Examples of liftings of Nichols algebras over racks ^{∗}

Nicolás Andruskiewitsch and Matías Graña

September 4, 2003

Abstract

We review known examples of finite-dimensional Nichols algebras associated to racks. We discuss pointed Hopf algebras whose infinitesimal Yetter-Drinfeld module is a realization of the rack of transpositions in 𝕊_{n} with constant cocycle −1.

2000 AMS Subject Classification: 16W30 Keywords: pointed Hopf algebras

1 Introduction

A general method for the classification of Hopf algebras whose coradical is a Hopf subalgebra was proposed in [AS1], see also [AS3]. In the case of pointed Hopf algebras of finite dimension, by [G1, Lemma 3.8] (see also [AG, Thm. 4.14]), we are first faced to the following question.

Question 1.1. For any finite rack X and for any 2-cocycle q : X ×X → ℂ^{×}, determine if the Nichols algebra of the braided vector space (ℂX, c^{q}) has finite dimension.

Recall that the Nichols algebra of a braided vector space (V, c) is given by 𝔅(V) = T (V)/J where J = ⊕_{n≥2}J_{n}, and where J_{n} ⊂ T^{n}(V) is the kernel of the quantum symmetrizer; see e. g. [AS3]. Let r ≥ 2; the rth. partial Nichols algebra of (V, c) is 𝔅̂_{r}(V) = T (V)/⟨⊕_{2≤n≤r}J_{n}⟩, see [AG, 6.3]. Recall also that a rack is a non-empty set endowed with an operation ▷ : X × X → X such that ϕ_{i} := i ▷ is a bijection for all i ∈ X, and the self-distributivity axiom holds: i ▷ (j ▷ k) = (i ▷ j) ▷ (i ▷ k), for all i, j, k ∈ X. The braiding c^{q} : ℂX ⊗ ℂX → ℂX ⊗ ℂX, on the vector space ℂX with a basis (x_{i})_{i∈X}, is given by c(x_{i} ⊗ x_{j}) = q_{ij} x_{i▷j} ⊗ x_{i}, for all i, j ∈ X; it is a solution of the braid equation by the cocycle condition q_{i▷j,i▷k}q_{i,k} = q_{i,j▷k}q_{j,k} ∀i, j, k ∈ X. See [AG, Ch. 1] for a survey on racks focused on our needs for classification of pointed Hopf algebras. Nichols algebras corresponding to a trivial rack (ϕ_{i} = id ∀i ∈ X) were considered in several

articles, see [AS3]. The examples of racks we consider below are:

• Subsets of a group stable under conjugation; then a ▷ b = aba^{−1}.

• Particularly, affine (or Alexander) racks; these are pairs (A, g) where A is a finite abelian group and g ∈ Aut A; then a ▷ b = g(b) + (id −g)(a) defines a rack structure on A. If A is a finite ring and g is multiplication by an invertible N ∈ A, we shall denote (A, N) := (A, g).

We collect the main examples of Nichols algebras which are known to be finite-dimensional; we exclude the diagonal case (= the rack is trivial). Other examples are in [AG, Prop. 6.8]; they are not of diagonal type but they arise from Nichols algebras of diagonal type by a kind of Fourier transform.

∗This work was partially supported by CONICET, Agencia Córdoba Ciencia, ANPCyT and Secyt (UNC). N. A.

thanks the kind invitation of R. Berger and LARAL to the University of St. Etienne, where his work was done. M. G.

thanks the warm hospitality of IRMA and ULP, Strasbourg, where his work was done.

Theorem 1.2. In all the examples below we take q_{ij} = −1 for all i, j.

(i). Let X be the set of transpositions in 𝕊_{n}, n = 3, resp. 4, 5. Then the Nichols algebra of the corresponding braided vector space (V, c) = (ℂX, c^{q}) is quadratic and has dimension 12, resp. 576, 8294400. Its Hilbert polynomial has degree 4, resp. 12, 40. The space J_{2} of relations in degree 2 has a basis

x_{σ}^{2}  ∀σ ∈ X,
x_{σ}x_{τ} + x_{τ} x_{σ}  ∀σ ≠ τ ∈ X s.t. στ = τσ,
x_{σ}x_{τ} + x_{ν}x_{σ} + x_{τ} x_{ν}  ∀σ ≠ τ ≠ ν ∈ X s.t. στ = νσ,

(consider only one vector for each pair σ, τ or triple σ, τ, ν).

