Pointed Hopf algebras of dimension 32

Matías Graña

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

                    e-mail: matiasg@dm.uba.ar

Abstract

We give a complete classification of the 32-dimensional pointed Hopf algebras over an algebraically closed field k with char k ≠ 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [AS1], [BDG] and [Ge] are given families of counterexamples for the tenth Kaplansky conjecture. Up to now, 32 is the lowest dimension where Kaplansky conjecture fails.

1. Introduction

In recent years there has been certain activity in classification problems of pointed Hopf algebras. In [AS1], [BDG], [Ge] independently the authors find infinite families of pointed Hopf algebras of dimension p^{4} (p an odd prime number), solving thus by the negative the tenth Kaplansky conjecture. In [D] there is a classification of pointed Hopf algebras of dimension p^{n} with coradical of dimension p^{n−1}.

In [AS1], [AS2], [AS3] there are classifications of pointed Hopf algebras of dimension p^{3}, p^{4} and a large number of strong results towards the classification of pointed Hopf algebras with commutative coradical. Using the techniques of these articles, namely the “lifting procedure” described below, we classify here the pointed Hopf algebras of dimension 32 over an algebraically closed field of characteristic ≠ 2. Related results can be found in [CDR] (the classification of the pointed Hopf algebras of dimension 16), in [Be] (classification of Ore extension Hopf algebras with coradical k(C_{2} × C_{2}) and kC_{4}), and in [G] (the description of the coradically graded pointed Hopf algebras of dimension p^{5}).

This work was partially supported by CONICET, CONICOR, Secyt (UNC), Secyt (UBA)

One of the key results is Corollary 2.14, where we compute the Nichols algebras 𝔅(V) (see 1.2 for the definition of Nichols algebras), V the Yetter– Drinfeld modules of dimension 2 over the groups of order 2^{j}, 1 ≤ j ≤ 4.

Our main result is

Theorem 1.0.1. Let A be a pointed Hopf algebra of dimension 32 over the algebraically closed field k, char k ≠ 2. Let Γ = G(A) be the group of grouplikes of A. If Γ = C_{2} then A is the algebra of section 3. If Γ = C_{2} × C_{2} then A is one of the 6 algebras of 4.1. If Γ = C_{4} then A is one of the 12 algebras of 4.2. If Γ = C_{2} × C_{2} × C_{2} then A is one of the 6 algebras of 5.1. If Γ = C_{2} × C_{4} then A is in the infinite family of algebras of 5.2. If Γ = C_{8} then A is in the infinite family of algebras of 5.3. If Γ = 𝔻_{4} the dihedral group, then A is one of the 7 algebras of 5.4. If Γ = ℍ the quaternionic group, then A is one of the 3 algebras of 5.5. If Γ = C_{2} × C_{2} × C_{2} × C_{2} then A is the algebra of 6.1. If Γ = C_{2} × C_{2} × C_{4} then A is one of the 7 algebras of 6.2. If Γ = C_{4} × C_{4} then A is one of the 4 algebras of 6.3. If Γ = C_{2} × C_{8} then A is one of the 14 algebras of 6.4. If Γ = C_{16} then A is one of the 7 algebras of 6.5. If Γ is one of the non abelian groups of order 16 then A is one of the 13 algebras of 6.6. If Γ has order 32 then A is a group algebra.

1.1. Some basic definitions and conventions. Throughout the article k shall be an algebraically closed field of characteristic ≠ 2. The degrees for graded algebras shall be denoted by parentheses, i.e. H(i) is the homogeneous component of degree i of the graded algebra H. The symbols i, ξ shall denote fixed roots of unity of orders 4, 8 respectively, such that i = ξ^{2}.

Let H be a Hopf algebra. We denote by _{H}^{H}YD the category of (left-left) Yetter–Drinfeld modules over H, i.e. M is an object of _{H}^{H}YD if it is a left Hmodule (with structure h ⊗ m ↦ h ⇀ m), a left H-comodule (with structure δ(m) = m_{(−1)} ⊗ m_{(0)}) and such that

δ(h ⇀ m) = h_{(1)}m_{(−1)}S(h_{(3)}) ⊗ h_{(2)}m_{(0)}.

We denote by c the braiding in _{H}^{H}YD, which is given by

c = c_{M,N} : M ⊗ N → N ⊗ M, m ⊗ n ↦ m_{(−1)} ⇀ n ⊗ m_{(0)}.

When H is clear from the context, we shall call a module in _{H}^{H}YD a YD-module. Unless explicitly stated, the letter V shall denote a YD-module.

For Γ a group and g ∈ Γ we denote by Γ_{g} the commutator subgroup

Γ_{g} = {h ∈ Γ | hg = gh}.

It is not difficult to prove (see for instance [AG]) that if H = kΓ and |Γ| ≠ 0 in k then _{H}^{H}YD is a semisimple category and the irreducible objects are given as follows: let g ∈ Γ and ρ : Γ_{g} → Aut(W) ∈ Γ̂_{g} an irreducible representation. Then the induced representation

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

with the usual module structure and the comodule structure given by

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

is an irreducible YD-module. Furthermore, let {g_{1}, …, g_{n}} be a subset of Γ consisting of one element for each conjugacy class, and let {ρ_{i}^{1}, …, ρ_{i}^{m_{i}}} be the irreducible representations of Γ_{g_{i}}. Then the objects M(g_{i}, ρ_{i}^{j}) (1 ≤ i ≤ n, 1 ≤ j ≤ m_{i}) gives a full collection of non-isomorphic irreducible objects of _{kΓ}^{kΓ}YD.

We extend the definition of M(g, ρ) to the case in which ρ is not irreducible. Observe that dim M(g, ρ) = deg ρ × [Γ : Γ_{g}]. Hence, if Γ is abelian, the irreducible modules of _{kΓ}^{kΓ}YD are 1-dimensional. In this case, let V = ⊕_{i}M(h_{i}, χ_{i}) where h_{i} ∈ Γ and χ_{i} ∈ Γ̂, and denote by x_{i} a generator of M(h_{i}, χ_{i}). By the definition of the braiding,

c(x_{i} ⊗ x_{j}) = χ_{j}(h_{i})x_{j} ⊗ x_{i}. (1.1.1)

Let q ∈ k. For n ≥ m ∈ ℕ, we use the standard notation

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

Let H be a graded Hopf algebra, H = ⊕_{i}H(i). Let π : H → H(0) and ι : H(0) → H be respectively the canonical projection and the canonical inclusion. Then

R = H^{coπ} = H^{coH(0)} = {x ∈ H | (id ⊗π)Δ(x) = x ⊗ 1}

is a braided Hopf algebra in _{H(0)}^{H(0)}YD. Furthermore, it is graded: R = ⊕_{i}R(i), with R(i) = R ∩ H(i). The algebra H can be reconstructed as a bosonization, or biproduct (see e.g. [AG])

H = R#H(0).

A braided Hopf algebra R in _{K}^{K}YD (K a Hopf algebra) is called Nichols algebra if

(1.1.2) R = ⊕_{i}R(i) is a graded braided Hopf algebra, R(0) = k1, (1.1.3) P(R) = R(1), (1.1.4) (1.1.5) R(1) generates R,

where P(R) stands for the space of primitive elements of R. It is not known whether in characteristic 0 and for R finite dimensional the condition (1.1.5), resp. (1.1.4), is superfluous, though there are some results in this direction. For instance, in [AS1, Thm. 3.2] is proved that if dim R(1) = 1 (R finite dimensional and char k = 0) then ((1.1.2)–(1.1.4))⇒(1.1.5). A Nichols algebra R is uniquely determined by its primitive elements P(R). We use hence the notation R = 𝔅(V) for the Nichols algebra R with R(1) = V, whenever V ∈ _{K}^{K}YD. We specify sometimes the graduation of 𝔅(V) by

𝔅(V) = ⊕_{i}𝔅^{i}(V).

The construction V ↦ 𝔅(V) can be made with the so called quantumshuffles (see for instance [AG]), though this construction does not depend on the braided category _{K}^{K}YD, but only on the pair (V, c). The quantum-shuffles construction applied to any pair (V, c) (c ∈ End(V ⊗V) a solution of the braid equation), the result has been called Quantum Shuffle Algebra in [Ros]. Nichols algebras have some particularly useful properties which distinguish them from this general concept of Quantum Shuffle Algebras, e.g. the kind of Poincaré duality stated in 2.7.

We shall use the following notation for finite abelian groups: let Γ = C_{n_{1}} × ⋯ × C_{n_{t}}. We denote by g_{j} a generator of the j-th factor, i.e.

Γ can be presented by generators g_{j} and relations g_{j}^{n_{j}} = [g_{j}, g_{k}] = 1. Furthermore, let α_{j} = −1, i, ξ, \sqrt{ξ} depending whether n_{j} = 2, 4, 8, 16 (recall that ξ^{2} = i, i^{2} = −1). We denote by ĝ_{i} the character of Γ given by ĝ_{i}(g_{j}) = α_{i}^{δ_{ij}}. Then Γ̂ ≃ Γ with generators ĝ_{i}, i = 1, …, t.

If x_{1}, …, x_{r} are vectors in a space V, we denote by (x_{1}, …, x_{r}) the linear span of them. If g_{1}, …, g_{r} are elements in Γ, we denote by (g_{1}, …, g_{r}) the subgroup generated by them.

If V ∈ _{kΓ}^{kΓ}YD and Γ is abelian then V = ⊕_{i=1}^{θ}M(h_{i}, χ_{i}). We denote by x_{i} a generator of M(h_{i}, χ_{i}), whence V = (x_{1}, …, x_{θ}) and c is given by the matrix (b_{ij}), with b_{ij} = χ_{j}(h_{i}).

1.2. Lifting Procedure. Let A be a finite dimensional pointed Hopf algebra. Let A_{0} ⊂ A_{1} ⊂ … be its coradical filtration, and consider

H = ⊕_{i}H(i) = gr A = ⊕_{i}(A_{i}/A_{i−1}) (A_{−1} = 0).

It is a coradically graded Hopf algebra, i.e. H_{n} = ⊕_{i=0}^{n}H(i). Take R = H^{coH(0)}. This is a coradically graded Hopf algebra in _{kΓ}^{kΓ}YD, where Γ = G(A) is the group of group-likes of A (and hence H(0) = kΓ). Notice that the order of Γ divides the dimension of A, by virtue of the Nichols–Zoeller theorem. Furthermore, R(0) = k1, from where R satisfies (1.1.2), (1.1.3) and (1.1.4). If moreover R satisfies (1.1.5) then it is a Nichols algebra, and hence R = 𝔅(R(1)). Since R is an invariant of A, we shall consider its “infinitesimal part” defined by

V = V (A) = R(1), (1.2.1)

and call dim V the rank of A. It is clear that the rank of A is equal to dim A_{1}/ dim A_{0} − 1.

The lifting procedure goes the other way: let n ∈ ℕ be fixed. We want to classify the n-dimensional pointed Hopf algebras over k. We can proceed as follows: for each group Γ such that |Γ| divides n,

1. Determine the modules V ∈ _{kΓ}^{kΓ}YD such that dim 𝔅(V) ≤ \frac{n}{|Γ|}.

2. Classify the isomorphism classes of the bosonizations 𝔅(V)#kΓ, where V runs over the modules in step 1. Note that two bosonizations 𝔅(V)#kΓ and 𝔅(V)′#kΓ may be isomorphic even though 𝔅(V) and 𝔅(V)′ (thus, V and V′) are not isomorphic (see 2.16).

3. For each isomorphism class 𝔅(V)#kΓ in step 2, classify all the liftings, i.e. all the pointed Hopf algebras (up to isomorphism) such that gr A = 𝔅(V)#kΓ.

4. Prove, if possible, that these are all the n-dimensional pointed Hopf algebras. Equivalently, show that a graded braided Hopf algebra R in _{kΓ}^{kΓ}YD such that dim R = n/|Γ| and satisfying (1.1.2)–(1.1.4), also satisfies (1.1.5) (for this purpose it is useful to classify those V in step 1 such that dim 𝔅(V) < n/|Γ|).

1.3. Contents. The article is organized as follows: in section 2 we compute the Nichols algebras that appear applying the procedure above for 32-dimensional pointed Hopf algebras.

From 3 to 6.6 we follow the steps 1, 2 and 3 of the procedure for each of the groups of order 2^{m}, m ≤ 4. In section 7 we prove the statement in step 4 of the procedure.

This work is part of the doctoral thesis of the author, who thanks N. Andruskiewitsch for his excellent guidance. I thank also H.-J. Schneider and M. Lamas for many valuable conversations.

Our main references on Hopf algebras are [S] and [M]. We refer to the survey article [AG] for definitions and properties of braided Hopf algebras, Nichols algebras (called there TOBAs) and for further references on the subject.

2. The Nichols algebras we shall need

For future reference, we give two families of examples of Nichols algebras. The first one corresponds to Taft algebras, and the second one to the more general case of Quantum Linear Spaces (QLS for brevity).

Example 2.1. Let V be 1-dimensional and let x be a generator of V. Suppose c(x ⊗ x) = qx ⊗ x, where q ∈ k^{×}. If q is a root of unity of order n > 1, or if q = 1 and char k = n > 0, then 𝔅(V) = k[x]/(x^{n}) as algebra (and hence dim 𝔅(V) = n). If q is not a root of unity (which can happen for instance for V ∈ _{kℤ}^{kℤ}YD), or if q = 1 and char k = 0, then 𝔅(V) = k[x] and hence it is infinite dimensional. The comultiplication in both cases is given by Δ(x^{d}) = ∑_{i=0}^{d} \binom{d}{i}_{q} x^{i} ⊗ x^{d−i}.

Remark 2.2. The above example shows the following useful result: let H be a finite dimensional Hopf algebra and R ∈ _{H}^{H}YD a braided Hopf algebra of dimension N. Suppose char k does not divide N. If x ∈ P(R) is such that kx is a sub-YD-module of R and c(x ⊗ x) = qx ⊗ x, where q ∈ k^{×}, then q ≠ 1.

Proof. Suppose q = 1. If char k = 0 then R would be infinite dimensional, if char k = p > 0 then x would generate an algebra of dimension a power of p, but then, taking the bosonization of both algebras, p would divide N.

Definition 2.3. Let q ∈ k^{×}. We define N(q) by

N(q) = \begin{cases}order of q & if q ≠ 1 and is a root of unity, \\ ∞ & if q is not a root of unity, \\ ∞ & if q = 1 and char k = 0, \\ char k & if q = 1 and char k > 0. \end{cases}

Definition 2.4. Let V ∈ _{H}^{H}YD be of dimension θ. We shall say that V (or c) has a matrix if there exists a basis {x_{1}, …, x_{θ}} of V and a matrix (b_{ij}) ∈ M_{θ}(k^{×}) such that the braiding c is given by

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

When we want to specify the basis, we say that V has a matrix (b_{ij}) in the basis {x_{1}, …, x_{θ}}. If H = kΓ and Γ is abelian, then every finite dimensional module has a matrix (see (1.1.1)).

Let V ∈ _{kΓ}^{kΓ}YD be finite dimensional. We shall say that V (or c) comes from the abelian case if there exists an invariant abelian subgroup Γ′ ⊂ Γ such that the image of δ : V → kΓ ⊗ V is included in kΓ′ ⊗ V. In this case, V can be considered as a module in _{kΓ′}^{kΓ′}YD (thus, it has a matrix) and the bosonization 𝔅(V)#kΓ can be reconstructed as an extension of Γ/Γ′ by 𝔅(V)#kΓ′. We note that the condition on Γ′ to be invariant can be dropped: it is not difficult to see that if there exists Γ′ abelian such that the image of δ lies in kΓ′ ⊗ V, then there exists Γ′′ ⊂ Γ′ invariant (and clearly abelian) such that the image of δ lies in kΓ′′ ⊗ V.

