aa r X i v : . [ m a t h . QA ] A ug 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 groupsis infinite. These results are consequences of [AHS]. We illustrate eachtechnique with applications.
Dedicado a Isabel Dotti y Roberto Miatello en su sexag´esimo cumplea˜nos . Introduction G be a finite group and let C G C G YD be the category of Yetter-Drinfeld modules over C G . 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 ∈ C G C G YD , decide when the Nichols algebra B ( V ) isfinite-dimensional. Recall that a Yetter-Drinfeld module over the group algebra C G (or over G for short) is a left C G -module and left C G -comodule M satisfying thecompatibility condition δ ( g.m ) = ghg − ⊗ 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 isa conjugacy class in G and ρ is an irreducible representation of the isotropygroup G s 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].
Date : October 24, 2018.2000
Mathematics Subject Classification.
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 dim B ( 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 group G and let V ∈ C G C G YD irreducible. If the underlying braided vector spacecontains a braided vector subspace of diagonal type, whose Nichols algebrahas infinite dimension, then dim B ( 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 = V ⊕ · · · ⊕ V θ ∈ C G C G YD , where the V i ’s areirreducible. Then the Nichols algebra of V is studied, under the assumptionthat the B ( V i ) 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 implies W finite. Although the picture is not yet complete,the previous result implies that, for a few G – explicitly, S , S , D n – 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 = V ⊕ V ∈ C Γ C Γ YD , where Γ = S , S or D n , such that dim B ( V ) = ∞ by [AHS, Section4]. Then there is a rack ( X, ⊲ ) and a cocycle q such that ( V, c ) ≃ ( C X, c q ).Let G be a finite group, let O be a conjugacy class in G , s ∈ O , ρ ∈ c G s and M ( O , ρ ) ∈ C G C G YD 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 ( C X, c q ); thus, necessarily, dim B ( 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. OINTED HOPF ALGEBRAS 3 O , ρ ) for S m 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 J , J , J , He and Suz are 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 dim B ( 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 ), where V is a vector space and c : V ⊗ V → V ⊗ V is a linear isomorphism such that c 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 ( v i ) ≤ i ≤ θ , let ( q ij ) ≤ i,j ≤ θ be a matrixof non-zero scalars and let c : V ⊗ V → V ⊗ V be given by c ( v i ⊗ v j ) = q ij v j ⊗ v i . 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 φ i : X → X , φ i ( j ) := i ⊲ j , is a bijection for all i ∈ 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 − . In this case, j ⊲ i = i whenever 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 if q i,j⊲k q j,k = q i⊲j,i⊲k q i,k , for all i , j , k ∈ X . Then ( C X, c q ) is a braided vectorspace, where C X is the vector space with basis e k , k ∈ X , and the braidingis given by c q ( e k ⊗ e l ) = q k,l e k⊲l ⊗ e k , for all k, l ∈ X. 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 C T is a braided vector subspace of ( C X, c q ) of diagonaltype. Definition 1.1.
Let X be a rack. Let X and X be two disjoint copiesof X , together with bijections ϕ i : X → X i , i = 1 ,
2. The square of X is the rack with underlying set the disjoint union X ` X and with rackmultiplication ϕ i ( x ) ⊲ ϕ j ( y ) = ϕ j ( x ⊲ y ) ,x, y ∈ X , 1 ≤ i, j ≤
2. We denote the square of X by X (2) . This is aparticular 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 b G the set of isomorphism classes of irreducible representations of G .We shall often denote a representant of a class in b G with the same symbolas the class itself.Here is an explicit description of the irreducible Yetter-Drinfeld module M ( O , ρ ). Let t = s , . . . , t M be a numeration of O and let g i ∈ G such that g i ⊲ s = t i for all 1 ≤ i ≤ M . Then M ( O , ρ ) = ⊕ ≤ i ≤ M g i ⊗ V , where V isthe vector space affording the representation ρ . Let g i v := g i ⊗ v ∈ M ( O , ρ ),1 ≤ i ≤ M , v ∈ V . If v ∈ V and 1 ≤ i ≤ M , then the action of g ∈ G is given by g · ( g i v ) = g j ( γ · v ), where gg i = g j γ , for some 1 ≤ j ≤ M and γ ∈ G s , and the coaction is given by δ ( g i v ) = t i ⊗ g i v . Then M ( O , ρ ) is abraided vector space with braiding c ( g i v ⊗ g j w ) = g h ( γ · w ) ⊗ g i v , for any1 ≤ i, j ≤ M , v, w ∈ V , where t i g j = g h γ for unique h , 1 ≤ h ≤ M and γ ∈ G s . Since s ∈ Z ( G s ), the center of G s , the Schur Lemma implies that(1.2) s acts by a scalar q ss on V. Lemma 1.2. If U is a subspace of W such that c ( U ⊗ U ) = U ⊗ U and dim B ( U ) = ∞ , then dim B ( W ) = ∞ . (cid:3) Lemma 1.3. [AZ, Lemma 2.2]
Assume that s is real (i. e. s − ∈ O ). If dim B ( O , ρ ) < ∞ , then q ss = − and s has even order. (cid:3) Let σ ∈ S m be a product of n j disjoint cycles of length j , 1 ≤ j ≤ m .Then the type of σ is the symbol (1 n , n , . . . , m n m ). We may omit j n j when n j = 0. The conjugacy class O σ of σ coincides with the set of allpermutations in S m 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 S m , like O mj for the conjugacy class of j -cycles in S m . OINTED HOPF ALGEBRAS 5 A technique from the dihedral group D n , n odd Let n > D n be the dihedral group of order 2 n ,generated by x and y with defining relations x = e = y n and xyx = y − . Let O x be the conjugacy class of x and let sgn ∈ c D xn be the sign representation( D xn = h x i ≃ Z ). 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 ( O x , sgn) ⊕ M ( O x , sgn)) has infi-nite dimension. (cid:3) Note that M ( O x , sgn) ⊕ M ( O x , sgn) is isomorphic as a braided vectorspace to ( C X n , q ), where • X n is the rack with 2 n elements s i , t j , i, j ∈ Z /n , and with structure s i ⊲ s j = s i − j , s i ⊲ t j = t i − j , t i ⊲ s j = s i − j , t i ⊲ t j = t i − j , i, j ∈ Z /n ; • q is the constant cocycle q ≡ − d divides n , then X d can be identified with a subrack of X n . Hence, itis enough to consider braided vector spaces ( C X p , q ), with p an odd prime.We fix a finite group G with the rack structure given by conjugation x ⊲ y = xyx − , x , y ∈ G . Let O be a conjugacy class in G . Definition 2.2.
Let p > µ i ) i ∈ Z /p of distinctelements of G is of type D p if(2.1) µ i ⊲ µ j = µ i − j , i, j ∈ Z /p. Let ( µ i ) i ∈ Z /p and ( ν i ) i ∈ Z /p be two families of type D p in G , such that µ i = ν j for all i, j ∈ Z /p . Then ( µ, ν ) := ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p is of type D (2) p if(2.2) µ i ⊲ ν j = ν i − j , ν i ⊲ µ j = µ i − j , i, j ∈ Z /p. It is useful to denote i ⊲ j = 2 i − j , for i, j ∈ Z /p .We state some consequences of this definition for further use. Remark . If ( µ i ) i ∈ Z /p is of type D p then µ − i ⊲ µ j = µ i − j , µ i ⊲ µ − j = µ − i − j , µ − i ⊲ µ − j = µ − i − j , (2.3) µ ki ⊲ µ j = µ i − j , µ i ⊲ µ kj = µ k i − j , µ ki ⊲ µ kj = µ k i − j , (2.4)for all i, j ∈ Z /p , and for all k odd. Remark . Assume that p is odd. If ( µ, ν ) = ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p is of type D (2) p , then for all i , j , µ i = µ j , ν i = ν j , µ i ν j = ν j µ i , ν i µ j = µ j ν i . (2.5)Indeed, µ h µ j = µ j µ h , hence µ h − j = µ h µ j µ − h = µ j . Take now h = i + j ANDRUSKIEWITSCH AND FANTINO
Lemma 2.5. If ( µ, ν ) = ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p is of type D (2) p , then (i) µ k µ l = µ t ( l − k )+ k µ t ( l − k )+ l , (ii) µ k ν l = µ t ( l − k )+ k ν t ( l − k )+ l , (iii) µ k ν l = ν (2 t +1)( l − k )+ k µ (2 t +1)( l − k )+ l ,for all k , l , t ∈ Z /p . Notice that we have the analogous relations interchanging µ by ν . Proof.
