensional Nichols algebra. As said in 1.4.10, such a module is of group type. We want, however, to keep track of the structure of the group. Lemma 3.1. Let Γ be a finite group, g ∈ Γ and ρ ∈ Γ̂_{g} an irreducible representation of Γ_{g}. Let q = ρ(g) (as noticed in 2.7, it is a scalar) and let N = N(q). Let V = M(g, ρ) and suppose V is of finite Nichols rank. Then

(1) if deg ρ ≥ 3 then N = 2 (i.e. q = −1);

(2) if deg ρ = 2 then 2 ≤ N ≤ 3 (i.e. q = −1 or it is a third root of unity).

Proof. Let W be the space of the representation ρ. For x ∈ W we denote also by x the element 1 ⊗ x ∈ Ind_{Γ_{g}}^{Γ} W = V. Let {x_{1}, …, x_{d}} be a basis of W. We have δ(x) = g ⊗ x ∀x ∈ W, and hence c(x_{i} ⊗ x_{j}) = g ⇀ x_{j} ⊗ x_{i} = qx_{j} ⊗ x_{i}. We have an inclusion 𝔅(W) ↪ 𝔅(V) (in fact, 𝔅(V) is free as a right 𝔅(W)-module by [G3, Th. 3.11]) and hence 𝔅(W) is finite dimensional. Now, W is of diagonal GT with matrix (q_{ij}), q_{ij} = q ∀i, j, and thus it is of Cartan type. Hence, the Cartan matrix of this braiding has 2 in the main diagonal and 2 − N outside the main diagonal. Since by a result of Lusztig (see [AS2, Thm. 3.1]) this matrix must correspond to a finite datum, the condition on N follows. □

As a consequence of this, we see that Nichols algebras which do not come from the abelian case are usually infinite dimensional. In the same vein, we have the following result which appeared as a conjecture in a former version of [AS2]. Proposition 3.2. If Γ has odd order, then there are finitely many modules in _{kΓ}^{kΓ}YD of finite Nichols rank.

Proof. Since _{kΓ}^{kΓ}YD has a finite number of simple modules up to isomorphism, it is enough to see that each simple module M(g, ρ) can appear no more than n times (n depending on M(g, ρ)) in a module V for it to be of finite Nichols rank. Let then W be the space affording ρ, 0 ≠ x ∈ W and q such that g ⇀ x = qx. Suppose V = n · M(g, ρ) + ⋯ (n > 0). Then we take Y = span{x_{1}, …, x_{n}}, where x_{i} is the element of the i-th copy of M(g, ρ) corresponding to x. It is immediate to see that Y is of diagonal GT with matrix (q_{ij}), q_{ij} = q ∀i, j. Now, q ≠ 1 since M(g, ρ) is of finite Nichols rank (since so is V), and q ≠ −1 since Γ is of odd order. The same argument as in the previous lemma tells that 1 ≤ n ≤ 2 if N(q) = 3 and n = 1 if N(q) > 3. □

Without the odd-order assumption the result is known to be false (for instance, for Γ = C_{2}).

Let now Γ be a finite group and O_{g} the conjugacy class of g ∈ Γ. Let ρ : Γ_{g} → Aut(W) be an irreducible representation, and take a basis of M(g, ρ) as follows: let {h_{1}, …, h_{s}} be a set of representatives of left cosets of Γ/Γ_{g} and let {x_{1}, …, x_{r}} be a basis of W. We consider the basis {z_{jl} = h_{j} ⊗ x_{l}}. By the definition of the braiding, it is useful to take into account the way in which O_{g} acts on itself by conjugation. To do this, we name t_{i} = h_{i}gh_{i}^{−1} and define the morphism

f_{g} : Γ → 𝕊_{s}, f_{g}(k)(i) = j if kt_{i}k^{−1} = t_{j}.

Notice that f_{g}(k) fixes i iff k commutes with t_{i}; in particular f_{g}(t_{i})(i) = i for all i. Remark 3.3. Let f : G → H be a morphism between two groups. For g ∈ G, the conjugacy class O_{g} has cardinality the index s = [G : G_{g}]. Let G̃ = f^{−1}H_{f(g)}. It is straightforward to see that the set f(O_{g}) has cardinality the index n = [G : G̃] and that for any h ∈ f(O_{g}) the fiber f^{−1}(h)∩O_{g} has cardinality the index m = [G̃ : G_{g}], whence the cardinality of the orbit O_{g} can be factorized by s = nm. Remark 3.4. Let V = M(g, ρ) where ρ : Γ_{g} → Aut(W) and let x ∈ W. Let q = ρ(g) (as noticed in 2.7, g acts by a scalar, on W). In particular, c(x ⊗ x) = g ⇀ x ⊗ x = qx ⊗ x. Let t ∉ Γ_{g} and y = t ⇀ x. Thus δ(y) = tgt^{−1} ⊗ y and tgt^{−1} acts on y by q, since

tgt^{−1} ⇀ y = tgt^{−1}t ⇀ x = tg ⇀ x = qt ⇀ x = qy.

Thus, c(y ⊗ y) = qy ⊗ y.

The preceding paragraphs suggest a different approach for classifying braided pairs of finite group type, as follows:

Definition 3.5. Let X be a finite set and ▷ : X × X → X a function. We say that (X, ▷) is a crossed set if

(1) for each i ∈ X, the function i ▷ • : X → X, j ↦ i ▷ j, is a bijection,

(2) i ▷ i = i ∀i ∈ X,

(3) j ▷ i = i whenever i ▷ j = j, and

(4) i ▷ (j ▷ k) = (i ▷ j) ▷ (i ▷ k) ∀i, j, k ∈ X.