Example 2.5. Let V ∈ _{H}^{H}YD be of dimension θ, and suppose V has a matrix (b_{ij}) in the basis {x_{1}, …, x_{θ}}. Let N_{i} = N(b_{ii}), see 2.3. Then it can be proved (see [AS1, Prop. 3.5]) that

dim 𝔅(V) ≥ ∏_{i=1}^{n} N_{i}.

Furthermore, if N_{i} < ∞ for all i, then the equality holds if and only if

b_{ij}b_{ji} = 1 for all i ≠ j. (2.6)

In this case, a PBW basis of 𝔅(V) is given by

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

and the multiplication is given by the following relations:

x_{i}^{N_{i}} = 0, x_{j}x_{i} = b_{ji}x_{i}x_{j} for i < j.

Hence, as an algebra, 𝔅(V) is a quantum linear space. Let now H = kΓ. In order to agree with the notation of [AS1], we shall say that 𝔅(V) is a Quantum Linear Space (or QLS) if V satisfies (2.6) and the lines kx_{i} are submodules of V in _{kΓ}^{kΓ}YD. If V can be considered as a module in _{kΓ′}^{kΓ′}YD for certain subgroup Γ′ ⊂ Γ (that is, if the image of the coaction δ lies in Γ′ ⊗ V) and it is a QLS in _{kΓ′}^{kΓ′}YD then we say that it is a QLS over Γ′.

We shall use an important result about Nichols algebras due to Nichols:

Proposition 2.7. Let R = ⊕_{i=0}^{N}R(i) be a Nichols algebra (in some braided rigid abelian category). Suppose that R(N) ≠ 0. Then

dim R(i) = dim R(N − i) ∀i = 0, …, N.

Proof. See for instance [AG, Pr. 3.2.2].

Lemma 2.8. Let A be a 2^{n}-dimensional pointed Hopf algebra with A_{0} = kΓ. Suppose |Γ| ≥ 2^{n−2} and let V = V (A) ∈ _{kΓ}^{kΓ}YD be the module defined by (1.2.1). Then dim V ≤ 2.

Proof. Since 2^{n} = |Γ| × dim 𝔅(V) we have dim 𝔅(V) ≤ 4; but if dim V ≥ 3, then dim 𝔅(V) ≥ 5 by 2.7.

In [N] the author introduces derivations for Nichols algebras, which turns out to be a useful tool for computing bases (and hence their dimensions).

Definition 2.9. Let V ∈ _{H}^{H}YD. For i + j = n, we denote by

Δ_{i,j} : 𝔅^{n}(V) → 𝔅^{i}(V) ⊗ 𝔅^{j}(V)

the (i, j)-component of the comultiplication of 𝔅(V). It is proved in [Sch] (or see [AG, Def. 3.2.10]) that Δ_{i,j} is injective ∀i, j. Let {x_{1}, …, x_{θ}} be a basis of V and let {x_{1}^{∗}, …, x_{θ}^{∗}} be its dual basis. We denote by ∂_{x_{i}} the differential operator on 𝔅(V) given by

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

By the injectivity of Δ_{i,j} it is clear that for z ∈ 𝔅^{n}(V) (n > 0) we have z = 0 if and only if ∂_{x_{i}}(z) = 0 for all i = 1, …, θ. Suppose now that V ∈ _{kΓ}^{kΓ}YD and some x_{i}^{∗} vanishes on the homogeneous components of degree ≠ g_{i} for some g_{i} ∈ Γ (that is, x_{i}^{∗}(y) = 0 if δ(y) = h ⊗ y and h ≠ g_{i}). Then it is easy to see that ∂_{x_{i}} verifies the Leibniz rule

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

Since usually we shall take bases {x_{1}, …, x_{θ}} of homogeneous elements (i.e. δ(x_{i}) = g_{i} ⊗ x_{i} for some g_{i} ∈ Γ), the previous condition will hold for all x_{i}^{∗}.

Definition 2.10. Let A be a Hopf algebra. We shall denote by Ad the adjoint action

Ad_{x}(y) = x_{(1)}yS(x_{(2)}).

Let B be a braided Hopf algebra with multiplication μ, let c ∈ End(B ⊗ B) be the braiding. We shall denote by Ad the braided adjoint action, given by

Ad_{x}(y) = μ(μ ⊗ S)(id ⊗c)(Δ ⊗ id)(x ⊗ y).

Note that if x ∈ P(B) then the adjoint is nothing but Ad_{x}(y) = μ(id −c)(x⊗y). Since in the cases we are interested in the endomorphism c is given by the context, we do not make a reference to c in the notation.

We need the following relation between both adjoint actions: let H be a graded Hopf algebra, π : H → H(0) the canonical projection and B ∈ _{H(0)}^{H(0)}YD the braided Hopf algebra of coinvariants (then H ≃ B#H(0)). Let x ∈ B(1) (it is a primitive element). Then it is straightforward to see that

Ad_{(b#1)}(c#1) = (Ad_{b}(c))#1 ∀c ∈ B,

where the left Ad is the classical one, while the right Ad is the braided one.

We shall compute now the Nichols algebras appearing in next sections.

Proposition 2.11. Let V ∈ _{H}^{H}YD with basis {x, y}. Suppose that c has a matrix (b_{ij}) in this basis, and suppose that b_{12}b_{21} ≠ 1, b_{11} ≠ 1, b_{22} ≠ 1. Let z = Ad_{x}(y) = xy − b_{12}yx. Let N_{i} = N(b_{ii}) (see 2.3) and suppose that N_{i} < ∞ (i.e., that b_{ii} is a root of unity, or b_{ii} = 1 and char k > 0). Let q = b_{11}b_{12}b_{21}b_{22}.

1. There exists n_{12} (respectively n_{21}) such that (Ad_{x})^{n_{12}}(y) = 0 (respectively (Ad_{y})^{n_{21}}(x) = 0). Let n_{12} (resp. n_{21}) be the least positive integer with such property. Then n_{12} ≤ N_{1} and n_{21} ≤ N_{2}. Suppose that there exists m_{12} (resp. m_{21}) such that b_{12}b_{21}b_{11}^{m_{12}} = 1 (resp. b_{12}b_{21}b_{22}^{m_{21}} = 1) and let m_{12} (resp. m_{21}) be the least non negative integer with such property. Then n_{12} = m_{12} + 1 (resp. n_{21} = m_{21} + 1).

2. If q = 1 and char k = 0 then z^{n} ≠ 0 ∀n and then 𝔅(V) is infinite dimensional. If q = 1 and char k ≠ 0 then dim 𝔅(V) ≥ N_{1}N_{2} char k.

3. Suppose b_{11} = b_{22} = −1 and let N = N(q). Then 𝔅(V) has dimension 4N, with a PBW basis {x^{a}y^{b}z^{c} | 0 ≤ a, b < 2, 0 ≤ c < N}.

Proof. 1. We compute inductively the derivatives of (Ad_{x})^{n}(y) (see 2.9):

∂_{x}((Ad_{x})^{1}(y)) = ∂_{x}(xy − b_{12}yx) = 0.

Suppose that ∂_{x}((Ad_{x})^{n}(y)) = 0, then

∂_{x}((Ad_{x})^{n+1}(y)) = ∂_{x} (x(Ad_{x})^{n}(y) − b_{11}^{n}b_{12}(Ad_{x})^{n}(y)x)
= b_{11}^{n}b_{12}(Ad_{x})^{n}(y) − b_{11}^{n}b_{12}(Ad_{x})^{n}(y) = 0,
∂_{y}((Ad_{x})^{1}(y)) = ∂_{y}(xy − b_{12}yx) = (1 − b_{12}b_{21})x.

Let σ_{n} = (1 − b_{12}b_{21}b_{11}^{n}). Suppose that ∂_{y}((Ad_{x})^{n}(y)) = σ_{0}σ_{1} ⋯ σ_{n−1}x^{n}. Then

∂_{y}((Ad_{x})^{n+1}(y)) = ∂_{y}(x(Ad_{x})^{n}(y) − b_{11}^{n}b_{12}(Ad_{x})^{n}(y)x)
= σ_{0} ⋯ σ_{n−1}x^{n+1} − b_{11}^{n}b_{12}σ_{0} ⋯ σ_{n−1}b_{21}x^{n+1}
= σ_{0} ⋯ σ_{n}x^{n+1}.

Thus, since x^{N_{1}} = 0, we have

∂_{x}((Ad_{x})^{N_{1}}(y)) = ∂_{y}((Ad_{x})^{N_{1}}(y)) = 0,

and hence (Ad_{x})^{N_{1}}(y) = 0, which proves that n_{12} exists and n_{12} ≤ N_{1}. Furthermore, if b_{12}b_{21}b_{11}^{m_{12}} = 1, we see that

∂_{x}((Ad_{x})^{m_{12}+1}(y)) = ∂_{y}((Ad_{x})^{m_{12}+1}(y)) = 0,

whence n_{12} ≤ m_{12} + 1 (notice that if m_{12} exists, then m_{12} < N_{1}). Moreover, the elements x^{i} do not vanish for i < N_{1}, whence if m_{12} exists then ∂_{y}((Ad_{x})^{m_{12}}(y)) ≠ 0, from where n_{12} > m_{12} and thus n_{12} is exactly m_{12} + 1.

2. We compute the derivatives of z^{n}. We do it for a general q = b_{11}b_{12}b_{21}b_{22} (keep the notation σ_{n} = (1 − b_{12}b_{21}b_{11}^{n})).

∂_{x}(z) = ∂_{x}(Ad_{x} y) = 0,
∂_{y}(z) = ∂_{y}(Ad_{x} y) = σ_{0}x,
∂_{x}∂_{y}(z) = ∂_{x}(σ_{0}x) = σ_{0},

∂_{y}(z^{n}) = ∑_{i=0}^{n−1}(b_{21}b_{22})^{i}σ_{0}z^{n−1−i}xz^{i},
∂_{x}∂_{y}(z^{n}) = ∑_{i=0}^{n−1}(b_{21}b_{22})^{i}(b_{11}b_{12})^{i}σ_{0}z^{n−1}
= (n)_{q}σ_{0}z^{n−1}.

Let now q = 1. If char k = 0, we see inductively that z^{n} ≠ 0 ∀n. Furthermore, since z^{n} ∈ 𝔅^{2n}(V), the set {z^{n} | n ≥ 0} is linearly independent, and thus 𝔅(V) is infinite dimensional. If char k > 0, we see inductively that z^{n} ≠ 0 ∀n < char k. Furthermore, with the same computations as in 3 below, we see that the set

{x^{a}y^{b}z^{c} | 0 ≤ a < N_{1}, 0 ≤ b < N_{2}, 0 ≤ c < char k}

is linearly independent, whence dim 𝔅(V) ≥ N_{1}N_{2} char k.

3. By the same computations than in 2, we have that the set {z^{i} | 0 ≤ i < N} is linearly independent. We first prove that z^{N} = 0:

∂_{x}(z^{N}) = 0 (as in 2)
∂_{y}(z^{N}) = ∑_{i=0}^{N−1}(b_{21}b_{22})^{i}σ_{0}z^{N−1−i}xz^{i},

but this element vanishes in 𝔅(V), since

∂_{x}(∂_{y}z^{N}) = (N)_{q}σ_{0}z^{N−1} = 0 (as in 2)
∂_{y}(∂_{y}z^{N}) = ∑_{i=0}^{N−1}(b_{21}b_{22})^{i}σ_{0}[ ∑_{j=0}^{N−2−i}(b_{21}b_{22})^{j+i}b_{21}z^{N−2−i−j}xz^{j}xz^{i}

+ ∑_{j=0}^{i−1}(b_{21}b_{22})^{j}z^{N−1−i}xz^{i−j−1}xz^{j}]

= ∑_{k+l≤N−2}(b_{21}b_{22})^{k+2l}σ_{0}b_{21}(1 + b_{22})z^{N−2−k−l}xz^{k}xz^{l}

= 0 (since b_{22} = −1).

Now, since b_{11} = b_{22} = −1, we have x^{2} = y^{2} = 0. Furthermore,

yx = b_{12}^{−1}xy − b_{12}^{−1}z,
zx = (xy − b_{12}yx)x = xyx = −b_{12}^{−1}x(xy − b_{12}yx) = −b_{12}^{−1}xz,
zy = (xy − b_{12}yx)y = −b_{12}yxy = −b_{12}y(xy − b_{12}yx) = −b_{12}yz.

Thus the set {x^{a}y^{b}z^{c} | 0 ≤ a, b < 2, 0 ≤ c < N} generates 𝔅(V) as a vector space. Let α_{b,c} ∈ k such that

∑_{\begin{matrix}0≤b<2 \\ 0≤c<N \end{matrix}} α_{b,c}y^{b}z^{c} = 0.

We compute

∂_{y}(yz^{c}) = (b_{21}b_{22})^{c}z^{c} + y∂_{y}z^{c},
∂_{x}∂_{y}(y) = 0,
∂_{x}∂_{y}(yz^{c}) = (c)_{q}σ_{0}yz^{c−1} (c > 0).

Applying (∂_{x}∂_{y})^{N−1}, we see that α_{0,N−1} = α_{1,N−1} = 0. Inductively, we see that α_{b,c} = 0 ∀b, c. In the same vein, let α_{a,b,c} ∈ k such that

∑_{\begin{matrix}0≤a,b<2 \\ 0≤c<N \end{matrix}} α_{a,b,c}x^{a}y^{b}z^{c} = 0.

We compute

∂_{x}(xy^{b}z^{c}) = b_{11}^{c}b_{22}^{b+c}y^{b}z^{c},

from where we see that α_{1,b,c} = 0 ∀b, c, and thus α_{a,b,c} = 0 ∀a, b, c, which proves the statement.

Remark 2.12. It can be proved in general that if V has a matrix (b_{ij}) in a basis {x_{1}, …, x_{θ}}, then the operator (∂_{x_{i}})^{N(b_{ii})} vanishes. Furthermore, the duality of [AG, Prop. 3.2.30] translates products into compositions.

Definition 2.13. Let V be a θ-dimensional YD module, and suppose V has a matrix (b_{ij}). According to [AS2], we say that V is of Cartan type (or CT) if there exists a matrix (a_{ij}) ∈ M_{θ}(ℤ) such that

b_{ij}b_{ji} = b_{ii}^{a_{ij}} ∀i, j.

If we choose a_{ii} = 2 and −order of b_{ii} < a_{ij} ≤ 0 for i ≠ j, then (a_{ij}) is a generalized Cartan matrix. We transfer to V the terminology on (a_{ij}).

Corollary 2.14.

1. Let V ∈ _{H}^{H}YD, and suppose that V has one of the following matrices in the basis {x, y} (recall from the beginning of section 1.1 that i, ξ denote fixed roots of unity of orders 4 and 8 respectively).

b_{q}^{1} = ( \begin{matrix}−1 & q^{−1} \\ q & −1 \end{matrix} ) (q ∈ k^{×}), b^{2} = ( \begin{matrix}−1 & −1 \\ 1 & −1 \end{matrix} ) ,

b_{±+}^{3} = ( \begin{matrix}−1 & 1 \\ ±i & −1 \end{matrix} ) , b_{±−}^{3} = ( \begin{matrix}−1 & −1 \\ ±i & −1 \end{matrix} ) ,

b_{±}^{4} = ( \begin{matrix}±i & ±i \\ −1 & −1 \end{matrix} ) , b_{±}^{5} = ( \begin{matrix}±i & ∓i \\ −1 & −1 \end{matrix} ) ,

b_{k}^{6} = ( \begin{matrix}−1 & 1 \\ ξ^{k} & −1 \end{matrix} ) (k = 1, 3, 5, 7).