We proceed by induction on t . We will prove (i); (ii) and (iii) aresimilar. The result is obvious when t = 0. Since µ k µ l = µ l µ l⊲k , then theresult holds for t = 1. Let us suppose that (i) holds for every s ≤ t . Now, µ k µ l = µ t ( l − k )+ k µ t ( l − k )+ l = µ t ( l − k )+ l µ ( t ( l − k )+ l ) ⊲ ( t ( l − k )+ k ) = µ ( t +1)( l − k )+ k µ ( t +1)( l − k )+ l by the recursive hypothesis. (cid:3) Lemma 2.6.
Assume that p is odd. If ( µ, ν ) is of type D (2) p , then for i ∈ Z /p , µ i ν i = µ ν , (2.6) ν i µ i = ν µ . (2.7) Proof.
Let i , j ∈ Z /p , with i = j . If we write (ii) of Lemma 2.5 with k = i , l = j and t = − / µ i ν j = µ i − j ν i . Thus, µ i ν i ν j = µ i ν j ν j ν i = µ i − j ν i ν i ν i − j = µ i − j ν i − j ν i , and, by (2.5), µ i ν i = µ i − j ν i − j . Now (2.6) follows taking j = 2 i . Now (2.7) follows from (2.6) by (2.2). (cid:3) We now set up some notation that will be used in the rest of this section.Let ( µ i ) i ∈ Z /p be a family of type D p in G , with p odd. Set g i = µ i/ , (2.8) α ij = g − i⊲j µ i g j = µ − i − j/ µ i µ j/ , (2.9)for all i, j ∈ Z /p . Then g i ⊲ µ = µ i , α ij ∈ G µ , i, j ∈ Z /p. Let now ( µ, ν ) be of type D (2) p and suppose that there exists g ∞ ∈ G suchthat g ∞ ⊲ µ = ν . Set f i = ν i/ g ∞ , (2.10) β ij = f − i⊲j µ i f j = g − ∞ ν − i − j/ µ i ν j/ g ∞ , (2.11) γ ij = g − i⊲j ν i g j = µ − i − j/ ν i µ j/ , (2.12) δ ij = f − i⊲j ν i f j = g − ∞ ν − i − j/ ν i ν j/ g ∞ . (2.13) OINTED HOPF ALGEBRAS 7
Then f i ⊲ µ = ν i , β ij , γ ij , δ ij ∈ G µ , i, j ∈ Z /p. We assume from now on that p is an odd prime. This is required in theproof of the next lemma, needed for the main result of the section.
Lemma 2.7.
Let ( µ, ν ) = ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p be of type D (2) p , and supposethat there exists g ∞ ∈ G such that g ∞ ⊲ µ = ν . Let g i and f i be as in (2.8) and (2.10) , respectively. Then, for all i, j ∈ Z /p , (a) α ij = δ ij = µ , (b) β ij = g − ∞ µ g ∞ , (c) γ ij = ν .Proof. Let k , l be in Z /p . Then, for all r ∈ Z /p , we have(2.14) µ k µ l = µ k + r µ l + r , µ k ν l = µ k + r ν l + r , µ k ν l = ν k + r µ l + r . This follows from (2.5) and Lemma 2.6 (when k = l ), and Lemma 2.5 (when k = l ). There are similar equalities interchanging µ ’s and ν ’s. Now α ij = µ − i − j/ µ i µ j/ = µ ,δ ij = g − ∞ ν − i − j/ ν i ν j/ g ∞ (2.14) = g − ∞ ν g ∞ = µ ,β ij = g − ∞ ν − i − j/ µ i ν j/ g ∞ (2.14) = g − ∞ µ g ∞ ,γ ij = µ − i − j/ ν i µ j/ = µ − i − j/ µ i − j/ ν = ν , and the Lemma is proved. (cid:3) We can now prove one of the main results of this paper.
Theorem 2.8.
Let ( µ, ν ) = ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p be a family of elements in G with µ ∈ O . Let ( ρ, V ) be an irreducible representation of the centralizer G µ . We assume that (H1) ( µ, ν ) is of type D (2) p ; (H2) ( µ, ν ) ⊆ O , with g ∞ ∈ G such that g ∞ ⊲ µ = ν ; (H3) q µ µ = − ; (H4) there exist v, w ∈ V − such that, ρ ( g − ∞ µ g ∞ ) w = − w, (2.15) ρ ( ν ) v = − v. (2.16) Then dim B ( O , ρ ) = ∞ . ANDRUSKIEWITSCH AND FANTINO
Proof.
We keep the notation (2.10)–(2.13) above. Let v, w ∈ V − W := span- { g i v : i ∈ Z /p } ∪ { f i w : i ∈ Z /p } . Let Ψ : C X p → W begiven by Ψ( s i ) = g i v , Ψ( t i ) = f i w , i ∈ Z /p . Since the elements µ i and ν j are all different, Ψ is a linear isomorphism. We claim that W is a braidedvector subspace of M ( O , ρ ) and that Ψ is an isomorphism of braided vectorspaces. We compute the braiding in W : c ( g i v ⊗ g j v ) = µ i g j v ⊗ g i v = g i⊲j α ij v ⊗ g i v (H3) = − g i⊲j v ⊗ g i v,c ( g i v ⊗ f j w ) = µ i f j w ⊗ g i v = f i⊲j β ij w ⊗ g i v (2.15) = − f i⊲j w ⊗ g i v,c ( f i w ⊗ g j v ) = ν i g j v ⊗ f i w = g i⊲j γ ij v ⊗ f i w (2.16) = − g i⊲j v ⊗ f i w,c ( f i w ⊗ f j w ) = ν i f j w ⊗ f i w = f i⊲j δ ij w ⊗ f i w (H3) = − f i⊲j w ⊗ f i w, by Lemma 2.7. The claim is proved. Hence, dim B ( W ) = ∞ by Theorem2.1. Now the Theorem follows from Lemma 1.2. (cid:3) As a consequence of Theorem 2.8, we can state a very useful criterion.
Corollary 2.9.