(ii). Let (A, g) = (𝔽_{p}, N) where p = 3, 4, 5, 7, N = 2, ω, 2, 3 respectively (here ω^{2} + ω + 1 = 0 ∈ 𝔽_{4}). Then:

• The Nichols algebra of the corresponding braided vector space (V, c) = (ℂX, c^{q}) has dimension pφ(p)(p − 1)^{p−2} (here φ is the Euler function).

• Its Hilbert polynomial has degree (p − 1)^{2}.

• The defining ideal J is generated by J_{2} + J_{v(p)(p−1)} (here v(p) = 2 for p = 4 and v(p) = 1 otherwise).

Furthermore, J_{2} has a basis (we write subindices in 𝔽_{p})

x_{i}^{2},  always  (1)
x_{i}x_{j} + x_{−i+2j}x_{i} + x_{j}x_{−i+2j}  for p = 3,  (2)
x_{i}x_{j} + x_{(ω+1)i+ωj}x_{i} + x_{j}x_{(ω+1)i+ωj}  for p = 4,  (3)
x_{i}x_{j} + x_{−i+2j}x_{i} + x_{3i−2j}x_{−i+2j} + x_{j}x_{3i−2j}  for p = 5,  (4)
x_{i}x_{j} + x_{−2i+3j}x_{i} + x_{j}x_{−2i+3j}  for p = 7,  (5)

(consider only one of repeated vectors). The relations in J_{v(p)(p−1)} are generated by J_{2} and the independent relation

x_{ω}x_{1}x_{0}x_{ω}x_{1}x_{0} + x_{1}x_{0}x_{ω}x_{1}x_{0}x_{ω} + x_{0}x_{ω}x_{1}x_{0}x_{ω}x_{1}  for p = 4,  (6)
x_{1}x_{0}x_{1}x_{0} + x_{0}x_{1}x_{0}x_{1}  for p = 5,  (7)
x_{2}x_{1}x_{0}x_{2}x_{1}x_{0} + x_{1}x_{0}x_{2}x_{1}x_{0}x_{2} + x_{0}x_{2}x_{1}x_{0}x_{2}x_{1}  for p = 7.  (8)

(iii). The Nichols algebra of the rack defined by the faces of the cube is quadratic and has the same dimension and Hilbert series as the Nichols algebra corresponding to the transpositions in 𝕊_{4} (see [AG, Thm. 6.12]).

Proof. The case of the transpositions in 𝕊_{3} (which is equal to (𝔽_{3}, 2)) and the transpositions in 𝕊_{4} are in [MS]. In the case of the transpositions in 𝕊_{5}, the quadratic approximation 𝔅̂_{2}(V) was computed with the help of the computer program bergman. The equality 𝔅̂_{2}(V) = 𝔅(V) was established in [G2] with a program in C there called Deriva, using [AG, Th. 6.4]. An algebra over the same rack but with a different cocycle appeared in [FK]. Both algebras have the same Hilbert polynomial, a coincidence also found in [MS] for 𝕊_{3} and 𝕊_{4}. The case of (𝔽_{4}, ω) is treated in [G1]. The cases of (𝔽_{5}, 2) and the faces of the cube appear in [AG]. The Nichols algebra corresponding to (𝔽_{7}, 3) was computed with

bergman and Deriva, see [G2].

Remark 1.3. The cases X = (𝔽_{p}, N) with N = 3, 5 for p = 5, 7 respectively are analogous to those stated here. In fact, these are the duals to those in the theorem. The other Nichols algebras in the theorem are self-dual.

In the rest of this Note, we consider the next steps of the Lifting method in the context of the Nichols algebras in 1.2.

2 Generation in degree one

Theorem 2.1. Any finite dimensional complex pointed Hopf algebra such that its infinitesimal braiding (see [AS3, Def. 1.15]) is one of those in 1.2 is generated as an algebra by its group-like and skewprimitive elements.

This result is in agreement with the Conjecture [AS2, Conj. 1.4].