The example we come from is X being a union of conjugacy classes of a finite group Γ and ▷ the conjugation, i ▷ j = iji^{−1}.

A crossed set provides a set-theoretical solution to the Braid Equation by

(3.6)  c : X × X → X × X, c(i, j) = (i ▷ j, i).

We remark that c being a solution to the Braid Equation is equivalent to the condition 4 in the above definition. The condition 1 is given to guarantee the bijectivity of c, and the conditions 2 and 3 are necessary conditions for X to be “injective”, in the language of [So].

Notice that any set-theoretical solution to the Braid Equation c gives rise to a (set-theoretic) representation of the braid group 𝔹_{n}, namely, X^{n} = X × ⋯ × X (n times), where σ_{i} acts by c in the coordinates i and i + 1.

Although [So, LYZ] consider more general set-theoretical solutions, in both articles the authors prove that any bijective solution (i, j) ↦ (^{i}j, i^{j}) is, in some sense, equivalent to a solution as in (3.6). Specifically, for any solution c̃ there exists a solution c as in (3.6) and a collection of bijective maps J_{n} : X^{n} → X^{n} which intertwine the representations of 𝔹_{n} associated to c̃ and c.

Let (X, ▷) be a crossed set and n ≥ 0. We define C^{n}(X) := {f : X^{n} → k^{×}} (X^{0} is a singleton). Let δ^{n} : C^{n}(X) → C^{n+1}(X) be given by

δ^{0} = 0
δ^{n}(f)(x_{0}, …, x_{n}) = ∏_{i=0}^{n−1} f(x_{0}, …, x_{i−1}, x_{i+1}, …, x_{n})^{(−1)^{i}}

× f(x_{0}, …, x_{i−1}, x_{i} ▷ x_{i+1}, …, x_{i} ▷ x_{n})^{(−1)^{i+1}}.

It is easy to prove that (C^{•}(X), δ) is a cochain complex (we remark that an analog definition for a general set-theoretical solution of the Braid Equation does not yield a complex). We define then the cohomology H^{n}(X) = H^{n}(C^{•}(X), δ) = ker(δ^{n})/ Im(δ^{n−1}).

Let f ∈ C^{2}(X). We define a map c^{f} : kX ⊗ kX → kX ⊗ kX by

c^{f}(i ⊗ j) = f(i, j)i ▷ j ⊗ i,

and extend it linearly. We denote by Braidings(X) the subset

Braidings(X) = {f ∈ C^{2}(X) | c^{f} verifies the Braid Equation}.

We then have Proposition 3.7. Let (X, ▷) be a crossed set and (C^{n}(X), δ) be given as before. Then

(1) H^{1}(X) = k^{π_{0}(X)}, where π_{0}(X) is the set of equivalence classes of the relation generated by j ∼ i ▷ j.

(2) H^{2}(X) = Braidings(X) modulo change of basis of the form i ↦ λ_{i}i for λ_{i} ∈ k^{×} ∀i ∈ X.

Proof. Straightforward (see [G4]). □

Notice that we always have an inclusion k^{×} ↪ H^{2}(X) by q ↦ (f(i, j) = q). Lemma 3.8. Let (X, ▷) be a crossed set and f a 2-cocycle in C^{2}(X). If f takes values on the roots of unity then (kX, c) is of finite GT.

Proof. We have to prove that the group-likes associated with the basis X of kX generate a finite group. These group-likes are nothing but g_{i} ∈ GL(kX), g_{i}(j) = f(i, j)i ▷ j. Let N_{ij} = N(f(i, j)), let N be the least common multiple of the N_{ij}’s and let G ⊂ k^{×} be the group of N-roots of unity. Let Y = X × G and 𝕊_{Y} the symmetric group on the (finite) set Y. It is clear then that there is an inclusion of the group generated by the g_{i}’s into 𝕊_{Y}. □

We can state 3.4 in this setting: if f is a 2-cocycle and k ▷ i = j, then we have f(i, i) = f(j, j). To see this, just compute

1 = δ(f)(k, i, i) = f(i, i)f(j, j)^{−1}f(k, i)^{−1}f(k, i).

4. Rank 3 Nichols algebras

In this section we show that if V is a 3-dimensional BP of GT, then it is either of diagonal GT or V ≃ V_{3,q}, q a root of unity (we give the definition of V_{3,q} in 4.1).