The information about 𝔅(V) that we need is given in the following table. We put z_{1} = Ad_{x}(y) or z_{1} = Ad_{y}(x) depending on the case.

TABLE 1: Matrices and algebras

b  dim 𝔅(V) CT?  z_{1}  |x| |y| |z_{1}|
b_{q}^{1}  4  Yes  0  2 2
b^{2}  8  Yes Ad_{x}(y) = xy + yx 2 2  2
b_{±+}^{3}  16  No Ad_{x}(y) = xy + yx 2 2  4
b_{±−}^{3}  16  No Ad_{x}(y) = xy + yx 2 2  4
b_{±}^{4}  16  No Ad_{y}(x) = xy + yx 4 2  2
b_{±}^{5} ≥ 8N(1) ≥ 24 No Ad_{y}(x) = xy + yx 4 2 N(1)
b_{k}^{6}  32  No Ad_{y}(x) = xy − yx 2 2  8

We denote by |w| the nilpotency order of w, i.e. w^{|w|} = 0, w^{|w|−1} ≠ 0.

2. Suppose V is 3-dimensional and has a matrix b_{q_{1},q_{2},q_{3}}^{1} in the basis {x, y, z}, where

b_{q_{1},q_{2},q_{3}}^{1} = ( \begin{matrix}−1 & q_{1}^{−1} & q_{3}^{−1} \\ q_{1} & −1 & q_{2}^{−1} \\ q_{3} & q_{2} & −1 \end{matrix} ) (q_{1}, q_{2}, q_{3} ∈ k^{×}).

Hence 𝔅(V) is an 8-dimensional QLS, with relations

x^{2} = y^{2} = z^{2} = 0, yx = q_{1}xy, zx = q_{3}xz, zy = q_{2}yz.

Proof. The cases b_{q}^{1}, b_{q_{1},q_{2},q_{3}}^{1} are particular cases of example 2.5.

The cases b^{2}, b_{±+}^{3}, b_{±−}^{3}, b_{±}^{5} and b_{k}^{6} are particular cases of 2.11. For b_{±}^{4}, it is straightforward to see that

zy = yz, zx = −ixz,

and, using the same techniques than in 2.11, that there is a PBW basis {x^{a}y^{b}z^{c} | 0 ≤ a < 4, 0 ≤ b, c < 2}.

Remark 2.15. Let V be an object in C, a braided abelian category. Suppose V has a matrix

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

in the basis {x, y}. Then it is clear that V has the matrix

[b]^{ρ} := ( \begin{matrix}b_{22} & b_{21} \\ b_{12} & b_{11} \end{matrix} )

in the basis {y, x}.

Consider now \overline{C} the category with same objects and morphisms as C but with the inverse braiding c̄ = c^{−1}. Consider V̄ = V in \overline{C}. Then V̄ has a matrix

[b]^{τ} := ( \begin{matrix}b_{11}^{−1} & b_{21}^{−1} \\ b_{12}^{−1} & b_{22}^{−1} \end{matrix} )

in the basis {x, y}. On the other hand, it can be shown that dim 𝔅(V̄) = dim 𝔅(V). Notice, for instance, that [b_{±+}^{3}]^{ρτ} = b_{∓+}^{3}.

Let M ∈ _{kΓ}^{kΓ}YD and f ∈ Aut(Γ). We write (M)^{f} for the YD-module with the same underlying space as M but with the structure given by

h ⇀^{f} m = f^{−1}(h) ⇀ m, δ^{f}(m) = (f ⊗ id)δ(m).

Note that M(g, ρ)^{f} = M(f(g), ρ ∘ f^{−1}). In order to classify non isomorphic algebras, the following criterium will be applied:

Lemma 2.16. Let V = ⊕_{i=1}^{n}M(g_{i}, ρ_{i}) and V′ = ⊕_{i=1}^{n′}M(g′_{i}, ρ′_{i}), where ρ_{i} and ρ′_{i} are irreducible. Then 𝔅(V)#kΓ ≃ 𝔅(V′)#kΓ if and only if n = n′ and there exist f ∈ Aut(Γ), σ ∈ 𝕊_{n} such that M(g′_{i}, ρ′_{i}) ≃ (M(g_{σ(i)}, ρ_{σ(i)}))^{f}.

Proof. Mimic [AS2, Prop. 6.3], where it is proved for Γ abelian.

For the last step of the lifting procedure we shall use the following Proposition, which gives an invariant way to lift the elements of the bosonized algebras.

Proposition 2.17. Let A be a pointed Hopf algebra, H = gr A be its associated graded Hopf algebra and Γ the group of group-likes of A. Let V = V (A) ∈ _{kΓ}^{kΓ}YD be the YD-module associated to A as in (1.2.1), that is, V = (H(1))^{coH(0)}. We have H(1) = V #kΓ. Take on H(1) the Γ action given by

g · (v#h) = (g ⇀ v)#ghg^{−1}

Consider on A the adjoint action of Γ. It restricts to the coradical filtration, and then A_{1} becomes a kΓ-module. Let p_{0} : A_{1} → H(1) = A_{1}/A_{0} be the canonical projection. Then

1. p_{0} is a morphism of kΓ-modules.

2. Let λ ∈ k^{Γ}, λ(g) = δ_{g,1}. Then p = (id #λ) ∘ p_{0} : A_{1} → V is a morphism of kΓ-modules.

3. There exists a section s : V → A_{1} of kΓ-modules to p such that if δ(x) = g ⊗ x then s(x) ∈ P_{g,1}(A).

Proof. The items 1 and 2 are straightforward looking at the definition of the action ⇀. We prove 3. For g, h ∈ Γ, let P′_{g,h} ⊂ A_{1} be subspaces such that P_{g,h}(A) = k(g − h) ⊕ P′_{g,h}. Then, by the Taft–Wilson theorem,

A_{1} = kΓ ⊕ ( ⨁_{g,h∈Γ} P′_{g,h}).

We have the exact sequence of vector spaces

0 → kΓ = A_{0} → A_{1} → V #kΓ → 0.

Consider now the inclusion

i : ⊕_{g∈Γ}P′_{g,1} ↪ A_{1}.

It is easy to verify that the composition p∘i is an isomorphism of vector spaces (a word of warning is needed here: ⊕_{g∈Γ}P′_{g,1} need not to be a kΓ-submodule of A_{1}). Let s_{0} : V → ⊕_{g∈Γ}P′_{g,1}, s_{0} = (p ∘ i)^{−1}, and consider as usual

s : V → A_{1}, s = \frac{1}{|Γ|} ∑_{h∈Γ}(h · s_{0}), i.e. s(x) = \frac{1}{|Γ|} ∑_{h∈Γ} h · (s_{0}(h^{−1} ⇀ x)).

It is clear that s is a kΓ section to p.

Let V^{g} = {x ∈ V | δ(x) = g ⊗x}; then V = ⊕_{g∈Γ}V^{g}. Moreover, let a ∈ P′_{g,1}. Then Δa = g ⊗ a + a ⊗ 1, and if b = a + kΓ ∈ A_{1}/A_{0} is the image of a under A_{1} → A_{1}/A_{0}, we have Δb = g ⊗ b + b ⊗ 1. Then b ∈ (V #kΓ)^{coH(0)}, whence b = v#1 for some v ∈ V such that Δ(v#1) = g ⊗ (v#1) + (v#1) ⊗ 1. This implies that δ(v) = g ⊗ v, and hence v = (id #λ)(v#1) ∈ V^{g}. Thus p(P′_{g,1}) ⊆ V^{g} for all g ∈ Γ. Now, since p ∘ i : ⊕_{g∈Γ}P′_{g,1} → ⊕_{g∈Γ}V^{g} is an isomorphism, we must have that p ∘ i|_{P′_{g,1}} : P′_{g,1} → V^{g} is an isomorphism, whence s_{0}(V^{g}) = P′_{g,1} for all g ∈ Γ.

Let now x ∈ V^{g}. We claim that s(x) ∈ P_{g,1}(A). This is true for, by the definition of s, we have

s(x) = \frac{1}{|Γ|} ∑_{h∈Γ} h · s_{0}(h^{−1} ⇀ x) ∈ ∑_{h∈Γ} h · s_{0}(V^{h^{−1}gh}) = ∑_{h∈Γ} h · P′_{h^{−1}gh,1}

⊆ ∑_{h∈Γ} h · P_{h^{−1}gh,1}(A) = ∑_{h∈Γ} P_{g,1}(A) = P_{g,1}(A).

We will use the above proposition as follows:

Tactic 2.18. If V has a basis {x_{1}, …, x_{θ}} such that x_{i} ∈ V^{g_{i}}, the following proposition enables us to consider a_{1}, …, a_{θ} such that a_{i} ∈ P_{g_{i},1}(A), the projection A_{1} → A_{1}/A_{0} sends a_{i} to (x_{i}#1) ∈ V #kΓ, and if g ∈ Γ acts on V by some matrix T in the basis {x_{i}}, then g acts on the linear span (a_{1}, …, a_{θ}) by the same matrix (in the basis {a_{i}}). Furthermore, since V generates 𝔅(V), the a_{i}’s together with Γ generate A. In general, suppose ∑_{i}(α_{i}x_{i,1} ⋯ x_{i,n}) = 0 ∈ 𝔅^{n}(V). Then b = ∑_{i}(α_{i}a_{i,1} ⋯ a_{i,n}) ∈ A lies in the kernel of the projection, i.e. b ∈ A_{n−1}. Usually, we shall have b ∈ P_{g,1}(A) for some g ∈ Γ, from where b shall lie in A_{1}.

The following is a very easy but useful remark:

Remark 2.19. Let A be a finite dimensional Hopf algebra such that char k does not divide dim A. Let g ∈ A be a group-like and a ∈ P_{g,1}(A) such that ga = ζag. Let n be the order of ζ. Then, by the same computations as in 2.1, a^{n} ∈ P_{g^{n},1}(A). If g^{n} = 1 we must have a^{n} = 0, for, if not, a would generate a sub Hopf algebra of dimension a power of char k (if char k > 0) or ∞ (if char k = 0), leading to a contradiction.

If we are lifting a QLS, we have a_{i} ∈ P_{g_{i},1}(A) and g_{i}a_{j}g_{i}^{−1} = b_{ij}a_{j}. If i ≠ j, b_{ij}b_{ji} = 1 and g_{i}g_{j} = g_{j}g_{i}. Then

Δ(a_{i}a_{j} − b_{ij}a_{j}a_{i}) = g_{i}g_{j} ⊗ a_{i}a_{j} + g_{i}a_{j} ⊗ a_{i} + a_{i}g_{j} ⊗ a_{j} + a_{i}a_{j} ⊗ 1

− b_{ij}g_{j}g_{i} ⊗ a_{j}a_{i} − b_{ij}a_{j}g_{i} ⊗ a_{i} − b_{ij}g_{j}a_{i} ⊗ a_{j} − b_{ij}a_{j}a_{i} ⊗ 1

= g_{i}g_{j} ⊗ (a_{i}a_{j} − b_{ij}a_{j}a_{i}) + (a_{i}a_{j} − b_{ij}a_{j}a_{i}) ⊗ 1,

from where a_{i}a_{j} −b_{ij}a_{j}a_{i} ∈ P_{g_{i}g_{j},1}(A). As above, if char k |̸ dim A and g_{i}g_{j} = 1, then a_{i}a_{j} − b_{ij}a_{j}a_{i} = 0.

3. Γ = C_{2}

Let V = ⊕_{i}M(h_{i}, χ_{i}). For 𝔅(V) to have dimension 16, we must have b_{ii} = −1 ∀i, where (b_{ij}) = (χ_{j}(h_{i})) is the matrix of V. Thus, V = ⊕_{i}M(g, ĝ), and hence (b_{ij}) = ĝ(g) = −1 ∀ij, from where 𝔅(V) is a QLS with generators x_{1}, …, x_{θ} and relations x_{i}^{2} = 0, x_{i}x_{j} = −x_{j}x_{i}. Thus, dim 𝔅(V) = 2^{θ}, and θ = 4. Let a_{1}, …, a_{4} be as in 2.18. Then ga_{i}g = −a_{i} and a_{i} ∈ P_{g,1}(A). We have by 2.19 that a_{i}^{2} ∈ P(A), and the same is true for a_{i}a_{j} + a_{j}a_{i}. Hence a_{i}^{2} = a_{i}a_{j} +a_{j}a_{i} = 0 ∀i, j. There exist thus only one Hopf algebra of dimension 32 with coradical kC_{2}. It is clear, using the same reasoning, that there exists a Hopf algebra with coradical kC_{2} and dimension n if and only if n = 2^{m} for some m ∈ ℕ. Furthermore, if n = 2^{m}, there exists only one such Hopf algebra. This is the content of [N, Thm. 4.2.1], but we included here a proof for completeness.

4. Γ with order 4

4.1. Γ = C_{2} × C_{2}. Necessarily the rank is 2 or 3: for rank 1 the root b_{11} has order 2, whence dim(𝔅(V)#kΓ) would be 8, and rank ≥ 4 is impossible by 2.5. The following is a list of the irreducible YD-modules giving Nichols algebras of dimension 2, with the action of Aut(Γ) as in the lemma 2.16. In this case, Aut(Γ) is isomorphic to 𝕊_{3}, generated by f_{1} and f_{2}, where, using the notation at the end of 1.1,

f_{1} = (g_{1} ↦ g_{1}g_{2}, g_{2} ↦ g_{2}), f_{2} = (g_{1} ↦ g_{1}, g_{2} ↦ g_{1}g_{2}).

Thus, we have

TABLE 2: modules / Aut(Γ), Γ = C^{2} × C^{2}
M(h, χ) h χ (M(h, χ))^{f_{1}} (M(h, χ))^{f_{2}}
Y_{2}^{1} g_{1}  ĝ_{1}  Y_{2}^{5}  Y_{2}^{2}
Y_{2}^{2} g_{1}  ĝ_{1}ĝ_{2}  Y_{2}^{6}  Y_{2}^{1}
Y_{2}^{3} g_{2}  ĝ_{2}  Y_{2}^{4}  Y_{2}^{6}
Y_{2}^{4} g_{2}  ĝ_{1}ĝ_{2}  Y_{2}^{3}  Y_{2}^{5}
Y_{2}^{5} g_{1}g_{2} ĝ_{1}  Y_{2}^{1}  Y_{2}^{4}
Y_{2}^{6} g_{1}g_{2} ĝ_{2}  Y_{2}^{2}  Y_{2}^{3}

Consider then V = M_{1} ⊕ M_{2} = M(h_{1}, χ_{1}) ⊕ M(h_{2}, χ_{2}). Since Aut(Γ) acts transitively, we may suppose that M_{1} = Y_{2}^{1}. For rank 2 this gives 6 possibilities, but the cases Y_{2}^{1} ⊕ Y_{2}^{4} and Y_{2}^{1} ⊕ Y_{2}^{6} are equivalent by means of f_{1} ∘ f_{2}. Hence, the possibilities are

TABLE 3: Rank 2, Γ = C^{2} × C^{2}
V h_{1} h_{2} χ_{1} χ_{2} (b_{ij}) dim 𝔅(V)
V_{2}^{1} g_{1} g_{1}  ĝ_{1} ĝ_{1} b_{−1}^{1}  4
V_{2}^{2} g_{1} g_{1}  ĝ_{1} ĝ_{1}ĝ_{2} b_{−1}^{1}  4
V_{2}^{3} g_{1} g_{2}  ĝ_{1} ĝ_{2} b_{1}^{1}  4
V_{2}^{4} g_{1} g_{2}  ĝ_{1} ĝ_{1}ĝ_{2} b^{2}  8
V_{2}^{5} g_{1} g_{1}g_{2} ĝ_{1} ĝ_{1} b_{−1}^{1}  4

For rank 3, V = ⊕_{i=1}^{3}M_{i} = ⊕_{i=1}^{3}M(h_{i}, χ_{i}). Since the module V_{2}^{4} must not appear as a submodule, we have the following possibilities:

TABLE 4: Rank 3, Γ = C^{2} × C^{2}
V h_{1} h_{2} h_{3} χ_{1} χ_{2} χ_{3}  (b_{ij}) dim 𝔅(V)
W_{2}^{1} g_{1} g_{1} g_{1}  ĝ_{1} ĝ_{1} ĝ_{1} b_{−1,−1,−1}^{1}  8
W_{2}^{2} g_{1} g_{1} g_{1}  ĝ_{1} ĝ_{1} ĝ_{1}ĝ_{2} b_{−1,−1,−1}^{1}  8
W_{2}^{3} g_{1} g_{1} g_{1}g_{2} ĝ_{1} ĝ_{1} ĝ_{1} b_{−1,−1,−1}^{1}  8
W_{2}^{4} g_{1} g_{1} g_{2}  ĝ_{1} ĝ_{1} ĝ_{2}  b_{1,−1,−1}^{1}  8

We compute now the liftings.