Let G be a finite group, µ i , ≤ i ≤ p − , distinct elementsin G , with p an odd prime. Let us suppose that there exists k ∈ Z such that µ k = µ and µ k ∈ O , the conjugacy class of µ . Let ρ = ( ρ, V ) ∈ d G µ .Assume further that (i) ( µ i ) i ∈ Z /p is of type D p , (ii) q µ µ = − .Then dim B ( O , ρ ) = ∞ .Proof. We may assume that 1 < k < | µ | . By hypothesis (ii), the order of µ is even; hence k is odd, say k = 2 t +1, with t ≥
1. Let ν i := µ ki , 0 ≤ i ≤ p − g ∞ ∈ G such that g ∞ ⊲ µ = µ k . Set ( µ, ν ) = ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p .Clearly ( µ, ν ) ⊆ O . We claim that ( µ, ν ) is of type D (2) p . Indeed, using (i)it is easy to see that ( µ i ) i ∈ Z /p ∪ ( ν i ) i ∈ Z /p are all distinct. Then the claimfollows by (2.4).It remains to check the hypothesis (H4) of Theorem 2.8. As g ∞ µ g − ∞ = µ k , g l ∞ µ g − l ∞ = µ k l , for all l ≥
0. In particular, g − ∞ µ g ∞ = g | g ∞ |− ∞ µ g −| g ∞ | +1 ∞ = µ k | g ∞|− . Then, since q µ µ = − k is odd, we see that ρ ( g − ∞ µ g ∞ ) = − id. Hence(2.15) holds, for any w ∈ V −
0. Also, ρ ( ν ) = ρ ( µ k ) = ( − id) k = − id,because k is odd; thus, (2.16) holds for any v ∈ V −
0. Thus, for any v , w in V −
0, we are in the conditions of Theorem 2.8. Then dim B ( O , ρ ) = ∞ . (cid:3) OINTED HOPF ALGEBRAS 9
Example 2.10.
Let m ≥ . Let σ ∈ S m of type (1 n , n , . . . , m n m ) , O theconjugacy class of σ and ρ ∈ c S σm . If there exists j , ≤ j ≤ m , such that • p divides j , for some odd prime p , and • n j ≥ ;then dim B ( O , ρ ) = ∞ . Before proving the Example, we state a more general Lemma that mightbe of independent interest. Here p is no longer an odd prime. Lemma 2.11.
Let m, p ∈ Z > . Let σ ∈ S m of type (1 n , n , . . . , m n m ) and O the conjugacy class of σ . If there exists j = 4 , ≤ j ≤ m , such that • p divides j , and • n j ≥ ;then O contains a subrack of type D (2) p .Proof. Let j = 2 p κ , with κ ≥
1. Let α = ( i i · · · i j ) be a j -cycle thatappears in the decomposition of σ as product of disjoint cycles and define I := ( i i i · · · i j − ) and P := ( i i i · · · i j ) . We claim that(a) I and P are disjoint pκ -cycles,(b) α = IP ,(c) α I α − = P , (and then σ I σ − = P ),(d) P t α P t = α t +1 , P t α − P t = α t − , for all integers t .The first two items are clear, while (c) follows from the well-known formula α ( l l . . . l k ) α − = ( α ( l ) α ( l ) . . . α ( l k )). (d). By (c), P t = α I t α − . Then P t α P t = α I t P t ; by (b), P t α P t = αα t , as claimed.We define σ i := P iκ σ P − iκ , ≤ i ≤ p − . (2.17)Notice that σ i = P iκ α P − iκ e σ , where e σ := α − σ . The elements ( σ i ) i ∈ Z /p are all distinct; indeed, if σ i = σ l , with i , l ∈ Z /p , then P iκ σ P − iκ = P lκ σ P − lκ , i. e. P ( i − l ) κ σ P − ( i − l ) κ = σ , which implies that i = σ ( i ) = P ( i − l ) κ σ P − ( i − l ) κ ( i ) = P ( i − l ) κ ( i ) = i i − l ) κ +2 , and this means that 2( i − l ) κ = 0 in Z /j . Thus i = l , as desired. We claim that ( σ i ) i ∈ Z /p is of type D p . If i , l ∈ Z /p , then σ i ⊲ σ l = P iκ σ P − iκ P lκ σ P − lκ P iκ σ − P − iκ = P iκ α P − iκ P lκ α P − lκ P iκ α − P − iκ e σ = P (2 i − l ) κ P ( l − i ) κ α P ( l − i ) κ α P ( i − l ) κ α − P ( i − l ) κ P − (2 i − l ) κ e σ = P (2 i − l ) κ α l − i ) κ +1 α α i − l ) κ − P − (2 i − l ) κ e σ = P (2 i − l ) κ α P − (2 i − l ) κ e σ = P (2 i − l ) κ σ P − (2 i − l ) κ = σ i⊲l , by (d), and the claim follows. Finally, the family of type D (2) p we are lookingfor is ( σ i ) i ∈ Z /p ∪ ( σ − i ) i ∈ Z /p . It remains to show that σ t = σ − l for all t , l ∈ Z p .If σ t = σ − l , then σ t ( i ) = σ − l ( i ), that is i = i j − , a contradiction to thehypothesis j = 4. (cid:3) Proof of the Example 2.10.
We may assume that q σσ = −
1, by Lemma 1.3.By Lemma 2.11, we have a family ( σ i ) i ∈ Z /p of type D p , with σ = σ . NowCorollary 2.9 applies, with µ = σ , k = | σ |−
1. Thus dim B ( O , ρ ) = ∞ . (cid:3) A technique from the symmetric group S We study separately the case p = 3 because of the many applicationsfound. In this setting, D ≃ S and O x = O = { (1 2) , (2 3) , (1 3) } is theconjugacy class of transpositions in S . The rack X is described as a set of6 elements X = { x , x , x , y , y , y } with the multiplication x i ⊲ x j = x k , y i ⊲ y j = y k , x i ⊲ y j = y k , y i ⊲ x j = x k , for i , j , k , all distinct or all equal.3.1. Families of type D and D (2)3 . We fix a finite group G and O aconjugacy class in G . Our aim is to give criteria to detect when O containsa subrack isomorphic to X . Definition 3.1.
Let σ , σ , σ ∈ G distinct. We say that ( σ i ) ≤ i ≤ is oftype D if σ i ⊲ σ j = σ k , where i , j , k are all distinct . (3.1)The requirement (3.1) consists of 6 identities, but actually 3 are enough. Remark . If σ ⊲ σ = σ , (3.2) σ ⊲ σ = σ , (3.3) σ ⊲ σ = σ , (3.4)then ( σ i ) ≤ i ≤ is of type D . (cid:3) OINTED HOPF ALGEBRAS 11
Here is a characterization of D families. Proposition 3.3.
Let σ , σ ∈ O . Define σ := σ ⊲ σ . Then ( σ i ) ≤ i ≤ isof type D if and only if σ G σ , (3.5) σ ∈ G σ , (3.6) σ = σ ⊲ ( σ ⊲ σ ) . (3.7) Proof.
The definition of σ is equivalent to (3.2) and (3.7) is equivalent to(3.4). Assume that ( σ i ) ≤ i ≤ is of type D . As σ = σ , σ G σ . Also, σ ⊲ σ = σ ⊲ ( σ ⊲ σ ) = σ ⊲ σ = σ . Hence σ ∈ G σ .Conversely, if σ G σ , then σ = σ , σ = σ . From (3.5) and (3.7), wesee that σ = σ . It remains to check (3.3): σ ⊲ σ = σ ⊲ σ = σ . (cid:3) Definition 3.4.
Let σ , σ , σ , τ , τ , τ ∈ G be distinct elements. We saythat ( σ, τ ) = ( σ , σ , σ , τ , τ , τ ) is of type D (2)3 , if ( σ i ) ≤ i ≤ and ( τ j ) ≤ j ≤ are of type D , and σ i ⊲ τ j = τ k , τ i ⊲ σ j = σ k , (3.8)where i , j , k are either all equal, or all distinct.The requirement (3.8) consists of 18 identities, but less are enough. Webegin by a first reduction. Lemma 3.5.