Let (V, c) be a 3-dimensional BP of group type which is not of abelian group type. Let {x̄_{0}, x̄_{1}, x̄_{2}} be a basis as in (1.4.11) and g_{i} ∈ GL(V) the associated automorphisms (i = 0, 1, 2). If two of the g_{i}’s were equal then (V, c) would be of abelian GT. For, suppose g_{1} = g_{2}. If g_{0} = g_{1}, the assertion is obvious. If not, g_{0}(x̄_{1}) ∈ span{x̄_{0}} or g_{0}(x̄_{1}) ∈ span{x̄_{1}, x̄_{2}}. In the first case, we have g_{0}g_{1}g_{0}^{−1} = g_{0} (which would imply that g_{0} = g_{1}), while in the second case we have g_{0}g_{1}g_{0}^{−1} = g_{1} (which would imply that g_{0} commutes with g_{1} and hence the assertion). Therefore, #{g_{0}, g_{1}, g_{2}} = 3. Consider now the subindices of the g_{i}’s and the x_{i}’s to be in ℤ/3. We have g_{i}g_{j}g_{i}^{−1} = g_{−i−j}, for if not we would fall in one of the contradictions of before. Furthermore, (1.4.13) implies that c(x̄_{i} ⊗ x̄_{j}) = f(i, j)x̄_{−i−j} ⊗ x̄_{i} for some f : {0, 1, 2} × {0, 1, 2} → k^{×}. That is, (V, c) can be constructed as in 3.5. Take q = f(0, 0) (i.e., g_{0}(x̄_{0}) = qx̄_{0}), and define x_{0} = x̄_{0}, qx_{1} = g_{2}(x_{0}), qx_{2} = g_{0}(x_{1}). Then it can be seen by hand that c(x_{i} ⊗ x_{j}) = qx_{−i−j} ⊗ x_{i}. To see this, we must compute g_{r}(x_{s}) for r, s ∈ ℤ/3, but this can be accomplished just by using the definition of the x_{i}’s and the relations g_{i}g_{j}g_{i}^{−1} = g_{−i−j}. Alternatively, we can verify that H^{2}(X) = k^{×} for X = ℤ/3 and i ▷ j = −i − j.

Definition 4.1. We denote this space by V_{3,q}. That is, V_{3,q} has a basis {x_{0}, x_{1}, x_{2}} (the subindices in ℤ/3) and the braiding is given by

c(x_{i} ⊗ x_{j}) = qx_{−i−j} ⊗ x_{i}.

Notice that c^{3} = q^{3}, from where the eigenvalues of c belong to {q, ξq, ξ^{2}q} for ξ a root of unity of order 3.

As a consequence, we have a classification of 3-dimensional BP of group type:

Lemma 4.2. Let (V, c) be a 3-dimensional BP of GT. Then either

• (V, c) is of abelian GT, or

• (V, c) = V_{3,q} for some q ∈ k^{×}.

Then, if Γ is a finite group and V ∈ _{kΓ}^{kΓ}YD is 3-dimensional, we have a basis {x_{0}, x_{1}, x_{2}} such that either

• c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i} for roots of unity q_{ij}, or

• c(x_{i} ⊗ x_{j}) = qx_{−i−j} ⊗ x_{i} (the subindices in ℤ/3) for q a root of unity.

The algebra 𝔅(V_{3,−1}) has been studied in [MS] and is also included in [AG]; it is 12-dimensional and can be presented with generators {x_{0}, x_{1}, x_{2}} and relations

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

Let q be a root of unity of order 3. By the remarks given above on the eigenvalues of c, the morphism 1 + c is injective, and hence 𝔅^{2}(V_{3,q}) has dimension 9. Since the image of ∂_{0} restricted to 𝔅^{2}(V_{3,q}) has dimension 3, its kernel has dimension 6. Furthermore, 2.7 part 3 says that the set {x_{1}^{i}x_{2}^{j}, 0 ≤ i, j < 3, i + j > 2} is linearly independent and it is clearly contained in ker ∂_{0}. Taking into account the vectors 1, x_{1}, x_{2} which lie also in ker ∂_{0}, we see that the Hilbert polynomial of ker ∂_{0} has coefficients greater or equal to P (t) = 1 + 2t + 6t^{2} + 2t^{3} + t^{4}, and then 2.7 part 1 says that dim 𝔅(V_{3,q}) ≥ 3P (1) = 36. Remark 4.3. By 2.7 part 3, dim(𝔅(V_{3,q})) ≥ N(q)^{3}, but the same result says that the equality does not hold. In fact, dim(𝔅(V_{3,−1})) > N(−1)^{3}. Suppose then q ≠ −1. If we had dim(𝔅(V_{3,q})) = N(q)^{3}, then 2.7 part 3 would tell that dim 𝔅^{2}(V_{3,1}) = 6, or equivalently, that dim ker S^{2} = 3. This is easily seen to be false, since for N(q) ≠ 6 we have ker S^{2} = ker(1 + c) = {0}, and for N(q) = 6 we have

ker S^{2} = span{x_{0}x_{1} − qx_{2}x_{0} + q^{2}x_{1}x_{2}, x_{1}x_{0} − qx_{2}x_{1} + q^{2}x_{0}x_{2}}.

The unique Nichols algebra of rank 3 which does not come from the abelian case and whose dimension is < 32 is hence 𝔅(V_{3,−1}). In particular, by 2.7 part 3 we see that it is the lowest dimensional Nichols algebra which does not come from the abelian case.

5. Rank 4 Nichols algebras

We classify now 4-dimensional braided pairs of finite GT. Let V ∈ _{kΓ}^{kΓ}YD be 4-dimensional and suppose V = ⊕_{i}M_{i} = ⊕_{i}M(g_{i}, ρ_{i}) is its decomposition as a sum of irreducible modules.

If M(g, ρ) is 2-dimensional, then it is of abelian GT. If it is 1-dimensional then g is central in Γ. Thus, a sum of 1-dimensional modules and at most one 2-dimensional module in _{kΓ}^{kΓ}YD is of abelian GT. We have the following possibilities:

(1) 4 = 1 + 1 + 1 + 1, i.e. a sum of four 1-dimensional modules. It is of abelian GT.

(2) 4 = 1 + 1 + 2, it is also of abelian GT.

(3) 4 = 1+3. If the module of dimension 3 is of abelian GT, then V is of abelian GT. The possibility we must consider is that the 3-dimensional module be V_{3,q} for some q.