4.1.1. V = V_{2}^{4}. We take a_{1} and a_{2} as in 2.18. Hence we have

Δ(a_{1}) = g_{1} ⊗ a_{1} + a_{1} ⊗ 1, g_{1}a_{1} = −a_{1}g_{1}, g_{2}a_{1} = a_{1}g_{2};
Δ(a_{2}) = g_{2} ⊗ a_{2} + a_{2} ⊗ 1, g_{1}a_{2} = −a_{2}g_{1}, g_{2}a_{2} = −a_{2}g_{2}.

Since a_{1}^{2} ∈ P(A), we must have a_{1}^{2} = 0 by 2.19. For the same reason, a_{2}^{2} = 0. Take now

a_{3} = Ad_{a_{1}}(a_{2}) = a_{1}a_{2} + g_{1}a_{2}S(a_{1}) = a_{1}a_{2} + a_{2}a_{1},

then

Δ(a_{3}) = g_{1}g_{2} ⊗ a_{1}a_{2} + g_{1}a_{2} ⊗ a_{1} + a_{1}g_{2} ⊗ a_{2} + a_{1}a_{2} ⊗ 1
= g_{1}g_{2} ⊗ a_{1}a_{2} − a_{2}g_{1} ⊗ a_{1} + a_{1}g_{2} ⊗ a_{2} + a_{1}a_{2} ⊗ 1.

It is straightforward to see that a_{3}^{2} ∈ P(A), and thus a_{3}^{2} = 0. Furthermore, it is clear from the definition of a_{3} that

a_{1}a_{3} = a_{3}a_{1}, a_{2}a_{3} = a_{3}a_{2},

and hence the unique lifting is the bosonization 𝔅(V_{2}^{2})#kΓ.

4.1.2. V = W_{2}^{1}. As in 2.18, we can take a_{1}, a_{2} and a_{3} such that

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

We have a_{i}^{2} ∈ P(A) and a_{i}a_{j} + a_{j}a_{i} ∈ P(A), whence a_{i}^{2} = a_{i}a_{j} + a_{j}a_{i} = 0 ∀i, j by 2.19. Hence, there is only one lifting: the bosonization 𝔅(W_{2}^{1})#kΓ.

4.1.3. V = W_{2}^{2}. This case is the same as the previous one, with the only difference that g_{2}a_{3} = −a_{3}g_{2}, while g_{2}a_{i} = a_{i}g_{2} for i = 1, 2. The same arguments say that there is no other lifting than the bosonization 𝔅(W_{2}^{2})#kΓ.

4.1.4. V = W_{2}^{3}. We have now g_{1}a_{i} = −a_{i}g_{1} and g_{2}a_{i} = a_{i}g_{2}. The difference with the case V = W_{2}^{1} is now that Δ(a_{3}) = g_{1}g_{2} ⊗ a_{3} + a_{3} ⊗ 1. As before, a_{i}^{2} ∈ P(A) and hence a_{i}^{2} = 0 by 2.19. Furthermore, a_{1}a_{2} + a_{2}a_{1} ∈ P(A), from where a_{1}a_{2} = −a_{2}a_{1}. However a_{i}a_{3} + a_{3}a_{i} ∈ P_{g_{2},1}(A) for i = 1, 2 and hence, there exist λ_{1}, λ_{2} ∈ k such that a_{i}a_{3} + a_{3}a_{i} = λ_{i}(g_{2} − 1). Now, let W′ = (a_{1}, a_{2}) and let T ∈ GL_{2}(k), T = (T_{ij}). If we replace the basis {a_{1}, a_{2}} by {T a_{1}, T a_{2}} (we denote also by T the automorphism of W′ with matrix T in the basis \{a_{1}, a_{2}\}), then the constants \binom{λ_{1}}{λ_{2}} are replaced by T \binom{λ_{1}}{λ_{2}}. Hence, there are 2 non isomorphic liftings: they can be characterized by the rank of the morphism W′ → k(g_{2} − 1), a ↦ aa_{3} + a_{3}a, which can be 0 or 1. Thus we have

1. The bosonization 𝔅(W_{2}^{3})#kΓ.

2. The algebra with the above relations plus a_{1}a_{3} + a_{3}a_{1} = g_{2} − 1, a_{2}a_{3} + a_{3}a_{2} = 0.

4.1.5. V = W_{2}^{4}. We have here Δ(a_{i}) = g_{1} ⊗ a_{i} + a_{i} ⊗ 1 for i = 1, 2, and Δ(a_{3}) = g_{2}⊗a_{3}+a_{3}⊗1, plus g_{1}a_{i} = −a_{i}g_{1}, g_{2}a_{i} = a_{i}g_{2} for i = 1, 2, g_{1}a_{3} = a_{3}g_{1}, g_{2}a_{3} = −a_{3}g_{2}. As before, a_{i}^{2} = 0 for i = 1, 2, 3, and also a_{1}a_{2} + a_{2}a_{1} = 0. Furthermore, a_{i}a_{3} − a_{3}a_{i} ∈ P_{g_{1}g_{2},1}(A), from where a_{i}a_{3} − a_{3}a_{1} = λ_{i}(g_{1}g_{2} − 1) for i = 1, 2. These last conditions may add some unwanted relation which would decrease the dimension of A. To see which λ_{i}’s give a 32-dimensional algebra, we use the diamond lemma:

g_{1}a_{3}a_{1} = g_{1}a_{1}a_{3} + λ_{1}g_{1}(g_{1}g_{2} − 1) = −a_{1}a_{3}g_{1} + λ_{1}g_{2} − λ_{1}g_{1}
g_{1}a_{3}a_{1} = −a_{3}a_{1}g_{1} = −a_{1}a_{3}g_{1} − λ_{1}(g_{1}g_{2} − 1)g_{1} = −a_{1}a_{3}g_{1} − λ_{1}g_{2} + λ_{1}g_{1},

from where λ_{1} = 0. An analogous work with the monomial g_{1}a_{3}a_{2} tells that λ_{2} = 0, and there is only one lifting: the bosonization 𝔅(W_{2}^{4})#kΓ.

4.2. Γ = C_{4}. By similar considerations as for Γ = C_{2} × C_{2}, the rank must be 2 or 3. Here Aut(Γ) ≃ C_{2}, generated by f(g) = g^{3}. The list of YD-modules giving Nichols algebras of dimension a power of 2 is

TABLE 5: modules / Aut(Γ), Γ = C^{4}

M(h, χ) h χ (M(h, χ))^{f} dim 𝔅(M(h, χ))
Y_{3}^{1} g ĝ  Y_{3}^{8}  4
Y_{3}^{2} g ĝ^{2}  Y_{3}^{7}  2
Y_{3}^{3} g ĝ^{3}  Y_{3}^{6}  4
Y_{3}^{4} g^{2} ĝ  Y_{3}^{5}  2
Y_{3}^{5} g^{2} ĝ^{3}  Y_{3}^{4}  2
Y_{3}^{6} g^{3} ĝ  Y_{3}^{3}  4
Y_{3}^{7} g^{3} ĝ^{2}  Y_{3}^{2}  2
Y_{3}^{8} g^{3} ĝ^{3}  Y_{3}^{1}  4

Hence, for rank 2 we have the following non isomorphic possibilities (we exclude the cases Y_{3}^{i} ⊕ Y_{3}^{j} with i, j ∈ {1, 3, 6, 8} by virtue of 2.5).

TABLE 6: Rank 2, Γ = C^{4}

V h_{1} h_{2} χ_{1} χ_{2} b_{11} b_{12} b_{21} b_{22} b_{ij}  dim 𝔅(V)
V_{3}^{1} g g ĝ ĝ^{2} i −1 i −1 [b_{−}^{4}]^{τ}  64
V_{3}^{2} g g ĝ^{3} ĝ^{2} −i −1 −i −1 [b_{+}^{4}]^{τ}  64
V_{3}^{3} g g ĝ^{2} ĝ^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{3}^{4} g g^{3} ĝ^{2} ĝ^{3} −1 −i −1 i [b_{−}^{5}]^{ρτ}  ≥ 24
V_{3}^{5} g g^{3} ĝ^{2} ĝ −1 i −1 −i [b_{+}^{5}]^{ρτ}  ≥ 24
V_{3}^{6} g g^{3} ĝ^{2} ĝ^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{3}^{7} g g^{2} ĝ ĝ i i −1 −1 b_{+}^{4} 64
V_{3}^{8} g g^{2} ĝ ĝ^{3} i −i −1 −1 b_{+}^{5}  ≥ 24
V_{3}^{9} g g^{2} ĝ^{3} ĝ −i i −1 −1 b_{−}^{5}  ≥ 24
V_{3}^{10} g g^{2} ĝ^{3} ĝ^{3} −i −i −1 −1 b_{−}^{4}  64
V_{3}^{11} g g^{2} ĝ^{2} ĝ −1 i 1 −1 [b_{++}^{3}]^{ρ}  16
V_{3}^{12} g g^{2} ĝ^{2} ĝ^{3} −1 −i 1 −1 [b_{−+}^{3}]^{ρ}  16
V_{3}^{13} g^{2} g^{2} ĝ ĝ −1 −1 −1 −1 b_{−1}^{1}  4
V_{3}^{14} g^{2} g^{2} ĝ ĝ^{3} −1 −1 −1 −1 b_{−1}^{1}  4

Thus, we have no rank 2 modules over C_{4} giving 8-dimensional Nichols algebras. For rank 3 the possible subcases are V_{3}^{3}, V_{3}^{6}, V_{3}^{13} and V_{3}^{14}. We have hence the following cases:

TABLE 7: Rank 3, Γ = C^{4}

V h_{1} h_{2} h_{3} χ_{1} χ_{2} χ_{3} b_{ij} dim 𝔅(V)
W_{3}^{1} g g g ĝ^{2} ĝ^{2} ĝ^{2} b_{−1,−1,−1}^{1}  8
W_{3}^{2} g g g^{3} ĝ^{2} ĝ^{2} ĝ^{2} b_{−1,−1,−1}^{1}  8
W_{3}^{3} g^{2} g^{2} g^{2} ĝ ĝ ĝ b_{−1,−1,−1}^{1}  8
W_{3}^{4} g^{2} g^{2} g^{2} ĝ ĝ ĝ^{3} b_{−1,−1,−1}^{1}  8

We compute now the liftings:

4.2.1. V = W_{3}^{1}. We take a_{i}, i = 1, 2, 3 as in 2.18, i.e. Δ(a_{i}) = g ⊗ a_{i} + a_{i} ⊗ 1, ga_{i} = −a_{i}g. Hence, by 2.19, a_{i}^{2} ∈ P_{g^{2},1}(A), and a_{i}a_{j} + a_{j}a_{i} ∈ P_{g^{2},1}(A), from where there exist λ_{i}, 1 ≤ i ≤ 6 such that

a_{1}^{2} = λ_{1}(g^{2} − 1), a_{2}^{2} = λ_{2}(g^{2} − 1), a_{3}^{2} = λ_{3}(g^{2} − 1),
a_{1}a_{2} + a_{2}a_{1} = λ_{4}(g^{2} − 1),
a_{1}a_{3} + a_{3}a_{1} = λ_{5}(g^{2} − 1),
a_{2}a_{3} + a_{3}a_{2} = λ_{6}(g^{2} − 1).

Thus, we have a quadratic form f : W = (a_{1}, a_{2}, a_{3}) → k(g^{2} − 1), f(a) = a^{2}. Furthermore, the basis {a_{1}, a_{2}, a_{3}} of W can be replaced by any other basis of W, as a result of what we must consider the equivalence class of the quadratic form f in order to have non isomorphic liftings. Since char k ≠ 2 and k is algebraically closed, f is characterized by its rank, and hence there are 4 liftings (up to isomorphism):

1. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{i}^{2} = 0 for i = 1, 2, 3. This is the bosonization 𝔅(W_{3}^{1})#kΓ.

2. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{i}^{2} = 0 for i = 1, 2, a_{3}^{2} = (g^{2} − 1).

3. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = 0, a_{i}^{2} = (g^{2} − 1) for i = 2, 3.

4. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{i}^{2} = (g^{2} − 1) for i = 1, 2, 3.

4.2.2. V = W_{3}^{2}. Similarly, we have a_{1}, a_{2} ∈ P_{g,1}(A), a_{3} ∈ P_{g^{3},1}(A) and such that ga_{i} = −a_{i}g for i = 1, 2, 3. Now, a_{i}a_{3} + a_{3}a_{i} ∈ P(A) (i = 1, 2), whence a_{i}a_{3} + a_{3}a_{i} = 0 for i = 1, 2. However, there exist λ_{i}, 1 ≤ i ≤ 4 such that

a_{i}^{2} = λ_{i}(g^{2} − 1) (i = 1, 2, 3), a_{1}a_{2} + a_{2}a_{1} = λ_{4}(g^{2} − 1).

Hence we have two quadratic forms:

f_{1} : W_{1} = (a_{1}, a_{2}) → k(g^{2} − 1), f_{1}(a) = a^{2},
f_{2} : W_{2} = (a_{3}) → k(g^{2} − 1), f_{2}(a) = a^{2}.

As before, we can replace {a_{1}, a_{2}} by another basis of W_{1}, and the same for {a_{3}}. As with W_{3}^{1}, f_{1} and f_{2} are characterized by their ranks, whence there are 3 possibilities for f_{1} and 2 for f_{2}, and finally there are 6 liftings:

1. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = 0, a_{2}^{2} = 0, a_{3}^{2} = 0. This is the bosonization 𝔅(W_{3}^{2})#kΓ.

2. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = 0, a_{2}^{2} = (g^{2} − 1), a_{3}^{2} = 0.

3. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = (g^{2} − 1), a_{2}^{2} = (g^{2} − 1), a_{3}^{2} = 0.

4. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = 0, a_{2}^{2} = 0, a_{3}^{2} = (g^{2} − 1).

5. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = 0, a_{2}^{2} = (g^{2} − 1), a_{3}^{2} = (g^{2} − 1).

6. a_{i}a_{j} + a_{j}a_{i} = 0 for i ≠ j, a_{1}^{2} = (g^{2} − 1), a_{2}^{2} = (g^{2} − 1), a_{3}^{2} = (g^{2} − 1).

