arXiv:0803.3486v2 [math.QA] 22 Aug 2008 NEW TECHNIQUES FOR POINTED HOPF ALGEBRAS NICOL ´ AS ANDRUSKIEWITSCH AND FERNANDO FANTINO Abstract. We present techniques that allow to decide that the dimen- sion of some pointed Hopf algebras associated with non-abelian groups is infinite. These results are consequences of [AHS]. We illustrate each technique with applications. Dedicado a Isabel Dotti y Roberto Miatello en su sexag´ esimo cumplea˜ nos. Introduction 0.1. Let G be a finite group and let C G C G YD be the category of Yetter- Drinfeld modules over CG. The most delicate of the questions raised by the Lifting Method for the classification of finite-dimensional pointed Hopf algebras H with G(H ) ≃ G [AS1, AS3], is the following: Given V ∈ C G C G YD, decide when the Nichols algebra B(V ) is finite-dimensional. Recall that a Yetter-Drinfeld module over the group algebra CG (or over G for short) is a left CG-module and left CG-comodule M satisfying the compatibility condition δ(g.m)= ghg −1 ⊗ g.m, for all m ∈ M h , g,h ∈ G. The list of all objects in C G C G YD is known: any such is completely reducible, and the class of irreducible ones is parameterized by pairs (O,ρ), where O is a conjugacy class in G and ρ is an irreducible representation of the isotropy group G s of a fixed s ∈ O. We denote the corresponding Yetter-Drinfeld module by M (O,ρ). In fact, our present knowledge of Nichols algebras is still preliminary. However, an important remark is that the Nichols algebra B(V ) depends (as algebra and coalgebra) just on the underlying braided vector space (V,c)– see for example [AS3]. This observation allows to go back and forth be- tween braided vector spaces and Yetter-Drinfeld modules. Indeed, the same braided vector space could be realized as a Yetter-Drinfeld module over dif- ferent groups, and even in different ways over the same group, or not at all. The braided vector spaces that do appear as Yetter-Drinfeld modules over some finite group are those coming from racks and 2-cocycles [AG]. Date : October 24, 2018. 2000 Mathematics Subject Classification. 16W30; 17B37. This work was partially supported by ANPCyT-Foncyt, CONICET, Ministerio de Cien- cia y Tecnolog´ ıa (C´ ordoba) and Secyt (UNC). 1
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NEW TECHNIQUES FOR POINTED HOPF ALGEBRAS
NICOLAS ANDRUSKIEWITSCH AND FERNANDO FANTINO
Abstract. We present techniques that allow to decide that the dimen-
sion of some pointed Hopf algebras associated with non-abelian groups
is infinite. These results are consequences of [AHS]. We illustrate each
technique with applications.
Dedicado a Isabel Dotti y Roberto Miatello en su sexagesimo cumpleanos.
Introduction
0.1. Let G be a finite group and let CGCGYD be the category of Yetter-
Drinfeld modules over CG. The most delicate of the questions raised bythe Lifting Method for the classification of finite-dimensional pointed Hopfalgebras H with G(H) ≃ G [AS1, AS3], is the following:
Given V ∈ CGCGYD, decide when the Nichols algebra B(V ) is
finite-dimensional.
Recall that a Yetter-Drinfeld module over the group algebra CG (or overG for short) is a left CG-module and left CG-comodule M satisfying thecompatibility condition δ(g.m) = ghg−1 ⊗ g.m, for all m ∈ Mh, g, h ∈ G.The list of all objects in CG
CGYD is known: any such is completely reducible,and the class of irreducible ones is parameterized by pairs (O, ρ), where O isa conjugacy class in G and ρ is an irreducible representation of the isotropygroup Gs of a fixed s ∈ O. We denote the corresponding Yetter-Drinfeldmodule by M(O, ρ).