(4) 4 = 2 + 2. Here V is of abelian GT unless ρ_{1} and ρ_{2} be characters and g_{1} do not commute with g_{2}.

(5) 4 = 4. V = M(g, ρ) is irreducible. There are three sub-cases:

(a) g is central and deg ρ = 4. It is of abelian GT (furthermore, by 3.1, ρ(g) = −1 and dim 𝔅(V) = 16).

(b) [Γ : Γ_{g}] = 2 and deg ρ = 2. It is of abelian GT (furthermore, by 3.1, N(ρ(g)) = 2 or 3. If N(ρ(g)) = 3 we have by 2.7 part 2 that dim 𝔅(V) ≥ 27^{2}).

(c) [Γ : Γ_{g}] = 4 and ρ is a character. We consider as before the situations which are not of abelian GT.

5.1. Case 3. Let V = V_{1} ⊕ V_{3,q} with dim V_{1} = 1, V_{1} = M(g, χ). By 2.7 part 2, we have

dim 𝔅(V) ≥ dim 𝔅(V_{1}) dim 𝔅(V_{3,q}) ≥ 36

unless χ(g) = q = −1, case in which dim 𝔅(V) ≥ 24. We denote by χ′, g′ the character and the element of Γ giving V_{3,−1}, i.e. V = M(g, χ) ⊕ M(g′, χ′). Since g is central, it acts on V_{3,−1} by a scalar, say q_{12}. We denote q_{21} = χ(g′). Let σ := q_{12}q_{21}. By 2.7 part 2, dim 𝔅(V) = 24 iff σ = 1. If σ ≠ 1, we consider x a generator of the space affording χ and x_{0}, x_{1}, x_{2} a basis of M(g′, χ′) as in section 4, and define for i = 0, 1, 2

z_{i} = Ad_{x_{i}}(x) = x_{i}x − q_{21}xx_{i}.

Let N = N(σ). It is straightforward (though tedious) to see, using derivations, that the set

{x^{i}z_{0}^{j_{0}} | 0 ≤ i < 2, 0 ≤ j_{0} < N}

is linearly independent. With this, we see also using derivations that the set

{x^{i}z_{0}^{j_{0}}z_{1}^{j_{1}} | 0 ≤ i < 2, 0 ≤ j_{0}, j_{1} < N}

is linearly independent, and finally we see that the set

{x^{i}z_{0}^{j_{0}}z_{1}^{j_{1}}z_{2}^{j_{2}} | 0 ≤ i < 2, 0 ≤ j_{0}, j_{1}, j_{2} < N}

is linearly independent. Furthermore, it is clear that this set is contained in K, the intersection of the kernels of ∂_{x_{0}}, ∂_{x_{1}}, ∂_{x_{2}}. Thus, by 2.7 part 1 (taking W = V_{3,q}), we see that dim 𝔅(V) ≥ 2N^{3} × 12 ≥ 192. Henceforth, in section 6 we shall not consider these cases, except for σ = 1.

5.2. Case 4. We have V = M_{1}⊕M_{2} = M(g_{1}, ρ_{1})⊕M(g_{2}, ρ_{2}), ρ_{1}, ρ_{2} are characters, g_{1}, g_{2} do not commute, and [Γ : Γ_{g_{1}}] = [Γ : Γ_{g_{2}}] = 2. Let g′_{1} = g_{2}g_{1}g_{2}^{−1}, g′_{2} = g_{1}g_{2}g_{1}^{−1}. The conjugacy class of g_{i} is then {g_{i}, g′_{i}} and, by [AG, Lemma 3.1.9], [g_{i}, g′_{i}] = 1. Let H be the subgroup of Γ generated by g_{1}, g′_{1}, which is commutative. Let σ ∈ Aut H be the restriction of the adjoint σ(h) = g_{2}hg_{2}^{−1}. We have σ^{2} = id, and let t = g′_{1}g_{1}^{−1}. Notice that

g′_{2} = g_{1}g_{2}g_{1}^{−1} = g_{2}g_{2}^{−1}g_{1}g_{2}g_{1}^{−1} = g_{2}g′_{1}g_{1}^{−1} = g_{2}t,
g_{2} = g_{1}g′_{2}g_{1}^{−1} = g′_{2}g′_{1}g_{1}^{−1} = g′_{2}t = g_{2}t^{2},

whence t^{2} = 1. Let then q_{1} = ρ_{1}(g_{1}), q_{2} = ρ_{2}(g_{2}), η_{1} = ρ_{1}(t), η_{2} = ρ_{2}(t) (thus η_{1} = ±1, η_{2} = ±1), and α_{1} = ρ_{1}(g_{2}^{2}), α_{2} = ρ_{2}(g_{1}^{2}). We have then the bases {x_{1}, x′_{1} = g_{2} ⇀ x_{1}} for M_{1} and {x_{2}, x′_{2} = g_{1} ⇀ x_{2}} for M_{2}, and the braidings are given respectively by

matrix of c in M_{1} = ( \begin{matrix}q_{1} & η_{1}q_{1} \\ η_{1}q_{1} & q_{1} \end{matrix} ) , matrix of c in M_{2} = ( \begin{matrix}q_{2} & η_{2}q_{2} \\ η_{2}q_{2} & q_{2} \end{matrix} ) ,

and on the other monomials the braiding acts as

c(x_{1} ⊗ x_{2}) = x′_{2} ⊗ x_{1},  c(x_{2} ⊗ x_{1}) = x′_{1} ⊗ x_{2},
c(x_{1} ⊗ x′_{2}) = α_{2}x_{2} ⊗ x_{1},  c(x_{2} ⊗ x′_{1}) = α_{1}x_{1} ⊗ x_{2},
c(x′_{1} ⊗ x_{2}) = η_{2}x′_{2} ⊗ x′_{1},  c(x′_{2} ⊗ x_{1}) = η_{1}x′_{1} ⊗ x′_{2},
c(x′_{1} ⊗ x′_{2}) = η_{2}α_{2}x_{2} ⊗ x′_{1},  c(x′_{2} ⊗ x′_{1}) = η_{1}α_{1}x_{1} ⊗ x′_{2}.