4.2.3. V = W_{3}^{3}. The situation here is easier: we have a_{i} ∈ P_{g^{2},1}(A) and ga_{i} = ia_{i}g for i = 1, 2, 3. Since a_{i}^{2} ∈ P(A) and a_{i}a_{j} + a_{j}a_{i} ∈ P(A) for all i, j, by 2.19 we have a_{i}^{2} = a_{i}a_{j} + a_{j}a_{i} = 0, and there is only one lifting: the bosonization 𝔅(W_{3}^{3})#kΓ.

4.2.4. V = W_{3}^{4}. Similar to the previous case, the only difference being that ga_{i} = ia_{i}g for i = 1, 2 and ga_{3} = −ia_{3}g. There is only one lifting: the bosonization 𝔅(W_{3}^{3})#kΓ.

5. Γ of order 8

By 2.8 we have dim V ≤ 2. If dim V = 1, we have V = M(h, χ), h ∈ Z(Γ) and χ a character such that χ(h) has order 4. If Γ is non abelian, we have Z(Γ) ⊆ [Γ; Γ], whence any irreducible representation of degree 1 vanishes on the center, and χ(h) = 1. This tells that there are no 32-dimensional pointed Hopf algebras with non abelian coradical of dimension 8.

Remark 5.0.1. In turn this explains why there are no 16-dimensional pointed Hopf algebras with non abelian coradical (see [CDR]).

If dim V = 2. Then, by [AG, Prop. 3.1.11], V comes from the abelian case, and hence it has a matrix (b_{ij}). By 2.5, V must be a QLS over some subgroup Γ′ ⊆ Γ, with b_{11} = b_{22} = −1. If Γ is non abelian, there are in principle three possibilities for V to be 2-dimensional:

5.0.2. Decomposition of V, Γ non abelian.

1. V = M(h_{1}, χ_{1}) ⊕ M(h_{2}, χ_{2}), where the h_{i}’s are central in Γ and the χ_{i}’s are characters, 2. V = M(h, χ), where [Γ : Γ_{h}] = 2 and χ is a character of Γ_{h},

3. V = M(h, ρ), where h is central in Γ and dim ρ = 2.

Case 1 does not arise for the same reasons as above: any irreducible representation of degree 1 of a non abelian group of order 8 vanishes on the center, and we would have b_{11} = b_{22} = 1, contradicting 2.2. We shall proceed as follows:

Tactic 5.0.3. In case 2, let x be a generator of the space affording χ. Note that χ(h) = b_{11} = −1. Let t ∉ Γ_{h}. The conjugacy class of h is {h,^{t}h = tht^{−1}}, and since [Γ : Γ_{h}] = 2 it is easy to prove that ^{t}h ∈ Γ_{h}. Notice that t^{−1} ∉ Γ_{h}, whence t^{−1}ht = tht^{−1} = ^{t}h. Let y = t ⇀ x and let ζ = χ(^{t}h). Then δ(y) = ^{t}h ⊗ y, and c can be computed as follows:

c(x ⊗ x) = (h ⇀ x) ⊗ x = −x ⊗ x,
c(x ⊗ y) = (h ⇀ y) ⊗ x = (ht ⇀ x) ⊗ x = (tt^{−1}ht ⇀ x) ⊗ x
= (ζt ⇀ x) ⊗ x = ζy ⊗ x,
c(y ⊗ x) = (^{t}h ⇀ x) ⊗ y = ζx ⊗ y,
c(y ⊗ y) = (^{t}ht ⇀ x) ⊗ y = (th ⇀ x) ⊗ y = −y ⊗ y.

Therefore, the matrix of c is ( \begin{matrix}−1 & ζ \\ ζ & −1 \end{matrix} ), whence 1 = b_{12}b_{21} = ζ^{2}, i.e. ζ = ±1.

To compute the liftings, let a, b be liftings of x, y as in 2.18. Then hah^{−1} = −a and b = tat^{−1}. The action of Γ on A_{1} may be computed from the action of Γ on V, as explained in 2.18.

Tactic 5.0.4. In case 3, let x be a generator of the space affording ρ, and let t ∈ Γ such that y = ρ(t)(x) is linearly independent with x. Since ρ is irreducible and h is central, ρ(h) = ζ id for some ζ ∈ k^{×}. Then the matrix of

c is ( \begin{matrix}ζ & ζ \\ ζ & ζ \end{matrix} ), from where ζ = b_{11} = −1. Let a, b be liftings of x, y as in 2.18.

Then b = tat^{−1} and hbh^{−1} = −b.

Remark 5.0.5. These cases do not arise when classifying pointed Hopf algebras of dimension p^{5}, p an odd prime number (see [G]), and there are no pointed Hopf algebras of dimension p^{5} with non commutative coradical of dimension p^{3}.

We consider now the case Γ abelian.

5.0.6. Use of diamond lemma

If Γ is abelian and dim V = 1, then V = M(h, χ) and, as in 2.18, we have a = a_{1} ∈ P_{h,1}(A). Furthermore, for 𝔅(V) to be 4-dimensional, we must have χ(h) of order 4. By 2.19, a^{4} ∈ P_{h^{4},1}(A) = k(h^{4} − 1) (χ(h) having order 4, this implies that h^{4} does not coincide neither with h, h^{2} nor h^{3}). We have hence A generated by Γ and a, with relations

tat^{−1} = χ(t)a ∀t ∈ Γ, a^{4} = λ(h^{4} − 1).

If h^{4} ≠ 1, taking a suitable scalar multiple of a we may suppose that λ ∈ {0, 1}. We use the diamond lemma to prove that A is 32-dimensional, but this adds some condition on λ. Let t ∈ Γ. We have

ta^{4} = χ^{4}(t)a^{4}t = χ^{4}(t)λ(h^{4} − 1)t,
ta^{4} = tλ(h^{4} − 1) = λ(h^{4} − 1)t,

whence

λ(χ^{4}(t) − 1)(h^{4} − 1) = 0 ∀t ∈ Γ. (5.0.7)

If h^{4} = 1 we can take λ = 0, and hence condition (5.0.7) can be written in the more compact form

λ(χ^{4} − 1) = 0. (5.0.8)

If Γ is abelian and dim V = 2, then V = M(h_{1}, χ_{1}) ⊕ M(h_{2}, χ_{2}). We have as in 2.18 a_{i} ∈ P_{h_{i},1}(A), ta_{i}t^{−1} = χ_{i}(t)a_{i}. As in 2.19, a_{i}^{2} ∈ P_{h_{i}^{2},1}(A) and Ad_{a_{1}}(a_{2}) = a_{1}a_{2} − χ_{2}(h_{1})a_{2}a_{1} ∈ P_{h_{1}h_{2},1}(A). Since 𝔅(V) is 4-dimensional, by 2.5 it is a QP, and we have χ_{2}(h_{1})χ_{1}(h_{2}) = 1, χ_{i}(h_{i}) = −1. Now, h_{1}h_{2} ≠ h_{1} and h_{1}h_{2} ≠ h_{2}, for if not h_{2} = 1 or h_{1} = 1, contradicting χ_{i}(h_{i}) = −1. Moreover, if h_{1}^{2} = h_{2}, we would have

−1 = χ_{2}(h_{2}) = χ_{2}(h_{1}^{2}) = (χ_{2}(h_{1}))^{2} = (χ_{1}(h_{2}))^{−2} = (χ_{1}(h_{1}))^{−4} = (−1)^{−4} = 1,

which is absurd. Thus h_{1}^{2} ≠ h_{2}, and analogously h_{2}^{2} ≠ h_{1}. Hence, P_{h_{i}^{2},1}(A) = k(h_{i}^{2} − 1) and P_{h_{1}h_{2},1}(A) = k(h_{1}h_{2} − 1), and we have

(5.0.9)a_{i}^{2} = λ_{i}(h_{i}^{2} − 1) (i = 1, 2),  a_{1}a_{2} − χ_{2}(h_{1})a_{2}a_{1} = λ_{3}(h_{1}h_{2} − 1).

Similarly to (5.0.8), we get

λ_{1}(χ_{1}^{2} − 1) = 0, λ_{2}(χ_{2}^{2} − 1) = 0. (5.0.10)

Doing the same with the monomials ta_{2}a_{1}, t ∈ Γ, we arrive to the condition

λ_{3}(χ_{1}χ_{2} − 1) = 0. (5.0.11)

We consider now group by group.

5.1. Γ = C_{2} × C_{2} × C_{2}. There are no 1-dimensional modules M(h, χ) in _{kΓ}^{kΓ}YD s.t. χ(h) has order 4, whence the rank must be 2. Now, the YD-modules giving 2-dimensional Nichols algebras are those M(h, χ) such that χ(h) = −1, but Aut(Γ) ≃ GL_{3}(ℤ/2), and this group acts transitively on these modules. Hence we can suppose that M_{1} = M(g_{1}, ĝ_{1}) (we are using the notation at the end of 1.1). The subgroup of Aut(Γ) which fixes M_{1} is isomorphic to 𝕊_{3}, and is generated by

f_{1} = (g_{1} ↦ g_{1}, g_{2} ↦ g_{2}g_{3}, g_{3} ↦ g_{3}), f_{2} = (g_{1} ↦ g_{1}, g_{2} ↦ g_{2}, g_{3} ↦ g_{2}g_{3}).

Thus we consider the action of this subgroup on the modules and choose one element in each orbit:

Y_{4}^{1} = M(g_{1}, ĝ_{1}), Y_{4}^{2} = M(g_{1}, ĝ_{1}ĝ_{2}), Y_{4}^{3} = M(g_{1}g_{2}, ĝ_{1}), Y_{4}^{4} = M(g_{1}g_{2}, ĝ_{1}ĝ_{3}), Y_{4}^{5} = M(g_{1}g_{2}, ĝ_{2}), Y_{4}^{6} = M(g_{2}, ĝ_{2}), Y_{4}^{7} = M(g_{2}, ĝ_{1}ĝ_{2}).

In order to give non isomorphic modules after taking the sums Y_{4}^{1} ⊕ Y_{4}^{i}, we must take into account the possibility of switching the order, as in lemma 2.16. The modules Y_{4}^{1} ⊕ Y_{4}^{5} and Y_{4}^{1} ⊕ Y_{4}^{7} give isomorphic algebras when its Nichols algebras are bosonized by means of the automorphism

(g_{1} ↦ g_{2}, g_{2} ↦ g_{1}g_{2}, g_{3} ↦ g_{3}),

and we consider only Y_{4}^{1} ⊕ Y_{4}^{5}. Hence, the list is

TABLE 8: Rank 2, Γ = C^{2} × C^{2} × C^{2}
V h_{1} h_{2} χ_{1} χ_{2} b_{11} b_{12} b_{21} b_{22} b_{ij} dim 𝔅(V)
V_{4}^{1} g_{1} g_{1}  ĝ_{1} ĝ_{1} −1 −1 −1 −1 b_{−1}^{1}  4
V_{4}^{2} g_{1} g_{1}  ĝ_{1} ĝ_{1}ĝ_{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{4}^{3} g_{1} g_{1}g_{2} ĝ_{1} ĝ_{1} −1 −1 −1 −1 b_{−1}^{1}  4
V_{4}^{4} g_{1} g_{1}g_{2} ĝ_{1} ĝ_{1}ĝ_{3} −1 −1 −1 −1 b_{−1}^{1}  4
V_{4}^{5} g_{1} g_{1}g_{2} ĝ_{1} ĝ_{2} −1 1 −1 −1 [b^{2}]^{ρ}  8
V_{4}^{6} g_{1} g_{2}  ĝ_{1} ĝ_{2} −1 1 1 −1 b_{1}^{1}  4

We compute the liftings:

5.1.1. V = V_{4}^{1}. We have, as in 2.18, a_{1}, a_{2} ∈ P_{g_{1},1}(A), and g_{1}a_{i} = −a_{i}g_{1}, g_{2}a_{i} = a_{i}g_{2}, g_{3}a_{i} = a_{i}g_{3} for i = 1, 2. By 2.19, we have a_{i}^{2} ∈ P(A) and also a_{1}a_{2} +a_{2}a_{1} ∈ P(A), whence a_{i}^{2} = a_{1}a_{2} +a_{2}a_{1} = 0, and there is only one lifting: the bosonization 𝔅(V_{4}^{1})#kΓ.

5.1.2. V = V_{4}^{2}. It is similar to the previous one, the only difference being that g_{2}a_{2} = −a_{2}g_{2}. The unique lifting is the bosonization 𝔅(V_{4}^{2})#kΓ

5.1.3. V = V_{4}^{3}. Here a_{1} ∈ P_{g_{1},1}(A), a_{2} ∈ P_{g_{1}g_{2},1}(A) and Γ acts by g_{1}a_{i} = −a_{i}g_{1}, g_{2}a_{i} = a_{i}g_{2}, g_{3}a_{i} = a_{i}g_{3}. As before, a_{i}^{2} ∈ P(A), whence a_{i}^{2} = 0 for i = 1, 2. However, a_{1}a_{2} + a_{2}a_{1} ∈ P_{g_{2},1}(A), from where as in (5.0.9) we have a_{1}a_{2} + a_{2}a_{1} = λ_{3}(g_{2} − 1). Taking a suitable scalar multiple of a_{1}, we may suppose that λ_{3} = 0 or λ_{3} = 1. There are hence two liftings:

1. a_{i}^{2} = 0, a_{1}a_{2} + a_{2}a_{1} = 0. This is the bosonization 𝔅(V_{4}^{3})#kΓ.

2. a_{i}^{2} = 0, a_{1}a_{2} + a_{2}a_{1} = (g_{2} − 1).

5.1.4. V = V_{4}^{4}. It is similar to V_{4}^{3}. We have a_{1} ∈ P_{g_{1},1}(A), a_{2} ∈ P_{g_{1}g_{2},1}(A) and Γ acts now by g_{1}a_{i} = −a_{i}g_{1}, g_{2}a_{i} = a_{i}g_{2}, g_{3}a_{1} = a_{1}g_{3}, g_{3}a_{2} = −a_{2}g_{3}. As before, a_{i}^{2} = 0, a_{1}a_{2} +a_{2}a_{1} = λ_{3}(g_{2} −1). Now, (5.0.11) tells that λ_{3} = 0. There is only one lifting: the bosonization 𝔅(V_{4}^{4})#kΓ.

5.1.5. V = V_{4}^{6}. Now a_{1} ∈ P_{g_{1},1}(A), a_{2} ∈ P_{g_{2},1}(A) and Γ acts by

g_{1}a_{1} = −a_{1}g_{1}, g_{1}a_{2} = a_{2}g_{1}, g_{2}a_{1} = a_{1}g_{2}, g_{2}a_{2} = −a_{2}g_{2}, g_{3}a_{i} = a_{i}g_{3}.

As before, a_{i}^{2} ∈ P(A), whence a_{i}^{2} = 0. Furthermore, a_{1}a_{2} − a_{2}a_{1} ∈ P_{g_{1}g_{2},1}(A), from where a_{1}a_{2} − a_{2}a_{1} = λ_{3}(g_{1}g_{2} − 1). Now, (5.0.11) tells that λ_{3} = 0. There is only one lifting: the bosonization 𝔅(V_{4}^{6})#kΓ.