In fact, our present knowledge of Nichols algebras is still preliminary.However, an important remark is that the Nichols algebra B(V ) depends (asalgebra and coalgebra) just on the underlying braided vector space (V, c)–see for example [AS3]. This observation allows to go back and forth be-tween braided vector spaces and Yetter-Drinfeld modules. Indeed, the samebraided vector space could be realized as a Yetter-Drinfeld module over dif-ferent groups, and even in different ways over the same group, or not atall. The braided vector spaces that do appear as Yetter-Drinfeld modulesover some finite group are those coming from racks and 2-cocycles [AG].
Thus, a comprehensive approach to the question above would be to solvethe following:
Given a braided vector space (V, c) determined by a rack anda 2-cocycle, decide when dimB(V ) < ∞.
But at the present moment and with the exception of the diagonal casementioned below, we know explicitly very few Nichols algebras of braidedvector spaces determined by racks and 2-cocycles; see [FK, MS, G1, AG, G2].
0.2. The braided vector spaces that appear as Yetter-Drinfeld modules oversome finite abelian group are the diagonal braided vector spaces. This leadsto the following question: Given a braided vector space (V, c) of diagonaltype, decide when the Nichols algebra B(V ) is finite-dimensional. The fullanswer to this problem was given in [H2], see [AS2, H1] for braided vectorspaces of Cartan type– and [AS4] for applications. These results on Nicholsalgebras of braided vector spaces of diagonal type were in turn used formore general pointed Hopf algebras. Let us fix a non-abelian finite groupG and let V ∈ CG
CGYD irreducible. If the underlying braided vector spacecontains a braided vector subspace of diagonal type, whose Nichols algebrahas infinite dimension, then dimB(V ) = ∞. In turns out that, for severalfinite groups considered so far, many Nichols algebras of irreducible Yetter-Drinfeld modules have infinite dimension; and there are short lists of thosenot attainable by this method. See [G1, AZ, AF1, AF2, FGV].
0.3. An approach of a different nature, inspired by [H1], was presented in[AHS]. Let us consider V = V1 ⊕ · · · ⊕ Vθ ∈ CG
CGYD, where the Vi’s areirreducible. Then the Nichols algebra of V is studied, under the assumptionthat the B(Vi) are known and finite-dimensional, 1 ≤ i ≤ θ. Under somecircumstances, there is a Coxeter group W attached to V , so that B(V )finite-dimensional impliesW finite. Although the picture is not yet complete,the previous result implies that, for a few G– explicitly, S3, S4, Dn– theNichols algebras of some V have infinite dimension. These applications relyon the lists mentioned at the end of 0.2.
0.4. The purpose of the present paper is to apply the results in 0.3 todiscard more irreducible Yetter-Drinfeld modules. Namely, let V = V1⊕V2 ∈CΓCΓYD, where Γ = S3, S4 or Dn, such that dimB(V ) = ∞ by [AHS, Section4]. Then there is a rack (X, ⊲) and a cocycle q such that (V, c) ≃ (CX, cq).
Let G be a finite group, let O be a conjugacy class in G, s ∈ O, ρ ∈ Gs andM(O, ρ) ∈ CG
CGYD the irreducible Yetter-Drinfeld module corresponding to(O, ρ). We give conditions on (O, ρ) such that M(O, ρ) contains a braidedvector subspace isomorphic to (CX, cq); thus, necessarily, dimB(O, ρ) = ∞.We illustrate these new techniques with several examples; see in particularExample 3.9 for one that can not be treated via abelian subracks.
POINTED HOPF ALGEBRAS 3
0.5. The facts glossed in the previous points strengthen our determinationto consider families of finite groups, in order to discard those irreducibleYetter-Drinfeld modules over them with infinite-dimensional Nichols algebraby the ‘subrack method’. Natural candidates are the families of simplegroups, or closely related; cf. the classification of simple racks in [AG]. Thecase of symmetric and alternating groups is treated in [AZ, AF1, AF2, AFZ];Mathieu groups in [F1]; other sporadic groups in [AFGV]; some finite groupsof Lie type with rank one in [FGV, FV]. Particularly, a list of only 9 typesof pairs (O, ρ) for Sm whose Nichols algebras might be finite-dimensional isgiven in [AFZ]; an analogous list of 7 pairs out of 1137 (for all 5 Mathieusimple groups) is given in [F1]; the sporadic groups J1, J2, J3, He and Suzare shown to admit no non-trivial pointed finite-dimensional Hopf algebrain [AFGV]. Our new techniques are crucial for these results.