Proof. We follow the lines of proof of an analogous statement in [AS4, Thm. 7.6]. It is equivalent to prove that a finite dimensional connected graded braided Hopf algebra R = ⊕_{n≥0}R(n) with braiding c which is generated in degree 1, and such that (R(1), c|_{R(1)⊗R(1)}) is dual to one of the braided vector spaces in 1.2, must be a Nichols algebra. Thus, let R be such an algebra with R(1) = V. We must prove that all the relations of 𝔅(V) hold in R. It is straightforward to see that in all the examples under consideration the elements in J_{2} are primitive, and if r ∈ J_{2} is one of the relations in the statement of 1.2 then c(r ⊗ r) = r ⊗ r. This implies that r = 0, or otherwise R would be infinite dimensional. We have to deal then with the relations in J_{4} and J_{6}. But those relations are primitive elements in the algebra T (V)/⟨J_{2}⟩, and then it is enough to prove that in T (V)/⟨J_{2}⟩ we have c(r ⊗ r) = r ⊗ r, r in J_{4} or J_{6}. Again, this is straightforward.

3 Liftings

3.1 The problem

Let X be a finite rack, q : X × X → ℂ^{×} be a 2-cocycle and (V, c) = (ℂX, c^{q}). A Y-D realization of (V, c) over a group G is a structure of Yetter-Drinfeld module over the group algebra ℂG on V such that c coincides with the braiding in _{k[G]}^{k[G]}YD and the elements of X are G-homogeneous; i. e., there exists a function g : X → G, i ↦ g_{i}, such that δ(i) = g_{i} ⊗ i, i ∈ X. Let A be a pointed Hopf algebra, G = G(A). The infinitesimal Yetter-Drinfeld module of A is the coinvariant part (A_{1}/A_{0})^{co ℂG} of the ℂG-Hopf bimodule A_{1}/A_{0}. Another fundamental step in the lifting method is to address the following question.

Question 3.1. Assume that the Nichols algebra of the braided vector space (V, c) = (ℂX, c^{q}) has finite dimension. Then describe all finite-dimensional pointed Hopf algebras A such that the infinitesimal Yetter-Drinfeld module of A is a Y-D realization of (V, c) over G = G(A).

See [AS3, AS4] for substantial partial answers in the case of trivial racks. Note that the related problem of describing all Y-D realizations of (V, c) presents hard computational aspects. The projection π : A_{1} → A_{1}/A_{0} being a morphism of ℂG-Hopf bimodules, we can choose a section σ of Hopf bimodules; set a_{i} := σ(x_{i}#1) ∈ A_{1}. Then

a_{i} is a (g_{i}, 1)-primitive and g_{i}a_{j}g_{i}^{−1} = q_{ij}a_{i▷j}, ∀i, j ∈ X.  (9)

To answer 3.1 we need to find the relations between the a_{i}’s, which are deformations of the relations in the Nichols algebra.

3.2 Faithful Y-D realizations

Let X be a finite rack, q : X × X → ℂ^{×} be a 2-cocycle and let (V, c) = (ℂX, c^{q}) as above. We shall say that a Y-D realization is faithful if g : X → G is injective.

The following definition, with a different notation, appears in [MS, S. 5].

Definition 3.2. A principal YD-realization for X, q over a finite group G is a collection (·, g : X → G, (χ_{i})_{i∈X}), where · is an action of G on X, g is an equivariant function with respect to the conjugation in G and χ_{i} : G → ℂ^{×}; such that the family (χ_{i})_{i∈X} is a 1-cocycle: χ_{i}(ht) = χ_{i}(t)χ_{t.i}(h), for all i ∈ X, h, t ∈ G; g_{i}.j = i ▷ j and χ_{i}(g_{j}) = q_{ij} for all i, j ∈ X.

Lemma 3.3. (a). If the rack X is faithful (that is, ϕ_{i} = ϕ_{j} only for i = j) then any Y-D realization is faithful.

(b). A principal Y-D realization over G defines a Y-D realization by

δ(x_{i}) = g_{i} ⊗ x_{i}, h.x_{i} = χ_{i}(h)x_{h.i}, i ∈ X, h ∈ G.

(c). Any faithful Y-D realization arises from a unique principal Y-D realization as in (b); and G acts by rack automorphisms on X.

(d). If X is faithful and indecomposable, and q is constant, then the 1-cocycle (χ_{i})_{i∈X} appearing in an arbitrary principal Y-D realization is constant: χ_{i} = χ for all i ∈ X, and χ is a multiplicative character of G.