5.2. Γ = C_{2} × C_{4}. We have, using the notation at the end of 1.1, Aut(Γ) ≃ 𝔻_{4} with generators

f_{1} = (g_{1} ↦ g_{1}g_{2}^{2}, g_{2} ↦ g_{1}g_{2}), f_{2} = (g_{1} ↦ g_{1}g_{2}^{2}, g_{2} ↦ g_{2}),

with relations f_{1}^{4} = f_{2}^{2} = 1, f_{1}f_{2} = f_{2}f_{1}^{3}. Hence there are 3 orbits in Γ under the action of Aut(Γ): {g_{2}, g_{1}g_{2}, g_{2}^{3}, g_{1}g_{2}^{3}}, {g_{1}, g_{1}g_{2}^{2}} and {g_{2}^{2}}. There are 7 orbits of irreducible YD-modules giving Nichols algebras of dimension a power of 2. The following is a list of one element per orbit and the subgroup of Aut(Γ) fixing the chosen element.

TABLE 9: modules/ Aut(Γ), Γ = C^{2} × C^{4}
M(h, χ) h χ dim 𝔅(M) (Aut(Γ))_{M}
Y_{5}^{1} g_{2} ĝ_{2}  4  1
Y_{5}^{2} g_{2} ĝ_{2}^{2}  2  (f_{2})
Y_{5}^{3} g_{2} ĝ_{2}^{3}  4  1
Y_{5}^{4} g_{2} ĝ_{1}ĝ_{2}^{2}  2  (f_{2})
Y_{5}^{5} g_{1} ĝ_{1}ĝ_{2}^{3}  2  (f_{2}f_{1})
Y_{5}^{6} g_{1} ĝ_{1}  2  (f_{1}^{2})
Y_{5}^{7} g_{2}^{2} ĝ_{2}  2  (f_{1}f_{2})

Hence there are two rank 1 Nichols algebras of dimension 4 over C_{2} × C_{4}. We go now to rank 2, i.e. V = M_{1} ⊕ M_{2}. Since dim 𝔅(M_{i}) = 2, we can suppose that M_{1} is one of Y_{5}^{2}, Y_{5}^{4}, Y_{5}^{5}, Y_{5}^{6} or Y_{5}^{7}. We give the list, which is the hardest to compute (we take into account 2.16).

TABLE 10: Rank 2, Γ = C^{2} × C^{4}
V h_{1} h_{2} χ_{1} χ_{2} b_{11} b_{12} b_{21} b_{22} b_{ij}  dim 𝔅(V)
V_{5}^{1} g_{2} g_{2}  ĝ_{2}^{2}  ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{2} g_{2} g_{2}  ĝ_{2}^{2}  ĝ_{1}ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{3} g_{2} g_{2}  ĝ_{1}ĝ_{2}^{2} ĝ_{1}ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{4} g_{2} g_{1}g_{2} ĝ_{2}^{2}  ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{5} g_{2} g_{1}g_{2} ĝ_{2}^{2}  ĝ_{1} −1 1 −1 −1 [b^{2}]^{ρ}  8
V_{5}^{6} g_{2} g_{1}g_{2} ĝ_{1}ĝ_{2}^{2} ĝ_{1} −1 1 1 −1 b_{1}^{1}  4
V_{5}^{7} g_{2} g_{2}^{3}  ĝ_{2}^{2}  ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{8} g_{2} g_{2}^{3}  ĝ_{1}ĝ_{2}^{2} ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{9} g_{2} g_{2}^{3}  ĝ_{1}ĝ_{2}^{2} ĝ_{1}ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{10} g_{2} g_{1}  ĝ_{2}^{2}  ĝ_{1} −1 1 1 −1 b_{1}^{1}  4
V_{5}^{11} g_{2} g_{1}  ĝ_{2}^{2}  ĝ_{1}ĝ_{2}^{2} −1 −1 1 −1 b^{2}  8
V_{5}^{12} g_{2} g_{1}  ĝ_{1}ĝ_{2}^{2} ĝ_{1} −1 1 −1 −1 [b^{2}]^{ρ}  8
V_{5}^{13} g_{2} g_{1}  ĝ_{1}ĝ_{2}^{2} ĝ_{1}ĝ_{2}^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{5}^{14} g_{2} g_{1}  ĝ_{2}^{2}  ĝ_{1}ĝ_{2}^{3} −1 −i 1 −1 [b_{−+}^{3}]^{ρ}  16
V_{5}^{15} g_{2} g_{1}  ĝ_{2}^{2}  ĝ_{1}ĝ_{2} −1 i 1 −1 [b_{++}^{3}]^{ρ}  16
V_{5}^{16} g_{2} g_{1}  ĝ_{1}ĝ_{2}^{2} ĝ_{1}ĝ_{2}^{3} −1 −i −1 −1 [b_{−−}^{3}]^{ρ}  16
V_{5}^{17} g_{2} g_{1}  ĝ_{1}ĝ_{2}^{2} ĝ_{1}ĝ_{2} −1 i −1 −1 [b_{+−}^{3}]^{ρ}  16
V_{5}^{18} g_{2} g_{2}^{2}  ĝ_{2}^{2}  ĝ_{1}ĝ_{2} −1 i 1 −1 [b_{++}^{3}]^{ρ}  16

Continues in next page

\begin{matrix}V & h_{1} & h_{2} & χ_{1} & χ_{2} & b_{11} & b_{12} & b_{21} & b_{22} & b_{ij} & dim 𝔅(V) \\ V_{5}^{19} & g_{2} & g_{2}^{2} & ĝ_{2}^{2} & ĝ_{1}ĝ_{2}^{3} & −1 & −i & 1 & −1 & [b_{−+}^{3}]^{ρ} & 16 \\ V_{5}^{20} & g_{2} & g_{2}^{2} & ĝ_{1}ĝ_{2}^{2} & ĝ_{2} & −1 & i & 1 & −1 & [b_{++}^{3}]^{ρ} & 16 \\ V_{5}^{21} & g_{2} & g_{2}^{2} & ĝ_{1}ĝ_{2}^{2} & ĝ_{2}^{3} & −1 & −i & 1 & −1 & [b_{−+}^{3}]^{ρ} & 16 \\ V_{5}^{22} & g_{2}^{2} & g_{2}^{2} & ĝ_{2} & ĝ_{2} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{23} & g_{2}^{2} & g_{2}^{2} & ĝ_{2} & ĝ_{1}ĝ_{2} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{24} & g_{2}^{2} & g_{2}^{2} & ĝ_{2} & ĝ_{2}^{3} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{25} & g_{2}^{2} & g_{1} & ĝ_{2} & ĝ_{1} & −1 & 1 & 1 & −1 & b_{1}^{1} & 4 \\ V_{5}^{26} & g_{2}^{2} & g_{1} & ĝ_{2} & ĝ_{1}ĝ_{2}^{3} & −1 & −1 & 1 & −1 & b^{2} & 8 \\ V_{5}^{27} & g_{2}^{2} & g_{1} & ĝ_{1}ĝ_{2}^{3} & ĝ_{1}ĝ_{2}^{3} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{28} & g_{2}^{2} & g_{1} & ĝ_{1}ĝ_{2}^{3} & ĝ_{1}ĝ_{2} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{29} & g_{2}^{2} & g_{1} & ĝ_{1}ĝ_{2} & ĝ_{1} & −1 & 1 & −1 & −1 & [b^{2}]^{ρ} & 8 \\ V_{5}^{30} & g_{2}^{2} & g_{1} & ĝ_{2}^{3} & ĝ_{1} & −1 & 1 & 1 & −1 & b_{1}^{1} & 4 \\ V_{5}^{31} & g_{2}^{2} & g_{1}g_{2}^{2} & ĝ_{2} & ĝ_{1} & −1 & 1 & −1 & −1 & [b^{2}]^{ρ} & 8 \\ V_{5}^{32} & g_{1} & g_{1} & ĝ_{1}ĝ_{2}^{3} & ĝ_{1}ĝ_{2}^{3} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{33} & g_{1} & g_{1} & ĝ_{1}ĝ_{2}^{3} & ĝ_{1}ĝ_{2} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{34} & g_{1} & g_{1} & ĝ_{1}ĝ_{2}^{3} & ĝ_{1} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{35} & g_{1} & g_{1} & ĝ_{1} & ĝ_{1} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{36} & g_{1} & g_{1} & ĝ_{1} & ĝ_{1}ĝ_{2}^{2} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{37} & g_{1} & g_{1}g_{2}^{2} & ĝ_{1}ĝ_{2}^{3} & ĝ_{2} & −1 & 1 & 1 & −1 & b_{1}^{1} & 4 \\ V_{5}^{38} & g_{1} & g_{1}g_{2}^{2} & ĝ_{1}ĝ_{2}^{3} & ĝ_{1} & −1 & −1 & 1 & −1 & b^{2} & 8 \\ V_{5}^{39} & g_{1} & g_{1}g_{2}^{2} & ĝ_{1} & ĝ_{1} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \\ V_{5}^{40} & g_{1} & g_{1}g_{2}^{2} & ĝ_{1} & ĝ_{1}ĝ_{2}^{2} & −1 & −1 & −1 & −1 & b_{−1}^{1} & 4 \end{matrix}

We compute the liftings. For rank 1 we have

5.2.1. V = Y_{5}^{1}. As in 2.18, we have a ∈ P_{g_{2},1}(A), g_{1}a = ag_{1}, g_{2}a = iag_{2}. By 2.19, we have a^{4} = 0 and there is only one lifting: the bosonization 𝔅(Y_{5}^{1})#kΓ.

5.2.2. V = Y_{5}^{3}. Analogous to the previous one, the only difference being that g_{2}a = −iag_{2}. There is only one lifting: the bosonization 𝔅(Y_{5}^{3})#kΓ.

We go now to rank 2. As explained in 5.0.6, we have a_{1}, a_{2} with a_{i} ∈ P_{h_{i},1}(A), ta_{i}t^{−1} = χ_{i}(t)a_{i},

a_{1}^{2} = λ_{1}(h_{1}^{2} − 1), a_{2}^{2} = λ_{2}(h_{2}^{2} − 1), a_{1}a_{2} − χ_{2}(h_{1})a_{2}a_{1} = λ_{3}(h_{1}h_{2} − 1).

Before computing the liftings, we notice some facts.

Remark 5.2.3. In certain cases, some element in Aut(Γ) interchanges M_{1} and M_{2}. In this situation the lifting corresponding to (λ_{1}, λ_{2}, λ_{3}) is isomorphic to that corresponding to (λ_{2}, λ_{1}, ±λ_{3}) (the ± sign would depend on χ_{2}(h_{1}), but in the cases in which this happens it will turn out that λ_{3} = 0).

Remark 5.2.4. In some cases M_{1} = M_{2}, h_{1}^{2} ≠ 1 and χ_{1}^{2} = 1. In those cases, let W be the linear span of a_{1}, a_{2}. We can change the basis {a_{1}, a_{2}} by any other basis of W, and (similarly to the case W_{3}^{1}) we have a quadratic form f : W → k(h_{1}^{2} − 1), f(a) = a^{2}. As in that case, A is characterized by the rank of this form, whence there shall be 3 liftings.

Remark 5.2.5. In the case V_{5}^{4} (and V_{6}^{2}, V_{6}^{4} for Γ = C_{8}) there is no restriction to λ_{1}, λ_{2} and λ_{3}. If we replace {a_{1}, a_{2}} by {sa_{1}, ta_{2}} (s, t ∈ k^{×}), then we get an isomorphic algebra but with (λ_{1}, λ_{2}, λ_{3}) replaced by (s^{2}λ_{1}, t^{2}λ_{2}, stλ_{3}). There are thus infinitely many liftings, which are parametrized by the quotient variety k^{3}/ ∼, where

(λ_{1}, λ_{2}, λ_{3}) ∼ (s^{2}λ_{1}, t^{2}λ_{2}, stλ_{3}), s, t ∈ k^{×}.

Remark 5.2.6. If for some reason (for instance 2.19 or (5.0.10)) λ_{1} = 0 or λ_{2} = 0 then taking suitable scalar multiples of a_{1} or a_{2} we can suppose also that λ_{3} ∈ {0, 1}.

Here is a list of the liftings up to isomorphism. We put

a_{3} = a_{1}a_{2} − χ_{2}(h_{1})a_{2}a_{1}.

TABLE 11: Rank 2, Γ = C^{2} × C^{4}, liftings
V Lift. −b_{12} a_{1}^{2}  a_{2}^{2} a_{3} Remarks
V_{5}^{1} 3 + 0 0 0 See 5.2.4
g_{2}^{2} − 1 0  0
g_{2}^{2} − 1 g_{2}^{2} − 1 0
V_{5}^{2} 4 + 0 0 0 See 5.0.11
g_{2}^{2} − 1 0  0
0 g_{2}^{2} − 1 0
g_{2}^{2} − 1 g_{2}^{2} − 1 0
V_{5}^{3} 3 + 0 0 0 See 5.2.4
g_{2}^{2} − 1 0  0
g_{2}^{2} − 1 g_{2}^{2} − 1 0

Continues in next page

V Lift. −b_{12} a_{1}^{2}  a_{2}^{2}  a_{3}  Remarks
V_{5}^{4} ∞ + λ_{1}(g_{2}^{2} − 1), λ_{2}(g_{2}^{2} − 1), λ_{3}(g_{1}g_{2}^{2} − 1) See 5.2.5

(λ_{1}, λ_{2}, λ_{3})/ ∼

V_{5}^{6} 3 − 0 0 0 See 5.0.11 and 5.2.3
g_{2}^{2} − 1 0  0
g_{2}^{2} − 1 g_{2}^{2} − 1  0
V_{5}^{7} 3 + 0 0 0 See 2.19 and 5.2.3
g_{2}^{2} − 1 0  0
g_{2}^{2} − 1 g_{2}^{2} − 1  0
V_{5}^{8} 4 + 0 0 0 See 2.19
g_{2}^{2} − 1 0  0
0 g_{2}^{2} − 1  0
g_{2}^{2} − 1 g_{2}^{2} − 1  0
V_{5}^{9} 3 + 0 0 0 See 5.2.3
g_{2}^{2} − 1 0  0
g_{2}^{2} − 1 g_{2}^{2} − 1  0
V_{5}^{10} 2 − 0 0 0 See 2.19 and 5.0.11
g_{2}^{2} − 1 0  0
V_{5}^{13} 4 + 0 0 0 See 2.19 and 5.2.6
g_{2}^{2} − 1 0  0
0 0 g_{1}g_{2} − 1
g_{2}^{2} − 1 0  g_{1}g_{2} − 1
V_{5}^{22} 1  +  0  0  0  See 2.19
V_{5}^{23} 1  +  0  0  0  See 2.19
V_{5}^{24} 1  +  0  0  0  See 2.19
V_{5}^{25} 1  −  0  0  0  See 2.19 and 5.0.11
V_{5}^{27} 1  +  0  0  0  See 2.19 and 5.0.11
V_{5}^{28} 2  +  0  0  0  See 2.19 and 5.2.6
0  0  g_{1}g_{2}^{2} − 1
V_{5}^{30} 1  −  0  0  0  See 2.19 and 5.0.11
V_{5}^{32} 1  +  0  0  0  See 2.19
V_{5}^{33} 1  +  0  0  0  See 2.19
V_{5}^{34} 1  +  0  0  0  See 2.19
V_{5}^{35} 1  +  0  0  0  See 2.19
V_{5}^{36} 1  +  0  0  0  See 2.19

Continues in next page

V Lift. −b_{12} a_{1}^{2} a_{2}^{2} a_{3}  Remarks
V_{5}^{37} 1  − 0 0  0 See 2.19 and 5.0.11
V_{5}^{39} 1  + 0 0  0 See 2.19 and 5.2.6
0 0 g_{2}^{2} − 1
V_{5}^{40} 1  + 0 0  0 See 2.19 and 5.0.11

5.3. Γ = C_{8}. We have Aut(Γ) ≃ C_{2} × C_{2}, generated by

f_{1} = (g ↦ g^{3}), f_{2} = (g ↦ g^{5}).