Let ( σ i ) ≤ i ≤ and ( τ j ) ≤ j ≤ such that (3.2) , (3.3) , (3.4) holdfor σ and for τ . If σ ⊲ τ = τ , (3.9) σ ⊲ τ = τ , (3.10) σ ⊲ τ = τ , (3.11) also hold, then σ i ⊲ τ i = τ i , ≤ i ≤ , and σ i ⊲ τ j = τ k , for all i , j , k distinct.Proof. We have to prove σ ⊲ τ = τ , (3.12) σ ⊲ τ = τ , (3.13) σ ⊲ τ = τ , (3.14) σ ⊲ τ = τ , (3.15) σ ⊲ τ = τ , (3.16) σ ⊲ τ = τ , (3.17) The identity (3.12) holds because σ ⊲ τ = σ ⊲ ( τ ⊲ τ ) = τ ⊲ τ = τ ; inturn, (3.13) and (3.14) hold because σ ⊲ τ = ( σ ⊲ σ ) ⊲ ( σ ⊲ τ ) = σ ⊲ ( σ ⊲ τ ) = σ ⊲ τ = τ ,σ ⊲ τ = ( σ ⊲ σ ) ⊲ ( σ ⊲ τ ) = σ ⊲ ( σ ⊲ τ ) = σ ⊲ τ = τ . Also, σ ⊲ τ = ( σ ⊲ σ ) ⊲ ( σ ⊲ τ ) = σ ⊲ ( σ ⊲ τ ) = σ ⊲ τ = τ , showing(3.15). Finally, σ ⊲τ = σ ⊲ ( σ ⊲τ ) = σ ⊲ ( σ ⊲τ ) = σ ⊲τ = σ ⊲ ( τ ⊲τ ) = τ ⊲ τ = τ , proving (3.16) and (3.17). (cid:3) Therefore, given 6 distinct elements σ , σ , σ , τ , τ , τ ∈ G , if the 12identities: (3.2), (3.3), (3.4), for σ and for τ , (3.9), (3.10), (3.11), and theanalogous identities τ ⊲ σ = σ , (3.18) τ ⊲ σ = σ , (3.19) τ ⊲ σ = σ , (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 σ , σ , σ , τ , τ , τ ∈ 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 τ = σ ⊲ τ = σ τ σ − . If τ acts on both sides of(3.11), then τ = τ ⊲ τ = ( τ ⊲ σ ) ⊲ ( τ ⊲ τ ) = σ ⊲ τ ; if now σ acts on thelast, then σ ⊲ τ = ( σ ⊲ σ ) ⊲ ( σ ⊲ τ ) = σ ⊲ τ = τ . Thus, (3.10) holds. We can now conclude from Lemma 3.5 that σ i ⊲ τ i = τ i ,1 ≤ i ≤
3, and σ i ⊲ τ j = τ k , for all i , j , k distinct. If now σ acts on (3.19),then σ = ( σ ⊲ τ ) ⊲ ( σ ⊲ σ ) = τ ⊲ σ , and (3.20) holds. (cid:3) Examples of D (2)3 type. We first spell out explicitly Theorem 2.8 andCorollary 2.9 for p = 3. Theorem 3.7.
Let σ , σ , σ , τ , τ , τ ∈ G distinct; denote ( σ, τ ) =( σ , σ , σ , τ , τ , τ ) . Let ρ = ( ρ, V ) ∈ d G σ . We assume that (H1) ( σ, τ ) is of type D (2)3 , (H2) ( σ, τ ) ⊆ O , with g ∈ G such that g ⊲ σ = τ , (H3) q σ σ = − , OINTED HOPF ALGEBRAS 13 (H4) there exist v, w ∈ V − such that, ρ ( g − σ g ) w = − w, (3.21) ρ ( τ ) v = − v, (3.22) Then dim B ( O , ρ ) = ∞ . (cid:3) Corollary 3.8.
Let σ , σ , σ ∈ O distinct. Assume that there exists k , ≤ k ≤ | σ | , such that σ k = σ and σ k ∈ O . Let ρ = ( ρ, V ) ∈ d G σ . Assumefurther that (1) ( σ i ) ≤ i ≤ is of type D , (2) q σ σ = − .Then dim B ( O , ρ ) = ∞ . (cid:3) Corollary 3.8 applies notably to a real conjugacy class of an element oforder greater than 2. We list several applications for G = S m . Example 3.9.
Let m ≥ . Let O be the conjugacy class of S m of type (1 n , n , . . . , m n m ) , where • n , n ≥ and • n j ≥ for some j , ≤ j ≤ m .Let σ ∈ O and ρ ∈ c S σm . Then dim B ( O , ρ ) = ∞ .Proof. By hypothesis, we can choose σ = (1 2) β where β fixes 1, 2 and 3. If q σσ = −
1, then dim B ( O , ρ ) = ∞ , by Lemma 1.3. Assume that q σσ = − x = (1 2) , y = (1 3) , z = (2 3) , σ = σ = xβ, σ = yβ, σ := zβ. Clearly ( σ i ) ≤ i ≤ is of type D , O is real and | σ | >
2. By Corollary 3.8,dim B ( O , ρ ) = ∞ . (cid:3) In particular, let O be the conjugacy class of S m of type (1 , , m − m ≥
6. By the preceding, dim B ( O , ρ ) = ∞ . But, if q σσ = −
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 ≥ . Let σ ∈ S m of type (1 n , n , . . . , m n m ) , O theconjugacy class of σ and ρ ∈ c S σm . Assume that • there exists j , ≤ j ≤ m , such that j = 2 k , with k ≥ and n j ≥ .Then dim B ( O , ρ ) = ∞ . Proof. If q σσ = −
1, then dim B ( O , ρ ) = ∞ , by Lemma 1.3. Assume that q σσ = −
1. Let α = ( i i · · · i j ) , α = ( i j +1 i j +2 · · · i j ) , α = ( i j +1 i j +2 · · · i j ) , be three j -cycles appearing in the decomposition of σ as product of disjointcycles and define I = ( i i i · · · i j − ) , B = ( i i j +1 )( i i j +2 ) · · · ( i j i j ) , P = ( i i i · · · i j ) , B = ( i j +1 i j +1 )( i j +2 i j +2 ) · · · ( i j i j ) . Then(a) I and P are disjoint 3 k -cycles,(b) I k P k = B B ,(c) α α α I α − α − α − = P , (and then σ I σ − = P ),(d) P k σ P k = σB B , and(e) P − k σ P − k = σB B .The first item is clear. To see (b), note that B B = ( i i j +1 i j +1 )( i i j +2 i j +2 ) · · · ( i j i j i j ) . (c) follows as in the proof of Lemma 2.11 (c). (d). By (b) and (c), wehave that σ − P k σ P k = I k P k = B B , as claimed. (e). By (b) and (c), σ − P − k σ P − k = I − k P − k = B B as claimed.Set now σ := σ , σ := P k σ P − k and σ := P − k σ P k . As in the proofof Example 2.10 we can see that σ , σ and σ are distinct. We check that( σ i ) ≤ i ≤ is of type D using Remark 3.2.By (d), P k σ P k ∈ S σm , i. e. P k σ P k σ P − k σ − P − k = σ , or σ P k σ P − k σ − = P − k σ P k . That is, σ ⊲σ = σ . Analogously, σ ⊲σ = σ is proved using (e).To check that σ ⊲σ = σ , note that σ ⊲σ = P k σ P − k P − k σ P k P k σ − P − k = σ , because P k σ P − k = P k σ P k P − k = σB B ∈ S σm , by (a) and (d).We now apply Corollary 3.8 and conclude that dim B ( O , ρ ) = ∞ . (cid:3) We shall need a few well-known results on symmetric groups.