0.6. If for some finite group G there is at most one irreducible Yetter-Drinfeld module V with finite-dimensional Nichols algebra, then [AHS, Th.4.2] can be applied again. If the conclusion is that dimB(V ⊕ V ) = ∞,then we can build a new rack together with a 2-cocycle realizing V ⊕V , andinvestigate when a conjugacy class in another group G′ contains this rack,and so on.
1. Notations and conventions
The base field is C (the complex numbers).
1.1. Braided vector spaces. A braided vector space is a pair (V, c), whereV is a vector space and c : V ⊗V → V ⊗V is a linear isomorphism such thatc satisfies the braid equation: (c⊗ id)(id⊗c)(c⊗ id) = (id⊗c)(c⊗ id)(id⊗c).
Let V be a vector space with a basis (vi)1≤i≤θ, let (qij)1≤i,j≤θ be a matrixof non-zero scalars and let c : V ⊗ V → V ⊗ V be given by c(vi ⊗ vj) =qijvj ⊗ vi. Then (V, c) is a braided vector space, called of diagonal type.
Examples of braided vector spaces come from racks. A rack is a pair(X, ⊲) where X is a non-empty set and ⊲ : X ×X → X is a function– calledthe multiplication, such that φi : X → X, φi(j) := i ⊲ j, is a bijection for alli ∈ X, and
i ⊲ (j ⊲ k) = (i ⊲ j) ⊲ (i ⊲ k) for all i, j, k ∈ X.(1.1)
For instance, a group G is a rack with x⊲y = xyx−1. In this case, j ⊲i = iwhenever i ⊲ j = j and i ⊲ i = i for all i ∈ G. We are mainly interested insubracks of G, e. g. in conjugacy classes in G.
Let (X, ⊲) be a rack. A function q : X × X → C× is a 2-cocycle ifqi,j⊲k qj,k = qi⊲j,i⊲k qi,k, for all i, j, k ∈ X. Then (CX, cq) is a braided vectorspace, where CX is the vector space with basis ek, k ∈ X, and the braidingis given by
cq(ek ⊗ el) = qk,l ek⊲l ⊗ ek, for all k, l ∈ X.
4 ANDRUSKIEWITSCH AND FANTINO
A subrack T of X is abelian if k ⊲ l = l for all k, l ∈ T . If T is an abeliansubrack of X, then CT is a braided vector subspace of (CX, cq) of diagonaltype.
Definition 1.1. Let X be a rack. Let X1 and X2 be two disjoint copies
of X, together with bijections ϕi : X → Xi, i = 1, 2. The square of X
is the rack with underlying set the disjoint union X1∐
X2 and with rack
multiplication
ϕi(x) ⊲ ϕj(y) = ϕj(x ⊲ y),
x, y ∈ X, 1 ≤ i, j ≤ 2. We denote the square of X by X(2). This is a
particular case of an amalgamated sum of racks, see e. g. [AG].
1.2. Yetter-Drinfeld modules. We shall use the notation given in [AF1].Let G be a finite group. We denote by |g| the order of an element g ∈ G;
and by G the set of isomorphism classes of irreducible representations of G.
We shall often denote a representant of a class in G with the same symbolas the class itself.
Here is an explicit description of the irreducible Yetter-Drinfeld moduleM(O, ρ). Let t1 = s, . . . , tM be a numeration of O and let gi ∈ G such thatgi ⊲ s = ti for all 1 ≤ i ≤ M . Then M(O, ρ) = ⊕1≤i≤M gi ⊗ V , where V isthe vector space affording the representation ρ. Let giv := gi⊗v ∈ M(O, ρ),1 ≤ i ≤ M , v ∈ V . If v ∈ V and 1 ≤ i ≤ M , then the action of g ∈ Gis given by g · (giv) = gj(γ · v), where ggi = gjγ, for some 1 ≤ j ≤ M andγ ∈ Gs, and the coaction is given by δ(giv) = ti ⊗ giv. Then M(O, ρ) is abraided vector space with braiding c(giv ⊗ gjw) = gh(γ · w) ⊗ giv, for any1 ≤ i, j ≤ M , v,w ∈ V , where tigj = ghγ for unique h, 1 ≤ h ≤ M andγ ∈ Gs. Since s ∈ Z(Gs), the center of Gs, the Schur Lemma implies that
(1.2) s acts by a scalar qss on V.