The orbits of Γ are {g, g^{3}, g^{5}, g^{7}}, {g^{2}, g^{6}} and {g^{4}}. There are 12 orbits of irreducible YD-modules giving Nichols algebras of dimension a power of 2. We choose an element in each:

TABLE 12: modules/ Aut(Γ), Γ = C^{8}

M(h, χ) h χ  dim 𝔅(M)
Y_{6}^{1} g ĝ  8
Y_{6}^{2} g ĝ^{3}  8
Y_{6}^{3} g ĝ^{5}  8
Y_{6}^{4} g ĝ^{7}  8
Y_{6}^{5} g ĝ^{2}  4
Y_{6}^{6} g ĝ^{6}  4
Y_{6}^{7} g ĝ^{4}  2
Y_{6}^{8} g^{2} ĝ  4
Y_{6}^{9} g^{2} ĝ^{3}  4
Y_{6}^{10} g^{2} ĝ^{2}  2
Y_{6}^{11} g^{2} ĝ^{6}  2
Y_{6}^{12} g^{4} ĝ  2

Hence there are 4 different 4-dimensional Nichols algebras of rank 1. Now, for rank 2, we suppose M_{1} is one of Y_{6}^{7}, Y_{6}^{10}, Y_{6}^{11} or Y_{6}^{12}. We have the following possibilities (we take into account 2.16).

TABLE 13: Rank 2, Γ = C^{8}

V h_{1} h_{2} χ_{1} χ_{2} b_{11} b_{12} b_{21} b_{22} b_{ij} dim 𝔅(V)
V_{6}^{1} g g ĝ^{4} ĝ^{4} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{2} g g^{3} ĝ^{4} ĝ^{4} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{3} g g^{7} ĝ^{4} ĝ^{4} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{4} g g^{5} ĝ^{4} ĝ^{4} −1 −1 −1 −1 b_{−1}^{1}  4

Continues in next page

V h_{1} h_{2} χ_{1} χ_{2} b_{11} b_{12} b_{21} b_{22} b_{ij}  dim 𝔅(V)
V_{6}^{5} g g^{2} ĝ^{4} ĝ^{2} −1 i 1 −1 [b_{++}^{3}]^{ρ}  16
V_{6}^{6} g g^{2} ĝ^{4} ĝ^{6} −1 −i 1 −1 [b_{−+}^{3}]^{ρ}  16
V_{6}^{7} g g^{6} ĝ^{4} ĝ^{2} −1 i 1 −1 [b_{++}^{3}]^{ρ}  16
V_{6}^{8} g g^{6} ĝ^{4} ĝ^{6} −1 −i 1 −1 [b_{−+}^{3}]^{ρ}  16
V_{6}^{9} g g^{4} ĝ^{4} ĝ −1 ξ 1 −1 [b_{1}^{6}]^{ρ}  32
V_{6}^{10} g g^{4} ĝ^{4} ĝ^{3} −1 ξ^{3} 1 −1 [b_{3}^{6}]^{ρ}  32
V_{6}^{11} g g^{4} ĝ^{4} ĝ^{7} −1 ξ^{7} 1 −1 [b_{7}^{6}]^{ρ}  32
V_{6}^{12} g g^{4} ĝ^{4} ĝ^{5} −1 ξ^{5} 1 −1 [b_{5}^{6}]^{ρ}  32
V_{6}^{13} g^{2} g^{2} ĝ^{2} ĝ^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{14} g^{2} g^{2} ĝ^{2} ĝ^{6} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{15} g^{2} g^{2} ĝ^{6} ĝ^{6} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{16} g^{2} g^{6} ĝ^{2} ĝ^{6} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{17} g^{2} g^{6} ĝ^{6} ĝ^{6} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{18} g^{2} g^{6} ĝ^{6} ĝ^{2} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{19} g^{4} g^{2} ĝ ĝ^{2} −1 1 i −1 b_{++}^{3}  16
V_{6}^{20} g^{4} g^{2} ĝ ĝ^{6} −1 1 i −1 b_{++}^{3}  16
V_{6}^{21} g^{4} g^{6} ĝ ĝ^{6} −1 1 −i −1 b_{−+}^{3}  16
V_{6}^{22} g^{4} g^{6} ĝ ĝ^{2} −1 1 −i −1 b_{−+}^{3}  16
V_{6}^{23} g^{4} g^{4} ĝ ĝ −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{24} g^{4} g^{4} ĝ ĝ^{3} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{25} g^{4} g^{4} ĝ ĝ^{7} −1 −1 −1 −1 b_{−1}^{1}  4
V_{6}^{26} g^{4} g^{4} ĝ ĝ^{5} −1 −1 −1 −1 b_{−1}^{1}  4

We compute now the liftings. For rank 1, we have the following algebras, where we take a as in 5.0.6.

TABLE 14: Rank 1, Γ = C^{8}

V h χ a^{4}  Remarks
Y_{6}^{5} g ĝ^{2} 0
g^{4} − 1
Y_{6}^{6} g ĝ^{6} 0
g^{4} − 1
Y_{6}^{8} g^{2} ĝ  0 See (2.19)
Y_{6}^{9} g^{2} ĝ^{3} 0 See (2.19)

For rank 2, as in 5.0.6, we have a_{1}, a_{2} with a_{i} ∈ P_{h_{i},1}(A), ta_{i}t^{−1} = χ_{i}(t)a_{i},

a_{i}^{2} = λ_{i}(h_{i}^{2} − 1) (i = 1, 2), a_{1}a_{2} + a_{2}a_{1} = λ_{3}(h_{1}h_{2} − 1)

(notice that b_{12} = χ_{2}(h_{1}) = −1 in all the 4-dimensional cases). Equations (5.0.10), (5.0.11) and remarks 5.2.3, 5.2.4, 5.2.5 and 5.2.6 apply here also. We put

a_{3} = a_{1}a_{2} + a_{2}a_{1}.

TABLE 15: Rank 2, Γ = C^{8}, liftings

V Lift. a_{1}^{2}  a_{2}^{2}  a_{3}  Remarks
V_{6}^{1} 3  0  0  0  See 5.2.4
g^{2} − 1 0  0
g^{2} − 1 g^{2} − 1  0
V_{6}^{2} ∞ λ_{1}(g^{2} − 1), λ_{2}(g^{6} − 1), λ_{3}(g^{4} − 1) See 5.2.5
(λ_{1}, λ_{2}, λ_{3})/ ∼
V_{6}^{3} 3 0 0 0 See 5.2.3
g^{2} − 1 0  0
g^{2} − 1 g^{6} − 1  0
V_{6}^{4} ∞ λ_{1}(g^{2} − 1), λ_{2}(g^{2} − 1), λ_{3}(g^{6} − 1) See 5.2.5
(λ_{1}, λ_{2}, λ_{3})/ ∼
V_{6}^{13} 1  0  0  0  See 5.0.10 and 5.0.11
V_{6}^{14} 2  0  0  0  See 5.0.10 and 5.2.6
0  0  g^{4} − 1
V_{6}^{15} 1  0  0  0  See 5.0.10 and 5.0.11
V_{6}^{16} 1  0  0  0  See 2.19 and 5.0.10
V_{6}^{17} 1  0  0  0  See 2.19 and 5.0.10
V_{6}^{18} 1  0  0  0  See 5.0.10 and 5.0.11
V_{6}^{23} 1  0  0  0  See 2.19
V_{6}^{24} 1  0  0  0  See 2.19
V_{6}^{25} 1  0  0  0  See 2.19
V_{6}^{26} 1  0  0  0  See 2.19

5.4. Γ = 𝔻_{4}. We take 𝔻_{4} generated by r and σ with relations r^{4} = σ^{2} = 1, σr = r^{3}σ. The conjugacy classes of Γ are {1}, {r^{2}}, {σ, r^{2}σ}, {rσ, r^{3}σ} and {r, r^{3}}. As explained in 5.0.2, there are two possibilities for V. For case 5.0.2(2), h must be in a conjugacy class with 2 elements, and hence we can take h = σ, h = rσ or h = r. However, the cases h = σ and h = rσ give isomorphic algebras when bosonized by means of the morphism (r ↦ r, σ ↦ rσ), and hence we take only the cases h = σ and h = r. The commutator subgroups are

Γ_{σ} = {1, σ, r^{2}, r^{2}σ} ≃ C_{2} × C_{2}, Γ_{r} = {1, r, r^{2}, r^{3}} ≃ C_{4}.

Case 5.0.2(3) arises taking h = r^{2} and ρ = ρ_{0} the irreducible representation given by

ρ_{0}(r) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , ρ_{0}(σ) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

The following is the list of irreducible YD-modules giving Nichols algebras of dimension 4:

TABLE 16: Γ = 𝔻_{4}

M(h, ρ) h  ρ  b_{12} = b_{21} dim 𝔅(M)
Y_{7}^{1}  σ σ ↦ −1, r^{2} ↦ 1  −1  4
Y_{7}^{2}  σ σ ↦ −1, r^{2} ↦ −1  1  4
Y_{7}^{3}  r  r ↦ −1  −1  4
Y_{7}^{4} r^{2}  ρ_{0}  −1  4

We compute the liftings. Though we can state some general conditions as in (5.0.11), it turns out to be simpler to use the diamond lemma in each case.

5.4.1. V = Y_{7}^{1}. As in 5.0.3, we have a ∈ P_{σ,1}(A) and b = rar^{3} ∈ P_{r^{2}σ,1}(A). Since

σ ⇀ x = −x, r ⇀ x = y, r ⇀ y = r^{2} ⇀ x = x,
σ ⇀ y = σr ⇀ x = r^{3}σ ⇀ x = r(r^{2}σ) ⇀ x = −r ⇀ x = −y,

we have

σaσ = −a, rar^{3} = b, σbσ = −b, rbr^{3} = a.

By 2.19, a^{2} = b^{2} = 0, and ab + ba ∈ P_{r^{2},1}(A), from where (taking a suitable scalar multiple of a or b), we have ab + ba = λ(r^{2} − 1), with λ ∈ {0, 1}. There are hence two liftings:

1. a^{2} = b^{2} = 0, ab + ba = 0. This is the bosonization 𝔅(Y_{7}^{1})#kΓ.

2. a^{2} = b^{2} = 0, ab + ba = (r^{2} − 1).

5.4.2. V = Y_{7}^{2}. It is similar to the case Y_{7}^{1}: we have a ∈ P_{σ,1}(A) and b = rar^{3} ∈ P_{r^{2}σ,1}(A) such that

σaσ = −a, rar^{3} = b, σbσ = b, rbr^{3} = −a.

As in that case, a^{2} = b^{2} = 0 by 2.19, and ab − ba = λ(r^{2} − 1) with λ ∈ {0, 1}, but now the diamond lemma applies:

0 = bba = bab − λb(r^{2} − 1) = abb − λ(r^{2} − 1)b − λb(r^{2} − 1) = 2λb,

from where λ = 0 for A to be 32-dimensional.

There is hence only one lifting: the bosonization 𝔅(Y_{7}^{2})#kΓ.

5.4.3. V = Y_{7}^{3}. By 5.0.3, we have here a ∈ P_{r,1}(A), b = σaσ ∈ P_{r^{3},1}(A), and

rar^{3} = −a, rbr^{3} = −b, σaσ = b, σbσ = a.

We have now a^{2} = λ_{1}(r^{2} − 1), b^{2} = λ_{2}(r^{2} − 1). Since ab + ba ∈ P(A) then ab + ba = 0 by 2.19. We can take λ_{i} ∈ {0, 1}, from where we have four liftings. However, the automorphism r ↦ r^{3}, σ ↦ σ gives an isomorphism between the algebra corresponding to (λ_{1} = 0, λ_{2} = 1) and that of (λ_{1} = 1, λ_{2} = 0), and thus there are three non isomorphic liftings:

1. a^{2} = 0, b^{2} = 0, ab + ba = 0. This is the bosonization 𝔅(Y_{7}^{3})#kΓ.

2. a^{2} = (r^{2} − 1), b^{2} = 0, ab + ba = 0.

3. a^{2} = (r^{2} − 1), b^{2} = (r^{2} − 1), ab + ba = 0.

5.4.4. V = Y_{7}^{4}. As in 5.0.4 (with t = r), we have here a, b ∈ P_{r^{2},1}(A), with

rar^{3} = b, rbr^{3} = −a, σaσ = b, σbσ = a.

By 2.19, we have a^{2} = b^{2} = ab + ba = 0, and thus there is only one lifting: the bosonization 𝔅(Y_{7}^{4})#kΓ.

5.5. Γ = ℍ. We take ℍ = \{e, (−e), \underline{i}, −\underline{i}, \underline{j}, −\underline{j}, \underline{k}, −\underline{k}\} with the standard relations \underline{i}\underline{j} = \underline{k}, etc. The conjugacy classes are \{e\}, \{(−e)\}, \{\underline{i}, −\underline{i}\}, \{\underline{j}, −\underline{j}\} and \{\underline{k}, −\underline{k}\}. Since the last three classes give isomorphic algebras via the morphisms

(\underline{i} ↦ \underline{j} ↦ \underline{k} ↦ \underline{i}) and (\underline{i} ↦ −\underline{i}, \underline{j} ↦ −\underline{j}, \underline{k} ↦ −\underline{k}),

there is only one case as in 5.0.2(2), which is given by h = \underline{i}, χ(\underline{i}) = −1. There is also only one case as in 5.0.2(3), which is given by h = (−e), ρ = ρ_{0}, where

ρ_{0}(\underline{i}) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , ρ_{0}(\underline{j}) = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , ρ_{0}(\underline{k}) = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) .

The list of 4-dimensional Nichols algebras giving non isomorphic algebras when bosonized is given hence by

TABLE 17: Γ = ℍ

M(h, ρ) h  ρ  b_{12} = b_{21} dim 𝔅(M)
Y_{8}^{1}  \underline{i} \underline{i} ↦ −1  −1  4
Y_{8}^{2} (−e) ρ_{0}  −1  4

The liftings are:

5.5.1. V = Y_{8}^{1}. As in 5.0.3 we have a ∈ P_{\underline{i},1}(A) and b = \underline{j}a(−\underline{j}) ∈ P_{−\underline{i},1}(A), with the action given by

\underline{i}a(−\underline{i}) = −a, \underline{j}a(−\underline{j}) = b, \underline{k}a(−\underline{k}) = −b,
\underline{i}b(−\underline{i}) = −b, \underline{j}b(−\underline{j}) = a, \underline{k}b(−\underline{k}) = −a.

We have λ_{1}, λ_{2} ∈ {0, 1} such that a^{2} = λ_{1}((−e) − 1), b^{2} = λ_{2}((−e) − 1), ab + ba ∈ P(A) ⇒ ab + ba = 0. The diamond lemma says, however, that λ_{1} = λ_{2}, since

\underline{j}a^{2} = λ_{1}\underline{j}((−e) − 1) = λ_{1}((−\underline{j}) − \underline{j}),
\underline{j}a^{2} = b\underline{j}a = b^{2}\underline{j} = λ_{2}((−e) − 1)\underline{j} = λ_{2}((−\underline{j}) − \underline{j}).

Hence, there are two liftings:

1. a^{2} = 0, b^{2} = 0, ab + ba = 0. This is the bosonization 𝔅(Y_{8}^{1})#kΓ.

2. a^{2} = ((−e) − 1), b^{2} = ((−e) − 1), ab + ba = 0.

5.5.2. V = Y_{8}^{2}. As in 5.0.4 we have a, b ∈ P_{−e,1}(A) with the action given by

\underline{i}a(−\underline{i}) = b, \underline{j}a(−\underline{j}) = ia, \underline{k}a(−\underline{k}) = ib,
\underline{i}b(−\underline{i}) = −a, \underline{j}b(−\underline{j}) = −ib, \underline{k}b(−\underline{k}) = ia.