We notice that, as in the proof of 3.1, the modules M_{i} (i = 1, 2) are of Cartan

type with associated matrices ( \begin{matrix}2 & 2 − N_{i} \\ 2 − N_{i} & 2 \end{matrix} ), where N_{i} = N(q_{i}). Thus 𝔅(V)

is infinite dimensional if N_{1} > 3 or N_{2} > 3. Furthermore, if N_{1} = 3 then 𝔅(M_{1}) has dimension 27 and analogously for M_{2}, whence, by 2.7 part 2, if N_{1} > 2 or N_{2} > 2 then dim 𝔅(V) ≥ 4 × 27 = 108.

We then consider q_{1} = q_{2} = −1. There are two cases: (η_{1}, η_{2}) = (1, 1) and (η_{1}, η_{2}) ≠ (1, 1). In the case (η_{1}, η_{2}) = (1, 1), we define for ϵ = ±1 the vectors

z_{ϵ} = \sqrt{α_{1}}x_{1} + ϵx′_{1}, z′_{ϵ} = \sqrt{α_{2}}x_{2} + ϵx′_{2},

which give a basis of V. It is straightforward to compute the braiding in this basis, which is given by

c(z_{ϵ} ⊗ z_{ϵ′}) = −z_{ϵ′} ⊗ z_{ϵ},
c(z_{ϵ} ⊗ z′_{ϵ′}) = \sqrt{α_{2}}ϵ′z′_{ϵ′} ⊗ z_{ϵ},
c(z′_{ϵ′} ⊗ z_{ϵ}) = \sqrt{α_{1}}ϵz_{ϵ} ⊗ z′_{ϵ′},
c(z′_{ϵ} ⊗ z′_{ϵ′}) = −z′_{ϵ′} ⊗ z′_{ϵ},

from where we see that 𝔅(V) comes from the abelian case. If we take as a subspace M_{ϵ,ϵ′} to be the linear span of {z_{ϵ}, z′_{ϵ′}} we have that the braiding has the matrix ( \begin{matrix}−1 & \sqrt{α_{2}}ϵ′ \\ \sqrt{α_{1}}ϵ & −1 \end{matrix} ). By virtue of 2.11, 𝔅(M_{ϵ,ϵ′}) has dimension 4N(ϵϵ′\sqrt{α_{1}}\sqrt{α_{2}}),

from where

dim(𝔅(V)) ≥ dim(𝔅(M_{ϵ,ϵ′})) dim(𝔅(M_{−ϵ,−ϵ′})) = 16(N(ϵϵ′\sqrt{α_{1}}\sqrt{α_{2}}))^{2} ≥ 64,

taking ϵ, ϵ′ in such a way that ϵϵ′\sqrt{α_{1}}\sqrt{α_{2}} ≠ 1. Thus we will not be interested in these algebras in section 6. Remark 5.2.1. A Nichols algebra as these can be found in [MS], where the authors consider a 4-dimensional Yetter–Drinfeld module over k𝔻_{4} which can be decomposed as a sum of two 2-dimensional ones. That algebra is associated to the constants q_{1} = q_{2} = −1, η_{1} = η_{2} = 1, α_{1} = α_{2} = 1. It can be seen taking M_{+,−} and M_{−,+} that each 𝔅(M_{±,∓}) is of type A_{2} and hence 8-dimensional, and that for the elements of M_{±,∓} ⊗ M_{∓,±} the braiding verifies c^{2} = id, from where, by 2.7 part 2, dim 𝔅(V) = 64. If (η_{1}, η_{2}) ≠ (1, 1), we suppose η_{1} = −1 (if (η_{1}, η_{2}) = (1, −1) we interchange M_{1} and M_{2}). We take z′_{ϵ} as before, and define

t_{+} = Ad_{x_{1}} z′_{+} = x_{1}z′_{+} − \sqrt{α_{1}}z′_{+}x_{1},
t_{−} = Ad_{x_{1}} z′_{−} = x_{1}z′_{−} + \sqrt{α_{1}}z′_{−}x_{1}.

It is straightforward (but tedious) to see using derivations that t_{+}, t_{−} are linearly independent, and that the set {t_{+}^{i_{+}}t_{−}^{i_{−}}(z′_{+})^{j_{+}}(z′_{−})^{′j_{−}} | 0 ≤ i_{+}, i_{−}, j_{+}, j_{−} < 2} is l.i. too. Since this set is contained in ker ∂_{x_{1}}∩ker ∂_{x′_{1}}, it is immediate by a generalization of 2.7 part 1 (see [G3]) that the set

{t_{+}^{i_{+}}t_{−}^{i_{−}}(z′_{+})^{j_{+}}(z′_{−})^{j_{−}}(x_{1})^{t_{1}}(x′_{1})^{t_{2}} | 0 ≤ i_{+}, i_{−}, j_{+}, j_{−}, t_{1}, t_{2} < 2}

is linearly independent, whence dim 𝔅(V) ≥ 64. Hence, in section 6 we shall not consider these algebras.