Proof. Since g_{i}.x_{j} = q_{ij}x_{i▷j} in any realization, (a) follows. To prove (b), the cocycle condition insures that the action is well-defined, whereas the equivariance of g insures the Yetter-Drinfeld compatibility. We now prove (c). If i ∈ X, h ∈ G then δ(h.x_{i}) = hg_{i}h^{−1} ⊗ h.x_{i}; thus there exists a unique j and a scalar χ_{i}(h) such that h.x_{i} = χ_{i}(h)x_{j}; set h.i := j. Since G acts well on V, this defines an action of G on X and χ_{i} is a 1-cocycle; since hg_{i}h^{−1} = g_{j}, g is equivariant. Let now i, j ∈ X, h ∈ G. Then g_{h.(i▷j)} = hg_{i▷j}h^{−1} = hg_{i}g_{j}g_{i}^{−1}h^{−1} = g_{h.i}g_{h.j}g_{h.i}^{−1} = g_{(h.i)▷(h.j)}, hence h.(i ▷ j) = (h.i) ▷ (h.j) by faithfulness. It is finally clear that χ_{i}(g_{j}) = q_{ij} for all i, j ∈ X. We prove (d), say with q_{ij} = ω i, j ∈ X. For h ∈ G, i, j ∈ X, we compute

ω = χ_{i}(g_{h.j}) = χ_{i}(hg_{j}h^{−1})
= χ_{i}(h^{−1})χ_{h^{−1}.i}(g_{j})χ_{(g_{j}h^{−1}).i}(h) = χ_{i}(h^{−1})ωχ_{j▷(h^{−1}.i)}(h).

If k = h^{−1}.i, this implies χ_{j▷k}(h) = χ_{h.k}(h^{−1})^{−1} = χ_{k}(h). Hence χ_{j▷k} = χ_{k} for all j, k ∈ X; since X is indecomposable the 1-cocycle is constant, and a fortiori a multiplicative character of G.

3.3 Some liftings

Let X, q and (V, c) = (ℂX, c^{q}) be as above. Let us fix a Y-D realization of (V, c) over a finite group G. Let K be the subgroup of G generated by the image of X; K acts by rack automophisms on X. Let R be the set of equivalence classes in X × X for the relation generated by (i, j) ∼ (i ▷ j, i); that is, the orbits of the set-theoretical solution of the braid equation associated to the rack. Then Aut_{▷} X also acts on R. If C ∈ R, then g_{C} := g_{i}g_{j} is well-defined, i. e. it does not depend on the choice of (i, j) ∈ C; and γg_{C}γ^{−1} = g_{γ·C} for any γ ∈ K; same for γ ∈ G if the realization is faithful, by 3.3. Assume that q_{ij} = −1, i, j ∈ X. Then the set of relations ∑_{(i,j)∈C} x_{i}x_{j} = 0, C ∈ R, is a basis of the space J_{2} of degree-2 relations in 𝔅(V). The lifting of these relations is given by the following result.

Lemma 3.4. Let A be a finite dimensional pointed Hopf algebra, G = G(A), whose infinitesimal Yetter-Drinfeld module is a realization of (V, c). Let (a_{i}) be as in (9). Then there exists λ_{C} ∈ ℂ, for all C ∈ R, normalized by

λ_{C} = 0  if g_{C} = 1,  (10)
λ_{C} = λ_{γ·C}, ∀γ ∈ K (or ∀γ ∈ Int_{▷} X),  (11)

such that the following relations hold in A:

∑_{(i,j)∈C} a_{i}a_{j} = λ_{C}(1 − g_{C}), C ∈ R.  (12)

If X is faithful and indecomposable and χ is as in 3.3 (d), then

χ^{2}(h)λ_{C} = λ_{h·C}, ∀h ∈ G.  (13)

Proof. Let C ∈ R. It is straightforward to see that b_{C} := ∑_{(i,j)∈C} a_{i}a_{j} is a (g_{C}, 1)-primitive. Thus b_{C} = λ_{C}(g_{C} − 1) + ∑_{i∈X s.t. g_{i}=g_{C}} λ_{i}a_{i}. But for all i ∈ X, g_{i} acts on V in the basis X by −1 times a permutation matrix, while g_{C} acts as a permutation matrix in the same basis. Thus, g_{i} ≠ g_{C}∀i ∈ X and we get (12). The relations (11), resp. (13), follow by applying ad γ, resp. ad h, to both sides of (12).