By 2.19, we have a^{2} = b^{2} = 0, and ab + ba = 0, and hence there is only one lifting: the bosonization 𝔅(Y_{8}^{2})#kΓ.

6. Γ of order 16

Now dim 𝔅(V) = 2, from where we must have V = M(h, χ), h ∈ Z(Γ) and χ a character such that χ(h) = −1. Hence, for Γ to have YD-modules V with dim 𝔅(V) = 2, it is necessary that Z(Γ) ⊄ [Γ; Γ]. There are 14 groups of order 16. Among them, 3 do not verify this condition.

As explained in 2.18, we take a ∈ P_{h,1}(A) such that ga = χ(g)ag for g ∈ Γ. Furthermore, there exists λ ∈ {0, 1} such that a^{2} = λ(h^{2} − 1). Analogously to (5.0.8), for A to be 32-dimensional we have

λg(h^{2} − 1) = ga^{2} = χ^{2}(g)a^{2}g = λχ^{2}(g)(h^{2} − 1)g ∀g ∈ Γ

whence λ(χ^{2}(g) − 1)(h^{2} − 1) = 0. Combining this with 2.19 we get

λ(χ^{2} − 1) = 0. (6.0.1)

6.1. Γ = C_{2} ×C_{2} ×C_{2} ×C_{2}. As in the case Γ = C_{2} ×C_{2} ×C_{2}, the group Aut(Γ) acts transitively on the YD-modules V such that dim 𝔅(V) = 2, from where we get only one algebra after bosonization. We take hence V = M(g_{1}, ĝ_{1}). By 2.19, there is only one lifting: the bosonization 𝔅(V)#kΓ.

6.2. Γ = C_{2}×C_{2}×C_{4}. There are 4 orbits with respect to the action of Aut(Γ): {1}, {g_{3}^{2}}, {g_{1}^{i}g_{2}^{j}g_{3}^{k}, (i, j) ≠ (0, 0), k = 0, 2}, {g_{1}^{i}g_{2}^{j}g_{3}^{k}, k = 1, 3} (we are using the notation at the end of 1.1). The first orbit, as always, does not give a Nichols algebra of dimension 2. For the others, we have the following list of YD-modules giving non isomorphic algebras when bosonized (we indclude only the cases dim 𝔅(V) = 2). We include also their corresponding liftings:

TABLE 18: Γ = C^{2} × C^{2} × C^{4}, liftings
M(h, χ) Liftings h χ  a^{2}  Remarks
Y_{9}^{1}  1  g_{3}^{2} ĝ_{3}  0  See 2.19
Y_{9}^{2}  1  g_{1}  ĝ_{1}  0  See 2.19
Y_{9}^{3}  1  g_{1} ĝ_{1}ĝ_{3}  0  See 2.19
Y_{9}^{4}  2  g_{3} ĝ_{3}^{2}  0
(g_{3}^{2} − 1)
Y_{9}^{5} 2 g_{3} ĝ_{1}ĝ_{3}^{2} 0
(g_{3}^{2} − 1)

6.3. Γ = C_{4} × C_{4}. The orbits are {1}, {g_{1}^{2}, g_{1}^{2}g_{2}^{2}, g_{2}^{2}}, {g_{1}^{i}g_{2}^{j}, 2 |̸i or 2 |̸j} (we are using the notation at the end of 1.1). The possibilities are

TABLE 19: Γ = C^{4} × C^{4}, liftings
M(h, χ) Liftings h χ  a^{2}  Remarks
Y_{10}^{1}  2  g_{1} ĝ_{1}^{2}  0
(g_{1}^{2} − 1)
Y_{10}^{2}  1  g_{1} ĝ_{1}^{2}ĝ_{2}  0  See 6.0.1
Y_{10}^{3}  1  g_{1}^{2} ĝ_{1}  0  See 2.19

6.4. Γ = C_{2} × C_{8}. The orbits are {1}, {g_{2}^{4}}, {g_{1}g_{2}^{2}, g_{1}g_{2}^{6}}, {g_{1}, g_{1}g_{2}^{4}}, {g_{2}^{2}, g_{2}^{6}}, {g_{1}^{i}g_{2}^{j}, j = 1, 3, 5, 7}. The list is

TABLE 20: Γ = C^{2} × C^{8}, liftings

M(h, χ) Liftings h  χ  a^{2}  Remarks
Y_{11}^{1}  1  g_{2}^{4}  ĝ_{2}  0  See 2.19
Y_{11}^{2}  2  g_{1}g_{2}^{2} ĝ_{1}  0
(g_{2}^{4} − 1)
Y_{11}^{3}  1  g_{1}g_{2}^{2} ĝ_{2}^{2}  0  See 6.0.1
Y_{11}^{4}  1  g_{1}  ĝ_{1}  0  See 2.19
Y_{11}^{5}  1  g_{1} ĝ_{1}ĝ_{2}^{2}  0  See 2.19
Y_{11}^{6}  1  g_{1}  ĝ_{1}ĝ_{2}  0  See 2.19
Y_{11}^{7}  1  g_{2}^{2}  ĝ_{2}^{2}  0  See 6.0.1
Y_{11}^{8}  1  g_{2}^{2}  ĝ_{2}^{6}  0  See 6.0.1
Y_{11}^{9}  1  g_{2}^{2} ĝ_{1}ĝ_{2}^{2}  0  See 6.0.1
Y_{11}^{10}  2  g_{2}  ĝ_{2}^{4}  0
(g_{2}^{2} − 1)
Y_{11}^{11} 2 g_{2} ĝ_{1}ĝ_{2}^{4} 0
(g_{2}^{2} − 1)

6.5. Γ = C_{16}. The orbits are

{1}, {g^{8}}, {g^{4}, g^{12}}, {g^{2}, g^{6}, g^{10}, g^{14}}, {g^{i}, 2 |̸i}.

The list is

TABLE 21: Γ = C^{16}, liftings

M(h, χ) Liftings h χ  a^{2}  Remarks
Y_{12}^{1}  1  g^{8} ĝ  0  See 2.19
Y_{12}^{2}  1  g^{4} ĝ^{2}  0  See 6.0.1
Y_{12}^{3}  1  g^{4} ĝ^{6}  0  See 6.0.1
Y_{12}^{4}  1  g^{2} ĝ^{4}  0  See 6.0.1
Y_{12}^{5}  1  g^{2} ĝ^{12}  0  See 6.0.1
Y_{12}^{6}  2  g ĝ^{8}  0
(g^{2} − 1)

6.6. Non abelian groups. As said above, the non abelian groups which verify Z(Γ) ⊄ [Γ; Γ] are 6, and can be presented in the following list:

TABLE 22: Non abelian groups of order 16

Group

Generators, relations Z(Γ) [Γ; Γ] Γ_{ab}

B_{1}  (g_{1}, g_{2})/(g_{1}^{8}, g_{2}^{2}, g_{2}g_{1}g_{2}g_{1}^{3}) (g_{1}^{2}) (g_{1}^{4})  C_{4} × C_{2}
B_{2}  (g_{1}, g_{2}, g_{3})/R_{2}  (g_{1}) (g_{1}^{2}) C_{2} × C_{2} × C_{2}
B_{3}  (g_{1}, g_{2})/(g_{1}^{4}, g_{2}^{4}, g_{2}g_{1}g_{2}^{3}g_{1}) (g_{1}^{2}, g_{2}^{2}) (g_{1}^{2})  C_{2} × C_{4}
B_{4} = 𝔻_{4} × C_{2}  (r, σ, t)  (t, r^{2}) (r^{2}) C_{2} × C_{2} × C_{2}
B_{5}  (g_{1}, g_{2}, g_{3})/R_{5}  (g_{1}^{2}, g_{2}) (g_{2})  C_{4} × C_{2}
B_{6} = ℍ × C_{2}  (\underline{i}, \underline{j}, t)  (t, −e) (−e) C_{2} × C_{2} × C_{2}

where we put for B_{2} and B_{5} the subgroups R_{2} and R_{5} respectively, generated by the elements

R_{2} : g_{1}^{4}, g_{2}^{2}, g_{3}^{2}, [g_{1}; g_{2}], [g_{1}; g_{3}], [g_{3}; g_{2}]g_{1}^{2},
R_{5} : g_{1}^{4}, g_{2}^{2}, g_{3}^{2}, [g_{1}; g_{2}], [g_{3}; g_{2}], [g_{1}; g_{3}]g_{2}.

The name B_{i} corresponds to the classification in [Bu, Sect. 118]. In these groups, we have [Γ; Γ] ⊂ Z(Γ), (Z(Γ) − [Γ; Γ]) = {h_{1}, h_{2}}. Hence, there are only two possibilities for h in each group.

For Γ = B_{4}, for instance, h_{1} = t and h_{2} = r^{2}t. Since the automorphism

(r ↦ r, σ ↦ σ, t ↦ r^{2}t)

carries h_{1} to h_{2}, we consider only h = t. There are 4 characters χ ∈ Γ̂ such that χ(t) = −1. They are given by

χ(t) = −1, χ(r) = ±1, χ(σ) = ±1.

However, the automorphisms

(t ↦ t, r ↦ t^{i}r, σ ↦ t^{j}σ) i, j = 0, 1

act transitively on these representations, whence we have only one algebra.

Doing the same analysis with the other groups, we arrive to the following list (we take a as explained at the beginning of the section):

TABLE 23: Γ non abelian of order 16, liftings

Group M(h, χ) Lift. h  χ  a^{2}  Remarks
B_{1}  Y_{13}^{1}  1 g_{1}^{2} χ(g_{1}) = i χ(g_{2}) = 1  0 See 6.0.1
B_{1}  Y_{13}^{2}  1 g_{1}^{2} χ(g_{1}) = i χ(g_{2}) = −1 0 See 6.0.1
B_{1}  Y_{13}^{3}  1 g_{1}^{2} χ(g_{1}) = −i χ(g_{2}) = 1  0 See 6.0.1
B_{1}  Y_{13}^{4}  1 g_{1}^{2} χ(g_{1}) = −i χ(g_{2}) = −1 0 See 6.0.1
B_{2}  Y_{13}^{5}  2 g_{1} χ(g_{1}) = −1 χ(g_{2}) = 1  0
χ(g_{3}) = 1 g_{1}^{2} − 1
B_{2} Y_{13}^{6} 2 g_{1} χ(g_{1}) = −1 χ(g_{2}) = −1 0
χ(g_{3}) = −1 g_{1}^{2} − 1
B_{3}  Y_{13}^{7}  1 g_{2}^{2} χ(g_{1}) = 1 χ(g_{2}) = i  0 See 2.19
B_{3}  Y_{13}^{8}  1 g_{1}^{2}g_{2}^{2} χ(g_{1}) = 1 χ(g_{2}) = i  0 See 2.19
B_{4}  Y_{13}^{9}  1  t χ(t) = −1 χ(r) = 1  0 See 2.19
χ(σ) = 1
B_{5} Y_{13}^{10} 1 g_{1}^{2} χ(g_{1}) = i χ(g_{2}) = 1 0 See 2.19
χ(g_{3}) = 1
B_{6} Y_{13}^{11} 1 t χ(t) = −1 χ(\underline{i}) = 1 0 See 2.19
χ(\underline{i}) = 1

7. The classification is complete

We use the techniques of [AS2, Sect. 8]. If A is a 32-dimensional pointed Hopf algebra, we apply the procedures described in 1.2 and get R a braided Hopf algebra in _{kΓ}^{kΓ}YD satisfying (1.1.2), (1.1.3) and (1.1.4). We may consider the (braided) dual R^{∗} as in [AG, Def. 2.2.2], which is a braided Hopf algebra in _{kΓ}^{kΓ}YD satisfying (1.1.2), (1.1.3) and (1.1.5). Let W = R^{∗}(1). We get in this way the sequence of braided Hopf algebras

T W → R^{∗} ↠ 𝔅(W),

whence 𝔅(W)#kΓ is an algebra of dimension 2^{n}, with n ≤ 5. If n = 5 then R^{∗} = 𝔅(W), from where R^{∗} satisfies (1.1.4) and hence R satisfies (1.1.5), whence R is a Nichols algebra and A is in the list above. We must prove thence that the above surjection R^{∗} ↠ 𝔅(W) is in fact a bijection. This is the same as to prove that all the relations in 𝔅(W) are relations in R^{∗}, i.e., if z ∈ T W vanishes in 𝔅(W), then it vanishes in R^{∗}. For this, notice first that for Γ non abelian there are no bosonizations of Nichols algebras over Γ with dimension 2^{n}, n < 5 (see 5.0.1).

Consider then Γ abelian. If W is 1-dimensional, let x be a generator of W. We have c(x ⊗ x) = qx ⊗ x for some q a root of unity. Let r be the order of q (q ≠ 1 because of 2.2). Thus, x^{r} ∈ P(R^{∗}) and c(x^{r} ⊗ x^{r}) = (x^{r} ⊗ x^{r}), whence by 2.2 we have x^{r} = 0 ∈ R^{∗}. This implies that R^{∗} verifies (1.1.4), and we are done.

Consider now dim W ≥ 2. By 2.5, dim 𝔅(W) ≥ 2^{dim W}, and then

dim 𝔅(W)#kΓ ≥ 2^{dim W} |Γ| ≥ 4|Γ|.

Thus, if R were not a Nichols algebra, |Γ| would be ≤ 4.

Let Γ = C_{2}. We have W = ⊕_{i=1}^{θ}M(g, ĝ). Then for y ∈ W we have y^{2} ∈ P(R^{∗}) and c(y^{2}⊗y^{2}) = y^{2}⊗y^{2}, from where by 2.2 we have y^{2} = 0 ∀y ∈ W. These are the relations for 𝔅(W), and we are done.

If |Γ| = 4, the bijectivity of the projection above is a consequence of the following

Proposition 7.1. Let Γ be a finite group, S = ⊕_{i}S(i) ∈ _{kΓ}^{kΓ}YD a graded braided Hopf algebra that verifies (1.1.3) and (1.1.5), and such that dim S = 8. Then S is a Nichols algebra.

Proof. Let V = S(1) and R′ = 𝔅(V). We have S ↠ R′ the canonical projection. If dim R′ = 8, we are done. Consider then dim R′ < 8. Since the projection induces an epimorphism S#kΓ ↠ R′#kΓ, we have dim R′ = 1, 2 or 4. Since

dim R ≥ dim R(0) + dim R(1) = dim S(0) + dim S(1) = 1 + dim S(1),

we have dim R′ = 2 or 4. If dim R′ = 2, then dim S(1) = 1 and by the considerations for the rank-1 case, we would have S = R′. Thus dim R′ = 4. Now 2.7 implies that dim R′(1) = 2, and then by [AG, Prop 3.1.11], R′(1) comes from the abelian case. Furthermore, by the same computations as in

5.0.3 and 5.0.4, the matrix of R′ is ( \begin{matrix}−1 & ζ \\ ζ & −1 \end{matrix} ) (with ζ = ±1) in some

basis {x, y}. Now, as in 2.1, we have x^{2} ∈ P(S) and y^{2} ∈ P(S), and it is straightforward to see that z = xy − ζyx ∈ P(S). Moreover,

c(x^{2} ⊗ x^{2}) = x^{2} ⊗ x^{2}, c(y^{2} ⊗ y^{2}) = y^{2} ⊗ y^{2}, c(z ⊗ z) = z ⊗ z.

Thus, thanks to 2.2, we have x^{2} = y^{2} = z = 0. Since S is generated by x and y, this implies dim S = 4, a contradiction.

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