5.3. Case 5c. We have V = M(g, ρ), with O_{g} = {t_{0} = g, t_{1}, t_{2}, t_{3}} and ρ a character. We consider as in the paragraph preceding 3.3, f_{g} : Γ → 𝕊_{4}. By 3.3 we have that the cardinality of f_{g}(O_{g}) may be 1, 2, 4. If #f_{g}(O_{g}) = 1 then O_{g} generates a commutative subgroup and hence V is of abelian GT.

If #f_{g}(O_{g}) = 2 we may index the t_{i}’s in such a way that

f_{g}(t_{0}) = f_{g}(t_{1}) = (2 3), f_{g}(t_{2}) = f_{g}(t_{3}) = (0 1)

(since f_{g}(t_{0}) = f_{g}(t_{1}) then t_{0} commutes with t_{1}, whence either f_{g}(t_{0}) = id –which would imply #f_{g}(O_{g}) = 1– or f_{g}(t_{0}) = (2 3). Analogously for f_{g}(t_{2}).). Let h ∈ Γ be such that ht_{0}h^{−1} = t_{2}; then f_{g}(h)(0) = 2. It is easy to see (multiplying h by the t_{i}’s on both sides if necessary) that ∃h′ ∈ Γ such that f_{g}(h′) = (0 2)(1 3). Let x_{0} be a generator of the space affording ρ, x_{1} = t_{2} ⇀ x_{0}, x_{2} = h′ ⇀ x_{0}, x_{3} = t_{0}h′ ⇀ x_{0}. It is straightforward to see that 𝔅(V) is a particular case of the algebras of 5.2. In fact, if q = ρ(t_{0}), η = ρ(t_{0}^{−1}t_{1}) and α = ρ(t_{2}^{2}) then 𝔅(V) ≃ 𝔅(M_{q,q,η,η,α,α}), and then dim 𝔅(V) ≥ 64. We shall not consider them in section 6.

If #f_{g}(O_{g}) = 4, then f_{g}(t_{i}) fixes i and does not fix j for j ≠ i, and then it is a tri-cycle. We may index the t_{i}’s in such a way that f_{g}(t_{0}) = (1 2 3). It can be seen that the hypothesis f_{g}(t_{1}) = (0 2 3) leads to a contradiction, and thence we have f_{g}(t_{1}) = (0 3 2). This implies f_{g}(t_{2}) = (0 1 3) and f_{g}(t_{3}) = (0 2 1). Take h = t_{2}t_{1}t_{0}^{−2}. Then f_{g}(h) = id, and thus h commutes with O_{g} (notice that for the same reason t_{i}^{3} commutes with O_{g} for i = 0, 1, 2, 3). We note that t_{1}^{3} = t_{2}t_{0}^{3}t_{2}^{−1} = t_{0}^{3}, and in the same way we see that t_{i}^{3} = t_{j}^{3}. Notice also that

h = t_{2}t_{1}t_{0}^{−2} = t_{0}^{−2}t_{0}^{2}(t_{2}t_{1})t_{0}^{−2} = t_{0}^{−2}t_{1}t_{3}, and
h^{2} = t_{0}^{−2}t_{1}t_{3}t_{2}t_{1}t_{0}^{−2} = t_{0}^{−2}t_{1}t_{3}t_{2}t_{3}^{−2}t_{1} = t_{0}^{−2}t_{1}t_{1}t_{3}t_{3}^{−2}t_{1}
= t_{0}^{−2}t_{1}^{2}t_{3}^{−1}t_{1} = t_{0}^{−2}t_{0}^{−1}t_{1}^{2}t_{1} = t_{0}^{−3}t_{1}^{3} = 1.

Take x_{0} a generator of the space affording ρ, let q = ρ(g) and take qx_{1} = t_{2} ⇀ x_{0}, qx_{2} = t_{0} ⇀ x_{1}, qx_{3} = t_{0} ⇀ x_{2}. Let α = ρ(h) = ±1; we compute