Remark . (i) If ρ is a faithful representation of S n , then ρ ( τ ) / ∈ C id, forevery τ ∈ S n , τ = id (since S n is centerless).(ii) If ρ = ( ρ, W ) ∈ c S n , with ρ = sgn, then for any involution τ ∈ S n (i. e., τ = id), there exists w ∈ W − ρ ( τ ) w = w (otherwise ρ ( τ ) = − id). Example 3.12.
Let m ≥ . Let σ ∈ S m of type (1 n , n , . . . , m n m ) , O theconjugacy class of σ and ρ ∈ c S σm . Assume that OINTED HOPF ALGEBRAS 15 • n ≥ and • there exists j , with j ≥ , such that n j ≥ .Then dim B ( O , ρ ) = ∞ .Proof. By Lemma 1.3, we may suppose that q σσ = −
1. Assume that ( i i ),( i i ) and ( i i ) are three transpositions appearing in the decomposition of σ as a product of disjoint cycles. We define x := ( i i )( i i )( i i ) , y := ( i i )( i i )( i i ) , z := ( i i )( i i )( i i )and α := xσ . It is easy to see, using for instance Proposition 3.3, that σ := σ, σ := yα, σ := zα, is of type D . Then dim B ( O , ρ ) = ∞ , by Corollary 3.8. Indeed, σ − ∈ O ,but σ = σ − because σ has order > (cid:3) In the proof of the next Example, we need some notation for the inducedrepresentation. Let H be a subgroup of a finite group G of index k , φ , . . . , φ k the left cosets of H in G , with representatives g φ , . . . , g φ k . Let θ = ( θ, W ) ∈ b H , and w , . . . w r a basis of W . Set V :=span- { g φ i w j | ≤ i ≤ k, ≤ j ≤ r } .For i , j , with 1 ≤ i ≤ k , 1 ≤ j ≤ r we define ρ : G → Aut( V ) by ρ ( g )( g φ i w j ) = g φ l θ ( h ) w j , where gg φ i = g φ l 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 ≥ . Let σ ∈ S m of type (1 n , n , . . . , m n m ) , O theconjugacy class of σ and ρ ∈ c S σm . If n ≥ , then dim B ( O , ρ ) = ∞ .Proof. By Lemma 1.3, we may suppose that q σσ = −
1. We denote the n transpositions appearing in the decomposition of σ as product of disjointcycles by A , , . . . , A n , and we define A = A , · · · A n , . Let us supposethat A , = ( i i ), A , = ( i i ), A , = ( i i ), A , = ( i i ), A , = ( i i )and A , = ( i i ). We define x := ( i i )( i i )( i i )( i i )( i i )( i i )and α := xσ .If there exists j , with j ≥
3, such that n j ≥
1, then the result followsfrom Example 3.12. Assume that n j = 0, for every j ≥
3, i. e. the type of σ is (1 n , n ). The centralizer of σ in S m is S σm = T × T , with T ≃ S n and T = Γ ⋊ Λ, withΓ := h A , , . . . , A n , i , Λ := h B , , . . . , B n − , i . Here B l, := ( i l − i l +1 )( i l i l +2 ), for 1 ≤ l ≤ n −
1. Note that Γ ≃ ( Z / n and Λ ≃ S n . Now, ρ = ρ ⊗ ρ , with ρ = ( ρ , V ) ∈ c T and ρ = ( ρ , V ) ∈ c T . For every t , 1 ≤ t ≤ n , we define χ t ∈ b Γ, by χ t ( A l, ) = ( − δ t,l , 1 ≤ l ≤ n . Then, the irreducible representations of Γ are χ t ,...,t J := χ t . . . χ t J , ≤ J ≤ n , ≤ t < · · · < t J ≤ n . The case J = 0 corresponds to the trivial representation of Γ.For every J , with 0 ≤ J ≤ n , we denote χ ( J ) := χ ,...,J . The action of Λon Γ induces a natural action of Λ on b Γ, namely ( λ · χ )( A l, ) := χ ( λ − A l, λ ),1 ≤ l ≤ n , λ ∈ Λ. The orbit and the isotropy subgroup of χ ( J ) ∈ b Γ are O χ ( J ) = { χ k ,...,k J : 1 ≤ k < · · · < k J ≤ n } , (3.24) Λ χ ( J ) = (Λ χ ( J ) ) × (Λ χ ( J ) ) (3.25) = h B , , . . . , B J − , i × h B J +1 , , . . . , B n − , i ≃ S J × S n − J . Thus, the characters χ ( J ) , 0 ≤ J ≤ n , form a complete set of representativesof the orbits in b Γ under the action of Λ.Since ρ ∈ \ Γ ⋊ Λ, we have that ρ = Ind Γ ⋊ ΛΓ ⋊ Λ χ ( J ) χ ( J ) ⊗ µ , with χ ( J ) asabove and µ = ( µ, W ) ∈ [ Λ χ ( J ) – see [S, Section 8.2]. By (3.25), µ = µ ⊗ µ ,with µ l = ( µ l , W l ) ∈ \ (Λ χ ( J ) ) l , l = 1, 2. Let { φ = Λ χ ( J ) , . . . , φ k } the leftcosets of Λ χ ( J ) in Λ, where k = [Λ : Λ χ ( J ) ] = n ! J !( n − J )! .Note that B , = ( i i )( i i ) , B , = ( i i )( i i ) and B , = ( i i )( i i ) . We define B := B , B , B , . Notice that the order of B is 2.Since q σσ = −
1, then J is odd. We will consider two cases. CASE (1): assume that J ≤
5. Then, B Λ χ ( J ) . This implies thatthe left coset φ of Λ χ ( J ) in Λ containing B is not the trivial coset φ . Wechoose as representatives of the cosets φ and φ to g φ = id and g φ = B ,respectively. We define v := g φ w + g φ w , with w ∈ W −
0. Notice that Bg φ = g φ id and Bg φ = g φ id. Using (3.23), we have that(3.26) ρ ( B ) v = ρ ( B )( g φ w ) + ρ ( B )( g φ w )= g φ µ (id) w + g φ µ (id) w = g φ w + g φ w = v . Let v := v ⊗ v , with v ∈ V −
0. Then, ρ ( B ) v = ( ρ ⊗ ρ )(id , B )( v ⊗ v ) = ρ (id) v ⊗ ρ ( B ) v = v ⊗ v = v, (3.27) OINTED HOPF ALGEBRAS 17 by (3.26). We define σ := σ , σ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,σ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,τ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,τ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,τ := ( i i )( i i )( i i )( i i )( i i )( i i ) α. We can check by straightforward computations that ( σ, τ ) is of type D (2)3 .Let g := ( i i )( i i )( i i ); thus, g ⊲ σ = τ . Moreover, τ = σB = gσg and σ τ = B = gσ τ g . Then, ρ ( τ ) v = − v = ρ ( gσ g ) v, by (3.27). Therefore, dim B ( O , ρ ) = ∞ , by Theorem 3.7. CASE (2): assume that J ≥
7. Then, B ∈ Λ χ ( J ) ; moreover, B ∈ (Λ χ ( J ) ) . Also, Bg φ = g φ B .Let v = g φ w , with w ∈ W −
0. Since W = W ⊗ W , we may assumethat w = w ⊗ w , with w ∈ W − w ∈ W −