Example 3.5. Let X be the rack of transpositions in 𝕊_{n}, n ≥ 3. The classes in R have either 1, 2 or 3 elements; Int_{▷} X permutes the classes with the same cardinality, so we have only scalars λ_{1}, λ_{2}, λ_{3}, the index corresponding to the cardinality; no λ_{2} for 𝕊_{3}. This notation is used in (16), (17), (18) below.

Example 3.6. Let X = (𝔽_{3}, 2), q_{ij} = −1, i, j ∈ X, and G = 𝔽_{3} ⋊ Γ, where Γ is a cyclic group of order 2P, denoted multiplicatively with a generator u; and u acts on 𝔽_{3} by −1. Let t be odd, 1 ≤ t ≤ P and set g_{i} := (i, u^{t}), i ∈ 𝔽_{3}; let ξ ∈ ℂ be such that ξ^{t} = −1 and let χ : G → ℂ^{×} be given by χ(i, u^{s}) = ξ^{s}, 0 ≤ i ≤ 2, 0 ≤ s < 2P. Then G acts on X and we have a principal YD-realization for X, q. Since g_{i}g_{j} is never 1, if i ≠ j, no conclusion on λ_{3} arises from (10) (we use here the notation from 3.5), but:

• g_{i}^{2} = 1 only if t = P; in this case necessarily λ_{1} = 0.

• If h = (0, u) then (13) implies λ_{1} = ξ^{2}λ_{1} and λ_{3} = ξ^{2}λ_{3}. Thus λ_{1} = λ_{3} = 0 if ξ ≠ −1.

Definition 3.7. Let X be the rack of transpositions in 𝕊_{n}, 3 ≤ n ≤ 5, q_{ij} = −1, i, j ∈ X. Let G be a finite group admitting a principal Y-D realization (·, g : X → G, χ) for X, q. Let λ_{1}, λ_{2}, λ_{3} be scalars satisfying (10), (13). Let A(G, ·, g, χ, λ_{i}) be the algebra presented by generators a_{σ}, σ ∈ X and H_{t}, t ∈ G; with relations

H_{t}H_{s} = H_{ts}, H_{e} = 1;  (14)
H_{t}a_{σ} = χ(t)a_{t·σ}H_{t}  ∀t ∈ G, σ ∈ X;  (15)
a_{σ}^{2} = λ_{1}(1 − g_{σ}^{2})  σ ∈ X;  (16)
a_{σ}a_{τ} + a_{τ} a_{σ} = λ_{2}(1 − g_{σ}g_{τ})  for στ = τσ, σ ≠ τ;  (17)
a_{σ}a_{τ} + a_{τ} a_{ν} + a_{ν}a_{σ} = λ_{3}(1 − g_{σ}g_{τ})  for στ = νσ, σ ≠ τ ≠ ν.  (18)

Theorem 3.8. Assume that n = 3, 4. Then A = A(G, ·, g, χ, λ_{i}) is a pointed Hopf algebra, with the comultiplication uniquely defined by declaring the elements H_{t}, t ∈ G to be group-like and a_{σ} to be (g_{σ}, 1)-primitive, σ ∈ X. The graded Hopf algebra associated to the coradical filtration of A is isomorphic to 𝔅(V)#ℂG; in particular, dim(A) = dim 𝔅(V)|G|. Conversely, any pointed Hopf algebra whose infinitesimal YD-module is a realization of (ℂX, c^{q}) is isomorphic to A(G, ·, g, χ, λ_{i}) for some principal Y-D realization (·, g : X → G, χ) and scalars λ_{1}, λ_{2}, λ_{3} satisfying (10), (13).

Proof. A routinary computation shows that the comultiplication is well-defined and admits a counit given by ε(H_{t}) = 1, ε(a_{σ}) = 0. The antipode is the unique algebra map S : A → A^{op} such that S(H_{t}) = H_{t^{−1}}, S(a_{σ}) = −g_{σ}^{−1}a_{σ}. We next check that dim A is the desired one. With the help of a computer program we found the Gröbner basis the of ideal generated by the relations when n = 4; the case n = 3 is easier and we omit it. We write a > b > c > d > e > f for (01), (02), (03), (12), (13), (23) and α, β, ϵ for λ_{2}, λ_{3}, λ_{1} respectively. With this order, the basis is given by the relations (15) together with