Lemma 1.2. If U is a subspace of W such that c(U ⊗ U) = U ⊗ U and
dimB(U) = ∞, then dimB(W ) = ∞. �
Lemma 1.3. [AZ, Lemma 2.2] Assume that s is real (i. e. s−1 ∈ O). If
dimB(O, ρ) < ∞, then qss = −1 and s has even order. �
Let σ ∈ Sm be a product of nj disjoint cycles of length j, 1 ≤ j ≤ m.Then the type of σ is the symbol (1n1 , 2n2 , . . . ,mnm). We may omit jnj
when nj = 0. The conjugacy class Oσ of σ coincides with the set of allpermutations in Sm with the same type as σ; we may use the type as asubscript of a conjugacy class as well. If some emphasis is needed, we adda superscript m to indicate that we are taking conjugacy classes in Sm, likeOmj for the conjugacy class of j-cycles in Sm.
POINTED HOPF ALGEBRAS 5
2. A technique from the dihedral group Dn, n odd
Let n > 1 be an odd integer. Let Dn be the dihedral group of order 2n,generated by x and y with defining relations x2 = e = yn and xyx = y−1. Let
Ox be the conjugacy class of x and let sgn ∈ Dxn be the sign representation
(Dxn = 〈x〉 ≃ Z2). The goal of this Section is to apply the next result, cf.
[AHS, Th. 4.8], or [AHS, Th. 4.5] for n = 3.
Theorem 2.1. The Nichols algebra B(M(Ox, sgn)⊕M(Ox, sgn)) has infi-
nite dimension. �
Note that M(Ox, sgn) ⊕ M(Ox, sgn) is isomorphic as a braided vectorspace to (CXn, q), where
• Xn is the rack with 2n elements si, tj , i, j ∈ Z/n, and with structure
by (d), and the claim follows. Finally, the family of type D(2)p we are looking
for is (σi)i∈Z/p∪(σ−1i )i∈Z/p. It remains to show that σt 6= σ−1
l for all t, l ∈ Zp.
If σt = σ−1l , then σ2
t (i1) = σ−2l (i1), that is i3 = ij−1, a contradiction to the
hypothesis j 6= 4. �
Proof of the Example 2.10. We may assume that qσσ = −1, by Lemma 1.3.By Lemma 2.11, we have a family (σi)i∈Z/p of type Dp, with σ0 = σ. NowCorollary 2.9 applies, with µ0 = σ0, k = |σ0|−1. Thus dimB(O, ρ) = ∞. �
3. A technique from the symmetric group S3
We study separately the case p = 3 because of the many applicationsfound. In this setting, D3 ≃ S3 and Ox = O3
2 = {(1 2), (2 3), (1 3)} is theconjugacy class of transpositions in S3. The rack X3 is described as a set of6 elements X3 = {x1, x2, x3, y1, y2, y3} with the multiplication
xi ⊲ xj = xk, yi ⊲ yj = yk, xi ⊲ yj = yk, yi ⊲ xj = xk,
for i, j, k, all distinct or all equal.
3.1. Families of type D3 and D(2)3 . We fix a finite group G and O a
conjugacy class in G. Our aim is to give criteria to detect when O containsa subrack isomorphic to X3.
Definition 3.1. Let σ1, σ2, σ3 ∈ G distinct. We say that (σi)1≤i≤3 is of
type D3 if
σi ⊲ σj = σk, where i, j, k are all distinct.(3.1)
The requirement (3.1) consists of 6 identities, but actually 3 are enough.