0. Then, using (3.23), ρ ( B ) v = ρ ( B )( g φ w ) = g φ µ ( B ) w = g φ ( µ ⊗ µ )( B, id)( w ⊗ w )= g φ (cid:16) µ ( B )( w ) ⊗ µ (id)( w ) (cid:17) = g φ (cid:16) ( µ ( B )( w ) ⊗ w (cid:17) . Notice that µ ∈ \ (Λ χ ( J ) ) . Since (Λ χ ( J ) ) ≃ S J , if µ = sgn, with sgnthe sign representation of S J , then there exists w ∈ W − µ ( B )( w ) = w , by Remark 3.11 (ii). In this case, we have ρ ( B ) v = g φ ( µ ( B )( w ) ⊗ w ) = g φ ( w ⊗ w ) = g φ w = v . (3.28)Taking v := v ⊗ v , with v ∈ V −
0, we have ρ ( B ) v = ( ρ ⊗ ρ )(id , B )( v ⊗ v ) = ρ (id) v ⊗ ρ ( B ) v = v ⊗ v = v, by (3.28). Considering σ i , τ i , 1 ≤ i ≤
3, as in the previous case, thehypothesis of Corollary 3.8 hold. Therefore, dim B ( O , ρ ) = ∞ .On the other hand, let us suppose that µ = sgn. Let w ∈ W , with w = w ⊗ w , w ∈ W − w ∈ W −
0. Let v = g φ w ; since µ ( B )( w ) = − w , we have ρ ( B ) v = − v . Choosing v := v ⊗ v , with v ∈ V −
0, we have that ρ ( B ) v = ( ρ ⊗ ρ )(id , B )( v ⊗ v ) = ρ (id) v ⊗ ρ ( B ) v = − v ⊗ v = − v. (3.29) We define σ := σ , σ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,σ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,τ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,τ := ( i i )( i i )( i i )( i i )( i i )( i i ) α,τ := ( i i )( i i )( i i )( i i )( i i )( i i ) α. It can be shown that ( σ, τ ) is of type D (2)3 . Let now g = ( i i )( i i )( i i );then, g ⊲ σ = τ . Furthermore, τ = B = gσg and σ τ = σB = g σ τ g .Then ρ ( τ ) v = − v = ρ ( gσg ) v and ρ ( σ τ ) v = v = ρ ( g σ τ g ) v, by (3.29). Therefore, dim B ( O , ρ ) = ∞ , by Theorem 3.7. (cid:3) A way to obtain a family of type D is to start from a monomorphism ρ : S → G and to consider the image by ρ of the transpositions. Anotherway is as follows. Remarks . Let G be a finite group and z ∈ Z ( G ).(i). Let ( σ i ) i ∈ Z / be of type D . Then ( zσ i ) i ∈ Z / is also of type D .(ii). Let ( σ, τ ) = ( σ i ) i ∈ Z / ∪ ( τ i ) i ∈ Z / be a family of type D (2)3 . Then( zσ, zτ ) = ( zσ i ) i ∈ Z / ∪ ( zτ i ) i ∈ Z / 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 = p m , m ∈ N , such that 3divides q − . Let ω ∈ F q be a primitive third root of 1.(i). If c ∈ F q , then ( µ i ) i ∈ Z / , where µ i = ω i ω i c ! , is a family of type D in GL (2 , F q ) . If c = − , then this is a family of type D in SL (2 , F q ) .The orbit of µ i is the set of matrices with minimal polynomial T − c .(ii). Let N > be an integer and let T be the subgroup of diagonalmatrices in GL ( N, F q ) . Let λ = diag( λ , λ , . . . , λ N ) ∈ T . Let O be theconjugacy class of λ . Assume that λ = − λ and let c = λ . Assume alsothat there exist i, j , with ≤ i, j ≤ N such that λ i = λ j ; say i = 3 , j = 4 ,for simplicity of the exposition. Then ( σ i ) i ∈ Z / ∪ ( τ i ) i ∈ Z / , where σ i = µ i
00 diag( λ , λ , . . . , λ N ) ! , τ i = µ i
00 diag( λ , λ , . . . , λ N ) ! , is a family of type D (2)3 in the orbit O ⊂ GL ( N, F q ) . OINTED HOPF ALGEBRAS 19
Let W = S N act on T in the natural way. Let χ : GL ( N, F q ) → C × be acharacter; it restricts to an irreducible representation ( χ, C ) of the centralizer GL ( N, F q ) σ . Fix a group isomorphism ϕ : F × q → G q − ⊂ C × , where G q − is the group of ( q − C . Recall that χ = ϕ (det h ) for someinteger h . Thus the restriction of χ to T is W -invariant. Proposition 3.16.
Keep the notation above. Assume that χ ( λ ) = − .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 = id N − is an involutionthat satisfies g ⊲ σ = τ . Because of the explicit form of χ , χ ( σ ) = − χ ( τ ), hence (H3) and (H4) hold. (cid:3) This example can be adapted to the setting of semisimple orbits in finitegroups of Lie type.4.
A technique from the symmetric group S The classification of the finite-dimensional Nichols algebras over S , givenin [AHS], relies on the fact (proved in loc. cit. ) that some Nichols algebras B ( V i ⊕ V j ) have infinite dimension. According to the general strategy pro-posed in the present paper, each of these pairs ( V i , V j ) 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 = { , , , , , } given by the verticesof the octahedron with the operation of rack given by the “right-hand rule”,i. e. if T i is the orthogonal linear map that fixes i and rotates the orthogonalplane by an angle of π/ i ), then we define ⊲ : X × X → X by i ⊲ j := T i ( j ) – see Figure 1.Explicitly,1 ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ , ⊲ . Let G be a finite group, σ , σ , σ , σ , σ , σ ∈ G distinct elements and O the conjugacy class of σ in G . r r r rrr ✡✡✡✡✡✡✡❅❅❅❅❅(cid:0)(cid:0)(cid:0)✁✁✁✁✁❇❇❇❇❇❇❇❅❅❅❅❅(cid:0)(cid:0)(cid:0)❇❇❇❇❇❇❇ ✁✁✁✁✁✡✡✡✡✡✡✡ Figure 1.
Octahedral rack.
Definition 4.1.
We will say that ( σ i ) ≤ i ≤ is of type O if the following holds σ i ⊲ σ j = σ i⊲j , ≤ i, j ≤ . Here and in the rest of this section, ⊲ in the subindex is the operation ofrack in the octahedral rack. In other words, ( σ i ) ≤ i ≤ is of type O if andonly if { σ i | ≤ i ≤ } is isomorphic to the octahedral rack via i σ i . Example 4.2.
Let m ≥ . Let us consider in S m the following 4-cycles (4.1) e σ = (1 2 3 4) , e σ = (1 2 4 3) , e σ = (1 3 2 4) , e σ = (1 3 4 2) , e σ = (1 4 2 3) , e σ = (1 4 3 2) . It is easy to see that ( e σ i ) ≤ i ≤ satisfy the relations given in the previousdefinition. Thus, ( e σ i ) ≤ i ≤ is of type O . Let χ − ∈ c S e σ be given by χ − (1 2 3 4) = −
1. The goal of this Section is toapply the next result, cf. [AHS, Theor. 4.7].
Theorem 4.3.
The Nichols algebra B (cid:0) M ( O , χ − ) ⊕ M ( O , χ − ) (cid:1) has infi-nite dimension. (cid:3) Remark . We note that M ( O , χ − ) ⊕ M ( O , χ − ) ≃ ( C Y, q ) as braidedvector spaces, where • Y = { x i , y j | ≤ i, j ≤ } ≃ X (2) , see Definition 1.1; • q is the constant cocycle q ≡ − Proof.