t_{0} ⇀ x_{0} = qx_{0}, t_{0} ⇀ x_{1} = qx_{2}, t_{0} ⇀ x_{2} = qx_{3},
t_{0} ⇀ x_{3} = q^{−2}t_{0}^{3} ⇀ x_{1} = q^{−3}t_{0}^{3}t_{2} ⇀ x_{0} = q^{−3}t_{2}t_{0}^{3} ⇀ x_{0} = t_{2} ⇀ x_{0} = qx_{1},
t_{1} ⇀ x_{0} = t_{0}^{2}t_{2}t_{0}^{−2} ⇀ x_{0} = q^{−2}t_{0}^{2}t_{2} ⇀ x_{0} = q^{−1}t_{0}^{2} ⇀ x_{1} = qx_{3},
t_{1} ⇀ x_{1} = qx_{1} by 3.4,
t_{1} ⇀ x_{2} = t_{2}^{−1}t_{2}t_{1}t_{0}^{−2}t_{0}^{2} ⇀ x_{2} = q^{2}t_{2}^{−1}h ⇀ x_{1} = q^{2}αt_{2}^{−1} ⇀ x_{1} = qαx_{0},
t_{1} ⇀ x_{3} = t_{0}^{2}t_{0}^{−2}t_{1}t_{3}t_{3}^{−1} ⇀ x_{3} = q^{−1}αt_{0}^{2} ⇀ x_{3} = qαx_{2},
t_{2} ⇀ x_{0} = qx_{1},
t_{2} ⇀ x_{1} = t_{2}t_{1}t_{0}^{−2}t_{0}^{2}t_{1}^{−1} ⇀ x_{1} = q^{−1}t_{2}t_{1}t_{0}^{−2}t_{0}^{2} ⇀ x_{1} = qαx_{3},
t_{2} ⇀ x_{2} = qx_{2} by 3.4,
t_{2} ⇀ x_{3} = t_{2}^{3}t_{2}^{−2} ⇀ x_{3} = q^{−1}α^{−1}t_{2}^{3}t_{2}^{−1} ⇀ x_{1} = q^{−2}α^{−1}t_{2}^{3} ⇀ x_{0}
= q^{−2}αt_{0}^{3} ⇀ x_{0} = qαx_{0},
t_{3} ⇀ x_{0} = t_{0}t_{2}t_{0}^{−1} ⇀ x_{0} = t_{0} ⇀ x_{1} = qx_{2},
t_{3} ⇀ x_{1} = t_{0}t_{2}t_{0}^{−1} ⇀ x_{1} = q^{−1}t_{0}t_{2} ⇀ x_{3} = αt_{0} ⇀ x_{0} = qαx_{0},
t_{3} ⇀ x_{2} = t_{0}t_{2}t_{0}^{−1} ⇀ x_{2} = q^{−1}t_{0}t_{2} ⇀ x_{1} = αt_{0} ⇀ x_{3} = qαx_{1},
t_{3} ⇀ x_{3} = qx_{3} by 3.4.

We shall denote by V_{4,q,α} the 4-dimensional module generated by {x_{0}, x_{1}, x_{2}, x_{3}} with the braiding of above. By 2.7 part 3, we have dim(𝔅(V_{4,q,α})) ≥ 81 if N(q) ≥ 3, whence we will be interested in the case q = −1.

5.3.2. If α = −1, it is easy to see that for V_{4,−1,−1} the restriction of c to the linear span of {x_{i} ⊗ x_{j} | i ≠ j} verifies c^{3} = 1, from where

ker S^{2} = ker(1 + c) = span{x_{i}^{2}, i = 0, 1, 2, 3}.

This implies that dim 𝔅^{2}(V_{4,−1,−1}) = 12. Now, since the image of the map ∂_{0} : 𝔅^{2}(V_{4,−1,−1}) → 𝔅^{1}(V_{4,−1,−1}) is 3-dimensional (we have x_{0} ∉ Im(∂_{0}) because x_{0}^{2} = 0), we have dim(ker(∂_{0}) ∩ 𝔅^{2}(V_{4,−1,−1})) = 9. Thus, the Hilbert polynomial of 𝔅(V_{4,−1,−1}) verifies

P_{𝔅(V_{4,−1,−1})}(t) ≥ (1 + t)(1 + 3t + 9t^{2} + 3t^{3} + t^{4})

by virtue of 2, whence dim 𝔅(V_{4,−1,−1}) ≥ 34. We shall not consider this algebra in section 6.

For the module V_{4,−1,1}, it is easy to find the relations in degree 2:

x_{i}^{2} = 0, i = 0, 1, 2, 3
x_{0}x_{1} + x_{2}x_{0} + x_{1}x_{2} = 0,
x_{0}x_{2} + x_{3}x_{0} + x_{2}x_{3} = 0,
x_{0}x_{3} + x_{1}x_{0} + x_{3}x_{1} = 0,
x_{1}x_{3} + x_{2}x_{1} + x_{3}x_{2} = 0.

With the order x_{0} > x_{1} > x_{2} > x_{3}, to give a Gröbner basis of the ideal generated we have to add the relations

x_{1}x_{2}x_{1} − x_{2}x_{1}x_{2} = 0,
x_{2}x_{3}x_{2} − x_{3}x_{2}x_{3} = 0.

It is not so easy^{3} to see that there is one more relation for 𝔅(V_{4,−1,1}):

x_{2}x_{3}x_{4}x_{2}x_{3}x_{4} + x_{3}x_{4}x_{2}x_{3}x_{4}x_{2} + x_{4}x_{2}x_{3}x_{4}x_{2}x_{3} = 0.

With this, it is straightforward (though a little bit tedious) to see that the dimension of 𝔅^{i}(V_{4,−1,1}) is 1, 4, 8, 11, 12, 12, 11, 8, 4, 1 for i = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively, and hence dim(𝔅(V_{4,−1,1})) = 72. We thus shall not consider this algebra.

6. Classification of Nichols algebras of dimension < 32

The Nichols algebras (and also their liftings) of dimension p and p^{2} are classified in [G3]. We shall consider now the other possibilities.