Remark 3.2. If
σ1 ⊲ σ2 = σ3,(3.2)
σ1 ⊲ σ3 = σ2,(3.3)
σ2 ⊲ σ3 = σ1,(3.4)
then (σi)1≤i≤3 is of type D3. �
POINTED HOPF ALGEBRAS 11
Here is a characterization of D3 families.
Proposition 3.3. Let σ1, σ2 ∈ O. Define σ3 := σ1 ⊲ σ2. Then (σi)1≤i≤3 is
of type D3 if and only if
σ1 6∈ Gσ2 ,(3.5)
σ21 ∈ Gσ2 ,(3.6)
σ1 = σ2 ⊲ (σ1 ⊲ σ2).(3.7)
Proof. The definition of σ3 is equivalent to (3.2) and (3.7) is equivalent to
(3.4). Assume that (σi)1≤i≤3 is of type D3. As σ3 6= σ2, σ1 6∈ Gσ2 . Also,
Therefore, given 6 distinct elements σ1, σ2, σ3, τ1, τ2, τ3 ∈ G, if the 12identities: (3.2), (3.3), (3.4), for σ and for τ , (3.9), (3.10), (3.11), and theanalogous identities
τ1 ⊲ σ1 = σ1,(3.18)
τ1 ⊲ σ2 = σ3,(3.19)
τ2 ⊲ σ1 = σ3,(3.20)
hold, then (σ, τ) is of type D(2)3 . But we can get rid of 3 of these 12 identities.
Proposition 3.6. Let σ1, σ2, σ3, τ1, τ2, τ3 ∈ G, all distinct, such that
(3.2), (3.3), (3.4), hold for σ and for τ , as well as the identities (3.9), (3.11)
and (3.19). Then (σ, τ) is of type D(2)3 .
Proof. By Lemma 3.5, it is enough to check (3.10), (3.18) and (3.20). First,
(3.18) holds because τ1 = σ1 ⊲ τ1 = σ1τ1σ−11 . If τ1 acts on both sides of
(3.11), then τ2 = τ1 ⊲ τ3 = (τ1 ⊲ σ2) ⊲ (τ1 ⊲ τ1) = σ3 ⊲ τ1; if now σ1 acts on the
Thus, (3.10) holds. We can now conclude from Lemma 3.5 that σi ⊲ τi = τi,
1 ≤ i ≤ 3, and σi ⊲ τj = τk, for all i, j, k distinct. If now σ3 acts on (3.19),
then σ3 = (σ3 ⊲ τ1) ⊲ (σ3 ⊲ σ2) = τ2 ⊲ σ1, and (3.20) holds. �
3.2. Examples of D(2)3 type. We first spell out explicitly Theorem 2.8 and
Corollary 2.9 for p = 3.
Theorem 3.7. Let σ1, σ2, σ3, τ1, τ2, τ3 ∈ G distinct; denote (σ, τ) =
(σ1, σ2, σ3, τ1, τ2, τ3). Let ρ = (ρ, V ) ∈ Gσ1 . We assume that
(H1) (σ, τ) is of type D(2)3 ,
(H2) (σ, τ) ⊆ O, with g ∈ G such that g ⊲ σ1 = τ1,
(H3) qσ1σ1 = −1,
POINTED HOPF ALGEBRAS 13
(H4) there exist v,w ∈ V − 0 such that,
ρ(g−1σ1g)w = −w,(3.21)
ρ(τ1)v = −v,(3.22)
Then dimB(O, ρ) = ∞. �
Corollary 3.8. Let σ1, σ2, σ3 ∈ O distinct. Assume that there exists k,
1 ≤ k ≤ |σ1|, such that σk1 6= σ1 and σk
1 ∈ O. Let ρ = (ρ, V ) ∈ Gσ1 . Assume
further that
(1) (σi)1≤i≤3 is of type D3,
(2) qσ1σ1 = −1.
Then dimB(O, ρ) = ∞. �
Corollary 3.8 applies notably to a real conjugacy class of an element oforder greater than 2. We list several applications for G = Sm.