We define e σ := (1 2 3 4) =: e τ , e σ := (1 2 4 3) =: e τ , e σ := (1 3 2 4) =: e τ , e σ := (1 3 4 2) =: e τ , e σ := (1 4 2 3) =: e τ , e σ := (1 4 3 2) =: e τ . OINTED HOPF ALGEBRAS 21
We will denote by ( e σ j ) ≤ j ≤ (resp. ( e τ j ) ≤ j ≤ ) the first copy (resp. the secondcopy) of O , with system of left cosets representatives of S (1 2 3 4)4 given by e g = e g = e σ , e g = e g = e σ , e g = e g = e σ , e g = e g = e σ , e g = e g = e σ , e g = e g = e σ e σ . The map M ( O , χ − ) ⊕ M ( O , χ − ) → ( C Y, q ) given by e g i x i and e g i +6 y i , ≤ i ≤ , is an isomorphism of braided vector spaces. (cid:3) Proposition 4.5.
A family ( σ i ) ≤ i ≤ of distinct elements in G is of type O if and only if the following identities hold: σ ⊲ σ = σ , σ ⊲ σ = σ , σ ⊲ σ = σ , σ ⊲ σ = σ , σ ⊲ σ = σ , (4.2) σ ⊲ σ = σ , σ ⊲ σ = σ , σ ⊲ σ = σ , σ ⊲ σ = σ , σ ⊲ σ = σ . (4.3) Proof.
If we apply σ ⊲ to the relations in (4.3), then we obtain therelations σ ⊲ σ j = σ ⊲j , 1 ≤ j ≤
6, because σ ⊲ σ = σ . Analogously, weobtain the relations σ i ⊲ σ j = σ i⊲j , 1 ≤ j ≤
6, for i = 3, 4; and the relations σ ⊲ σ j = σ ⊲j , 1 ≤ j ≤
6, follow by applying σ ⊲ to the ones in (4.3). (cid:3) Lemma 4.6. If ( σ i ) ≤ i ≤ is of type O , then (i) σ = σ = σ = σ = σ = σ , (ii) σ σ = σ σ = σ σ , (iii) σ σ = σ σ = σ σ , (iv) σ σ = σ σ = σ σ .Proof. (i). Since σ i ⊲ ( σ i ⊲ ( σ i ( ⊲ ( σ i ⊲ σ j )))) = σ j , then σ i ∈ G σ j , 1 ≤ i, j ≤ σ = ( σ σ σ − ) = σ σ σ − = σ , and the rest is similar. (ii). ByDefinition 4.1, we see that σ σ = σ σ σ σ − = σ σ σ σ − σ σ − = σ σ σ σ − = σ σ ,σ σ = σ σ σ σ − = σ σ σ σ − σ σ − = σ σ σ σ − = σ σ . Then, σ σ = σ σ = σ σ , as claimed.(iii). By (ii), we have that σ σ = σ σ σ σ = σ σ σ σ = σ σ σ σ = σ σ σ = σ σ σ = σ σ . Then, σ σ = σ σ . We apply σ ⊲ ( σ ⊲ ( σ ⊲ )) to the last expression andwe have σ σ = σ σ .(iv) follows from (iii) applying σ ⊲ ( σ ⊲ ). (cid:3) Definition 4.7.
Let σ i , τ i ∈ G , 1 ≤ i ≤
6, all distinct. We say that ( σ, τ )is of type O (2) if ( σ i ) ≤ i ≤ and ( τ j ) ≤ j ≤ are both of type O , and σ i ⊲ τ j = τ i⊲j , τ i ⊲ σ j = σ i⊲j , ≤ i, j ≤ . (4.4) Lemma 4.8. If ( σ, τ ) is of type O (2) , then (i) σ τ = σ τ = σ τ = σ τ = σ τ = σ τ , (ii) σ − j τ j = σ − τ , ≤ j ≤ , (iii) τ − σ τ = τ − σ , (iv) τ − σ τ = σ τ − , (v) σ − σ τ = σ − τ σ , (vi) σ − σ τ = τ σ − .Proof. (i). First, σ τ = σ σ τ σ − = τ σ τ − τ σ τ − τ σ − = τ σ σ σ − = τ σ = σ τ . (4.5)Applying now σ ⊲ to (4.5) we get σ τ = τ σ . Applying σ ⊲ to thislast identity, we have σ τ = τ σ . The rest is similar.(ii). By (i) and Lemma 4.6 (ii) for ( τ i ) ≤ i ≤ , we have that σ − τ = σ − τ − τ τ = σ − τ − τ τ = σ − τ . The other relations can be obtained in an analogous way.(iii). It is easy to see that τ − σ τ = τ − τ τ τ σ = τ − τ τ τ σ = τ − τ τ τ σ = τ − τ τ σ τ = τ − τ σ τ τ = τ − τ σ τ = τ − σ . (iv) follows from (iii) applying σ ⊲ ( σ ⊲ ).(v). Clearly, σ − σ τ = σ − σ σ σ τ = σ − σ σ σ τ = σ − σ σ σ τ = σ − σ σ τ σ = σ − σ τ σ σ = σ − σ τ σ σ = σ − τ = σ − τ σ . (vi) follows from (v) applying σ ⊲ ( σ ⊲ ). (cid:3) Applications.
Let G be a finite group, O a conjugacy class of G . Let( σ i ) ≤ i ≤ ⊂ O be of type O . We define g := σ , g := σ , g := σ , g := σ , g := σ , g := σ σ ;(4.6) OINTED HOPF ALGEBRAS 23 then, σ i = g i ⊲ σ , 1 ≤ i ≤
6. It is easy to see that following relations hold σ g = g σ , σ g = g σ , σ g = g σ ,σ g = g σ , σ g = g σ , σ g = g σ − ,σ g = g σ , σ g = g σ , σ g = g σ ,σ g = g σ , σ g = g σ , σ g = g σ ,σ g = g σ , σ g = g σ − σ , σ g = g σ ,σ g = g σ , σ g = g σ , σ g = g σ ,σ g = g σ , σ g = g σ , σ g = g σ ,σ g = g σ , σ g = g σ , σ g = g σ ,σ g = g σ − , σ g = g σ , σ g = g σ ,σ g = g σ , σ g = g σ σ − , σ g = g σ σ ,σ g = g σ , σ g = g σ , σ g = g σ σ ,σ g = g σ , σ g = g σ , σ g = g σ . Let ρ = ( ρ, V ) ∈ d G σ and v ∈ V −
0. Assume that v is an eigenvector of ρ ( σ ) with eigenvalue λ . We define W := span- { g i v | ≤ i ≤ } . Then, W is a braided vector subspace of M ( O , ρ ). Lemma 4.9.
Let ( σ i ) ≤ i ≤ , ( g i ) ≤ i ≤ , ( ρ, V ) ∈ d G σ , W , λ as above. Assumethat q σ σ = λ = − . Then W ≃ M ( O , χ − ) as braided vector spaces.Proof. Since q σ σ = − ρ ( σ i ) = id, 1 ≤ i ≤
6, from Lemma(4.6) (i). Let e σ i be as in (4.1). If we choose e g = e σ , e g = e σ , e g = e σ , e g = e σ , e g = e σ , e g = e σ e σ , then e g i ⊲ e σ = e σ i , 1 ≤ i ≤
6. Thus, M ( O , χ − ) = span- { e g i v , | ≤ i ≤ } ,with v ∈ V −
0, where V is the vector space affording the representation χ − of S (1 2 3 4)4 . Now, the map W → M ( O , χ − ) given by g i v e g i v , 1 ≤ i ≤ (cid:3) The next lemma is needed for the main result of the section.
Lemma 4.10.