Let (V, c) be a BP of finite GT with Nichols rank p^{3}. We have r(n) = 3 (see 2.8 for the definition of r(n)), from where dim V ≤ 3. By the results in section 4, V is of abelian GT, for, if not, we would have V = V_{3,q}, but then by 2.7 part 2, N(q) = p and by 4.3, the Nichols rank of V would be > p^{3}, a contradiction. Hence, if dim V = 3, 𝔅(V) is a QLS because of 2.7 part 2, i.e., V has a basis {x_{1}, x_{2}, x_{3}} such that the set {x_{1}^{n_{1}}x_{2}^{n_{2}}x_{3}^{n_{3}} | 0 ≤ n_{i} < p} is a basis of 𝔅(V). If dim V = 2

then c has a matrix ( \begin{matrix}q_{11} & q_{12} \\ q_{21} & q_{22} \end{matrix} ) where the q_{ii}’s are either p-roots or p^{2}-roots of

unity. If one of them has order p^{2}, say q_{22}, then N(q_{11}) = p and 𝔅(V) is a QLS. If N(q_{11}) = N(q_{22}) = p then by 2.11 we have two possibilities:

(1) p = 2 and q_{12}q_{21} ≠ 1. Furthermore, for 𝔅(V) to be 8-dimensional we must have q_{12}q_{21} = −1. The algebra 𝔅(V) is of type A_{2}.

(2) p > 2 and q_{12}q_{21}q_{22} = 1. Interchanging q_{11} with q_{22} we see that also q_{12}q_{21}q_{11} = 1, whence q_{11} = q_{22} and 𝔅(V) is also of type A_{2}.

Thus we reach in both cases to the same situation. The algebra 𝔅(V) has a PBW basis given by {x_{1}^{m_{1}}z^{n}x_{2}^{m_{2}} | 0 ≤ m_{1}, m_{2}, n < p}, z = x_{2}x_{1} − q_{21}x_{1}x_{2}. The last case to be considered is dim V = 1. We have simply {x} a basis of V such that c(x ⊗ x) = qx ⊗ x, N(q) = p^{3}. Thus 𝔅(V) = k[x]/(x^{p^{3}}).

We then have proved:

3I used a computer program by E. Müller [Mü] to see that there should be one more relation in degree 6.

Theorem 6.1. Let (V, c) be a BP of finite group type with Nichols rank p^{3} (p a prime number). Then V is of diagonal GT and it fits in one of the following possibilities:

dim V Type
3 QLS, N(q_{11}) = N(q_{22}) = N(q_{33}) = p
2 QLS, N(q_{11}) = p, N(q_{22}) = p^{2}
2 A_{2}, q_{11} = q_{22} = (q_{12}q_{21})^{−1}, N(q_{11}) = p
1 QLS, N(q_{11}) = p^{3}
□

Let n = p_{1}p_{2} a product of two distinct primes such that p_{1} < p_{2} < p_{1}^{2}. Then r(n) < 3, whence dim V ≤ 2 and it is of abelian GT. If dim V = 2 and c has matrix

( \begin{matrix}q_{11} & q_{12} \\ q_{21} & q_{22} \end{matrix} ), then by 2.7 part 2 we must have (reversing the basis if necessary)

N(q_{11}) = p_{1}, N(q_{22}) = p_{2}, and q_{12}q_{21} = 1, i.e., 𝔅(V) is a QLS with PBW basis {x_{1}^{m_{1}}x_{2}^{m_{2}} | 0 ≤ m_{i} < p_{i}, i = 1, 2}. If dim V = 1 then as before x generates V, c(x ⊗ x) = qx ⊗ x with N(q) = p_{1}p_{2} and 𝔅(V) = k[x]/(x^{p_{1}p_{2}}).

Since we have computed the dimension of 𝔅(V_{3,−1}), together with the fact that we have shown that dim 𝔅(V_{3,q}) ≥ 32 if q ≠ −1 and dim 𝔅(V) ≥ 32 for V a 4dimensional BP of finite GT which is not of abelian GT, this last argument applies also for p_{1} = 2 and p_{2} < 16 (i.e., n = 10, n = 14, n = 22 and n = 26). In these cases, if dim V were 3, then V would be of abelian GT because if not V = V_{3,−1}, which would give a Nichols algebra of dimension 12, a contradiction. Then remark 2.9 applies and dim V ≤ 2.

We have hence the classification of the Nichols algebras of dimensions 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31. For the remaining dimensions it is routine to make the computations, and we refer to [G4]. The result can be stated as follows:

Type  Dimensions

QLS, rank 1

All dimensions

QLS, rank 2  ≠ p
QLS, rank 3 8, 12, 16, 18, 20, 24, 27, 28, 30
QLS, rank 4  16, 24
A_{2}  8, 12, 16, 20, 27, 28
A_{2} × A_{1}  16, 24
V_{3,−1}  12
V_{3,−1} × A_{1}  24

When applying the lifting procedure, the computations in the previous section are the first step in order to classify pointed Hopf algebras of index < 32. The following lemmas, which can be proved as in [AS4, Lemma 7.2] or [G1, §7] and are referred to [G4], show that any such Hopf algebra is generated by group-likes and skew-primitive elements, completing the second step of the procedure. Lemma 6.2. Let (V = (x_{1}, x_{2}), c) be an BP of abelian GT with matrix (q_{ij}), and suppose dim 𝔅(V) < 32. If S is a graded braided Hopf algebra between T V and 𝔅(V), i.e.

T V ↠ S ↠ 𝔅(V),

with graded maps and categorical maps (that is, there exists a braided category such that T V, S and 𝔅(V) belong to it and the projections are maps in the category), then either S = 𝔅(V) or dim S ≥ 32. □ Lemma 6.3. Let S be a finite dimensional graded braided Hopf algebra with graded maps T (V_{3,−1}) ↠ S ↠^{p} 𝔅(V_{3,−1}). Then p is an isomorphism. □

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Matías Graña Depto. de Matemática Pab I, FCEyN, Ciudad Universitaria (1428) Buenos Aires, Argentina

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