Example 3.9. Let m ≥ 6. Let O be the conjugacy class of Sm of type
(1n1 , 2n2 , . . . ,mnm), where
• n1, n2 ≥ 1 and
• nj ≥ 1 for some j, 3 ≤ j ≤ m.
Let σ ∈ O and ρ ∈ Sσm. Then dimB(O, ρ) = ∞.
Proof. By hypothesis, we can choose σ = (1 2)β where β fixes 1, 2 and 3. If
qσσ 6= −1, then dimB(O, ρ) = ∞, by Lemma 1.3. Assume that qσσ = −1.
Now set
x = (1 2), y = (1 3), z = (2 3), σ1 = σ = xβ, σ2 = yβ, σ3 := zβ.
Clearly (σi)1≤i≤3 is of type D3, O is real and |σ1| > 2. By Corollary 3.8,
dimB(O, ρ) = ∞. �
In particular, let O be the conjugacy class of Sm of type (1, 2,m−3), withm ≥ 6. By the preceding, dimB(O, ρ) = ∞. But, if qσσ = −1, then M(O, ρ)has negative braiding; that is, it is not possible to decide if the dimensionof B(O, ρ) is infinite via abelian subracks. See [F2] for details.
Example 3.10. Let m ≥ 6. Let σ ∈ Sm of type (1n1 , 2n2 , . . . ,mnm), O the
conjugacy class of σ and ρ ∈ Sσm. Assume that
• there exists j, 1 ≤ j ≤ m, such that j = 2k, with k ≥ 2 and nj ≥ 3.
Then dimB(O, ρ) = ∞.
14 ANDRUSKIEWITSCH AND FANTINO
Proof. If qσσ 6= −1, then dimB(O, ρ) = ∞, by Lemma 1.3. Assume that
(c) follows as in the proof of Lemma 2.11 (c). (d). By (b) and (c), we
have that σ−1PkσPk = IkPk = B1B2, as claimed. (e). By (b) and (c),
σ−1P−kσP−k = I−kP−k = B2B1 as claimed.
Set now σ1 := σ, σ2 := PkσP−k and σ3 := P−kσPk. As in the proof
of Example 2.10 we can see that σ1, σ2 and σ3 are distinct. We check that
(σi)1≤i≤3 is of type D3 using Remark 3.2.
By (d), PkσPk ∈ Sσm, i. e. PkσPkσP−kσ−1P−k = σ, or σPkσP−kσ−1 =
P−kσPk. That is, σ1⊲σ2 = σ3. Analogously, σ1⊲σ3 = σ2 is proved using (e).
To check that σ2⊲σ3 = σ1, note that σ2⊲σ3 = PkσP−kP−kσPkPkσ−1P−k =
σ, because PkσP−2k = PkσPkP−3k = σB1B2 ∈ Sσm, by (a) and (d).
We now apply Corollary 3.8 and conclude that dimB(O, ρ) = ∞. �
We shall need a few well-known results on symmetric groups.
Remark 3.11. (i) If ρ is a faithful representation of Sn, then ρ(τ) /∈ C id, for
every τ ∈ Sn, τ 6= id (since Sn is centerless).
(ii) If ρ = (ρ,W ) ∈ Sn, with ρ 6= sgn, then for any involution τ ∈ Sn
(i. e., τ2 = id), there exists w ∈ W − 0 such that ρ(τ)w = w (otherwise
ρ(τ) = − id).
Example 3.12. Let m ≥ 6. Let σ ∈ Sm of type (1n1 , 2n2 , . . . ,mnm), O the
conjugacy class of σ and ρ ∈ Sσm. Assume that
POINTED HOPF ALGEBRAS 15
• n2 ≥ 3 and
• there exists j, with j ≥ 3, such that nj ≥ 1.
Then dimB(O, ρ) = ∞.