bab − aba − βb + βa = 0, cac − aca − βc + βa = 0,
cbc − bcb − βc + βb = 0, ede − ded − βe + βd = 0,
cabc + bcab + abca + αcb − βca − βbc + αba + αac − βab + ϵβg_{f}^{2}
− (β − α)βg_{f}^{3}g_{d} − (ϵ + α − β)β = 0,
cbac + bacb + acba − βcb + αca + αbc − βba − βac + αab + ϵβg_{f}^{2}
− (β − α)βg_{f}^{3}g_{e} − (ϵ + α − β)β = 0,

cabaca + bcabac + ϵ^{2}abg_{f}^{4} − βcaba − βbcab − αbacb − βbaca − αacba
− βabca − βabac − (β − α)βcbg_{f}^{3}g_{d} − ϵαcag_{f}^{2} − ϵαbcg_{f}^{2} + ϵβacg_{f}^{2}
− 2ϵ^{2}abg_{f}^{2} + (β^{2} − α^{2} + αϵ)ca − (α − ϵ)αbc + β^{2}ba + (ϵ^{2} − α^{2} + β^{2})ab
+ (β − ϵ)βac + (β − α)β^{2}g_{f}^{3}g_{d} + (β − α)αβg_{f}^{3}g_{e} − ϵ^{2}βg_{f}^{4}
+ (2ϵ − β)ϵβg_{f}^{2} − ((ϵβ) + (β − ϵ)^{2} − α^{2})β = 0,
cabacb + acabac − βcaba − αbcab − βbacb − βacba − βacab − αabca
− βabac − (β − α)βcag_{f}^{3}g_{e} + ϵ^{2}bag_{f}^{4} − ϵαcbg_{f}^{2} + ϵβcag_{f}^{2} − 2ϵ^{2}bag_{f}^{2}
− ϵαacg_{f}^{2} − (α − ϵ)αcb + (β − ϵ)βca + (ϵ^{2} − α^{2} + β^{2})ba
+ (β^{2} − α^{2} + αϵ)ac + β^{2}ab + (β − α)αβg_{f}^{3}g_{d} + (β − α)β^{2}g_{f}^{3}g_{e}
− ϵ^{2}βg_{f}^{4} + (2ϵ − β)ϵβg_{f}^{2} − ((ϵβ) + (β − ϵ)^{2} − α^{2})β = 0.

If ϵ = α = β = 0, the leading terms of the relations remain unchanged; thus dim A = dim(𝔅(V)#ℂG). One concludes that gr A ≃ 𝔅(V)#ℂG. The converse follows from 2.1, 3.3 (d) and the first part of the Theorem.

The determination of isomorphisms between different Hopf algebras of this type can be easily performed with available techniques, see e. g. [AS3]. Under suitable choices of YD-data, new infinite families of non-isomorphic Hopf algebras of the same dimension are obtained.

References

[AG] N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math., to appear. Also in

      arXiv/0202084.

[AS1] N. Andruskiewitsch and H.-J. Schneider, Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p^{3}, J. Algebra 209 (1998), 658–691.

[AS2] N. Andruskiewitsch and H.-J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 1–45.

[AS3] N. Andruskiewitsch and H.-J. Schneider, Pointed Hopf Algebras, in “New directions in Hopf algebras”, 1–68, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, Cambridge, 2002.

[AS4] N. Andruskiewitsch and H.-J. Schneider, Finite quantum groups over abelian groups of prime exponent, Ann. Sci. Ec. Norm. Super. 35 (2002), 1–26.

[FK] S. Fomin and A.N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus. (English), Prog. Math. 172, 147-182 (1999).

[G1] M. Graña, On Nichols algebras of low dimension, in New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math. 267 (2000), 111–134.

[G2] M. Graña, Zoo of finite dimensional Nichols algebras of nonabelian group type, document available at

      http://mate.dm.uba.ar/∼matiasg/zoo.html.

[MS] A. Milinski and H-J. Schneider, Pointed Indecomposable Hopf Algebras over Coxeter Groups, Contemp. Math. 267 (2000), 215–236.

N. A.: FaMAF - Univ. Nacional de Córdoba - CIEM - CONICET - (5000) Ciudad Universitaria - Córdoba, Argentina, andrus@mate.uncor.edu M. G.: Depto de Matemática - FCEyN - Universidad de Buenos Aires - Pab. I - Ciudad Universitaria - (1428) Buenos Aires - Argentina, matiasg@dm.uba.ar

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