Proof. By Lemma 1.3, we may suppose that qσσ = −1. Assume that (i1 i2),
(i3 i4) and (i5 i6) are three transpositions appearing in the decomposition of
σ as a product of disjoint cycles. We define
x := (i1 i2)(i3 i4)(i5 i6), y := (i1 i4)(i3 i6)(i2 i5), z := (i1 i6)(i2 i3)(i4 i5)
and α := xσ. It is easy to see, using for instance Proposition 3.3, that
σ1 := σ, σ2 := yα, σ3 := zα,
is of type D3. Then dimB(O, ρ) = ∞, by Corollary 3.8. Indeed, σ−1 ∈ O,
but σ 6= σ−1 because σ has order > 2. �
In the proof of the next Example, we need some notation for the inducedrepresentation. LetH be a subgroup of a finite groupG of index k, φ1, . . . , φk
the left cosets of H in G, with representatives gφ1 , . . . , gφk. Let θ = (θ,W ) ∈
H, and w1, . . . wr a basis of W . Set V :=span-{gφiwj | 1 ≤ i ≤ k, 1 ≤ j ≤ r}.
For i, j, with 1 ≤ i ≤ k, 1 ≤ j ≤ r we define ρ : G → Aut(V ) by
ρ(g)(gφiwj) = gφl
θ(h)wj , where ggφi= gφl
h, with h ∈ H.(3.23)
Thus ρ = (ρ, V ) is a representation of G and deg ρ = [G : H] deg θ.
Example 3.13. Let m ≥ 12. Let σ ∈ Sm of type (1n1 , 2n2 , . . . ,mnm), O the
conjugacy class of σ and ρ ∈ Sσm. If n2 ≥ 6, then dimB(O, ρ) = ∞.
Proof. By Lemma 1.3, we may suppose that qσσ = −1. We denote the n2
transpositions appearing in the decomposition of σ as product of disjoint
cycles by A1,2, . . . , An2,2 and we define A2 = A1,2 · · ·An2,2. Let us suppose
It can be shown that (σ, τ) is of typeD(2)3 . Let now g = (i2 i3)(i6 i7)(i10 i11);
then, g ⊲ σ = τ1. Furthermore, τ1 = B = gσg and σ2 τ2 = σB = g σ2 τ2 g.
Then
ρ(τ1)v = −v = ρ(gσg)v and ρ(σ2 τ2)v = v = ρ(g σ2 τ2 g)v,
by (3.29). Therefore, dimB(O, ρ) = ∞, by Theorem 3.7. �
A way to obtain a family of type D3 is to start from a monomorphismρ : S3 → G and to consider the image by ρ of the transpositions. Anotherway is as follows.
Remarks 3.14. Let G be a finite group and z ∈ Z(G).
(i). Let (σi)i∈Z/3 be of type D3. Then (zσi)i∈Z/3 is also of type D3.
(ii). Let (σ, τ) = (σi)i∈Z/3 ∪ (τi)i∈Z/3 be a family of type D(2)3 . Then
(zσ, zτ) = (zσi)i∈Z/3 ∪ (zτi)i∈Z/3 is also a family of type D(2)3 .
Here is a combination of these two ways.
Example 3.15. Let p be a prime number and q = pm, m ∈ N, such that 3
divides q − 1. Let ω ∈ Fq be a primitive third root of 1.
(i). If c ∈ Fq, then (µi)i∈Z/3, where µi =
(0 ωi
ω2ic 0
), is a family of type
D3 in GL(2,Fq). If c = −1, then this is a family of type D3 in SL(2,Fq).
The orbit of µi is the set of matrices with minimal polynomial T 2 − c.
(ii). Let N > 3 be an integer and let T be the subgroup of diagonal
matrices in GL(N,Fq). Let λ = diag(λ1, λ2, . . . , λN ) ∈ T. Let O be the
conjugacy class of λ. Assume that λ1 = −λ2 and let c = λ21. Assume also
that there exist i, j, with 3 ≤ i, j ≤ N such that λi 6= λj; say i = 3, j = 4,
for simplicity of the exposition. Then (σi)i∈Z/3 ∪ (τi)i∈Z/3, where
σi =
(µi 0
0 diag(λ3, λ4, . . . , λN )
), τi =
(µi 0
0 diag(λ4, λ3, . . . , λN )
),
is a family of type D(2)3 in the orbit O ⊂ GL(N,Fq).
POINTED HOPF ALGEBRAS 19
Let W = SN act on T in the natural way. Let χ : GL(N,Fq) → C× be acharacter; it restricts to an irreducible representation (χ,C) of the centralizerGL(N,Fq)
σ0 . Fix a group isomorphism ϕ : F×q → Gq−1 ⊂ C×, where Gq−1
is the group of (q − 1)-th roots of 1 in C. Recall that χ = ϕ(deth) for someinteger h. Thus the restriction of χ to T is W-invariant.
Proposition 3.16. Keep the notation above. Assume that χ(λ) = −1.
Then the dimension of the Nichols algebra B(O, χ) is infinite.
Proof. The result follows from Theorem 3.7. Indeed, hypothesis (H1) and
(H2) clearly hold. The matrix g =
id2 0 0 0
0 0 1 0
0 1 0 0
0 0 0 idN−4
is an involution
that satisfies g ⊲ σ0 = τ0. Because of the explicit form of χ, χ(σ0) = −1 =
χ(τ0), hence (H3) and (H4) hold. �
This example can be adapted to the setting of semisimple orbits in finitegroups of Lie type.
4. A technique from the symmetric group S4
The classification of the finite-dimensional Nichols algebras over S4, givenin [AHS], relies on the fact (proved in loc. cit.) that some Nichols algebrasB(Vi ⊕ Vj) have infinite dimension. According to the general strategy pro-posed in the present paper, each of these pairs (Vi, Vj) gives rise to a rackand a cocycle, and to a technique to discard Nichols algebras over othergroups. Here we study one of these possibilities, and leave the others for afuture publication.
The octahedral rack is the rack X = {1, 2, 3, 4, 5, 6} given by the verticesof the octahedron with the operation of rack given by the “right-hand rule”,i. e. if Ti is the orthogonal linear map that fixes i and rotates the orthogonalplane by an angle of π/2 with the right-hand rule (pointing the thumb toi), then we define ⊲ : X ×X → X by i ⊲ j := Ti(j) – see Figure 1.
then, σi = gi ⊲ σ1, 1 ≤ i ≤ 6. It is easy to see that following relations hold
σ1g1 = g1σ1, σ1g2 = g5σ1, σ1g3 = g2σ1,
σ2g1 = g3σ1, σ2g2 = g2σ1, σ2g3 = g6σ−11 ,
σ3g1 = g4σ1, σ3g2 = g1σ6, σ3g3 = g3σ1,
σ4g1 = g5σ1, σ4g2 = g2σ6, σ4g3 = g1σ6,
σ5g1 = g2σ1, σ5g2 = g6σ−21 σ6, σ5g3 = g3σ6,
σ6g1 = g1σ6, σ6g2 = g3σ6, σ6g3 = g4σ6,
σ1g4 = g3σ1, σ1g5 = g4σ1, σ1g6 = g6σ6,
σ2g4 = g4σ6, σ2g5 = g1σ6, σ2g6 = g5σ36,
σ3g4 = g6σ−16 , σ3g5 = g5σ6, σ3g6 = g2σ
31,
σ4g4 = g4σ1, σ4g5 = g6σ1σ−26 , σ4g6 = g3σ
21σ6,
σ5g4 = g1σ6, σ5g5 = g5σ1, σ5g6 = g4σ1σ26 ,
σ6g4 = g5σ6, σ6g5 = g2σ6, σ6g6 = g6σ1.
Let ρ = (ρ, V ) ∈ Gσ1 and v ∈ V − 0. Assume that v is an eigenvector ofρ(σ6) with eigenvalue λ. We define W := span- {giv | 1 ≤ i ≤ 6}. Then, Wis a braided vector subspace of M(O, ρ).
Lemma 4.9. Let (σi)1≤i≤6, (gi)1≤i≤6, (ρ, V ) ∈ Gσ1 , W , λ as above. Assume
that qσ1σ1 = λ = −1. Then W ≃ M(O44, χ−) as braided vector spaces.
Proof. Since qσ1σ1 = −1 we have that ρ(σ4i ) = id, 1 ≤ i ≤ 6, from Lemma