Let σ i , τ i , ≤ i ≤ , be distinct elements in G , O a conjugacyclass of G . Assume that ( σ, τ ) ⊆ O is of type O (2) , with g ∈ G such that g ⊲ σ = τ . Let (4.7) g := σ , g := σ , g := σ , g := σ ,g := σ , g := σ σ , g := gσ , g := τ g,g := τ g, g := τ g, g := τ g, g := τ gσ . Then, the following relations hold: τ g = g σ , τ g = g σ , τ g = g σ ,τ g = g σ , τ g = g σ , τ g = g σ − ,τ g = g σ , τ g = g g − τ g, τ g = g σ ,τ g = g σ , τ g = g g − τ g, τ g = g g − τ g,τ g = g σ , τ g = g σ − g − τ g, τ g = g g − τ g,τ g = g g − τ g, τ g = g g − τ g, τ g = g g − τ g,τ g = g σ , τ g = g σ , τ g = g g − τ g,τ g = g g − τ g, τ g = g g − τ g, τ g = g ( g − τ g ) ,τ g = g ( g − τ g ) − , τ g = g g − τ g, τ g = g σ ,τ g = g σ , τ g = g σ ( g − τ g ) − , τ g = g σ g − τ g,τ g = g g − τ g, τ g = g σ , τ g = g σ ( g − τ g ) ,τ g = g g − τ g, τ g = g g − τ g, τ g = g σ ,σ g = g g − σ g, σ g = g g − σ g, σ g = g g − σ g,σ g = g g − σ g, σ g = g g − σ g, σ g = g σ − ( g − σ g ) ,σ g = g g − σ g, σ g = g g − σ g, σ g = g g − σ g,σ g = g g − σ g, σ g = g g − σ g, σ g = g g − σ g,σ g = g g − σ g, σ g = g σ − g − σ g, σ g = g g − σ g,σ g = g g − σ g, σ g = g g − σ g, σ g = g g − σ g,σ g = g g − σ g, σ g = g g − σ g, σ g = g g − σ g,σ g = g g − σ g, σ g = g g − σ g, σ g = g γ , ,σ g = g γ , , σ g = g g − σ g, σ g = g σ ( g − σ g ) ,σ g = g g − σ g, σ g = g γ , , σ g = g σ g − σ g,σ g = g g − σ g, σ g = g g − σ g, σ g = g γ , ,σ g = g g − σ g, σ g = g g − σ g, σ g = g g − σ g, where γ , = σ ( g − σ g ) − ( g − σ g ) , γ , = σ − ( g − σ g ) ( g − σ g ) − , γ , = σ − ( g − σ g ) ( g − σ g ) − and γ , = σ ( g − σ g ) − ( g − σ g ) , OINTED HOPF ALGEBRAS 25 τ g = g τ , τ g = g τ , τ g = g τ ,τ g = g τ , τ g = g τ , τ g = g σ − τ ,τ g = g τ , τ g = g τ , τ g = g τ ,τ g = g τ , τ g = g τ , τ g = g τ ,τ g = g τ , τ g = g σ − τ , τ g = g τ ,τ g = g τ , τ g = g τ , τ g = g τ ,τ g = g τ , τ g = g τ , τ g = g τ ,τ g = g τ , τ g = g τ , τ g = g σ τ σ ,τ g = g σ − τ σ − , τ g = g τ , τ g = g σ τ ,τ g = g τ , τ g = g τ σ − , τ g = g σ τ σ ,τ g = g τ , τ g = g τ , τ g = g τ σ ,τ g = g τ , τ g = g τ , τ g = g τ . Proof.
The proof follows by straightforward computations, Lemma 4.6 for σ and τ , and Lemma 4.8. (cid:3) Here is the main result of this section.
Theorem 4.11.
Let σ i , τ i ∈ G , ≤ i ≤ , distinct elements in G , O aconjugacy class of G and ρ = ( ρ, V ) ∈ d G σ . Let us suppose that (H1) ( σ, τ ) is of type O (2) , (H2) ( σ, τ ) ⊆ O , with g ∈ G such that g ⊲ σ = τ , (H3) q σ σ = − ,there exists v ∈ V − such that (H4) ρ ( σ ) v = − v , (H5) ρ ( τ ) v = − v ,and there exists w ∈ V − such that (H6) ρ ( g − σ g ) w = − w , (H7) ρ ( g − σ g ) w = − w ,Then dim B ( O , ρ ) = ∞ .Proof. Let g j ∈ G , 1 ≤ j ≤
12, as in (4.7). Then, g j ⊲ σ = σ j , 1 ≤ j ≤ g j ⊲ σ = τ j − , 7 ≤ j ≤
12. By Lemma 4.10, we have that(a) if 1 ≤ i, j ≤
6, then g − i⊲j σ i g j = σ r σ s , with r + s odd,(b) if 7 ≤ i, j ≤
12, then g − i⊲j τ i − g j = σ r ( g − τ g ) s , with r + s odd, (c) if 1 ≤ i ≤ ≤ j ≤
12, then g − i⊲j σ i g j = σ r ( g − σ g ) s ( g − σ g ) t ,with r + s + t odd,(d) if 1 ≤ j ≤ ≤ i ≤
12, then g − i⊲j τ i − g j = σ r τ s σ t , with r + s + t odd, because τ = σ − τ σ .Let W := span- { g i v, | ≤ i ≤ } and W ′ := span- { g i w, | ≤ i ≤ } ,with v , w ∈ V −
0, where v satisfies (H4)-(H5) and w satisfies (H6)-(H7).Then, W and W ′ are braided vector subspaces of M ( O , ρ ). We will provethat W ⊕ W ′ ≃ M ( O , χ − ) ⊕ M ( O , χ − ) , as braided vector spaces. Hence dim B ( W ⊕ W ′ ) = ∞ , by Theorem 4.3, andthe result follows from Lemma 1.2.By Remark 4.4, we only need to see that the isomorphism of linear vectorspaces W ⊕ W ′ → M ( O , χ − ) ⊕ M ( O , χ − ) given by g i v e g i and g i +6 w e g i +6 ≤ i ≤ , respects the braiding, and this is just a matter of the cocycle. For this, wecompute explicitly the braiding in the basis { g i v, g j +6 w, | ≤ i, j ≤ } of W ⊕ W ′ .By (a), (H3) and (H4), if 1 ≤ i, j ≤
6, then c ( g i v ⊗ g j v ) = g i⊲j ρ ( g − i⊲j σ i g j )( v ) ⊗ g i v = − g i⊲j v ⊗ g i v. From Lemma 4.8 (i), τ = σ − τ σ . Thus, g − τ g = ( g − σ g ) − σ ( g − σ g ).By (b), (H3), (H6) and (H7), if 7 ≤ i, j ≤
12, then c ( g i w ⊗ g j w ) = g i⊲j ρ ( g − i⊲j τ i − g j )( w ) ⊗ g i w = − g i⊲j w ⊗ g i w. By (c), (H3), (H6) and (H7), if 1 ≤ i ≤ ≤ j ≤
12, then c ( g i v ⊗ g j w ) = g i⊲j ρ ( g − i⊲j σ i g j )( w ) ⊗ g i v = − g i⊲j w ⊗ g i v. By (d), (H3), (H4) and (H5), if 1 ≤ j ≤ ≤ i ≤
12, then c ( g i w ⊗ g j v ) = g i⊲j ρ ( g − i⊲j τ i − g j )( v ) ⊗ g i w = − g i⊲j v ⊗ g i w. This completes the proof. (cid:3)
As an immediate consequence we have the following result.
Corollary 4.12.
Let σ i , τ i ∈ G , ≤ i ≤ all distinct, O a conjugacy classof G and ρ = ( ρ, V ) ∈ d G σ with q σ σ = − . Assume that ( σ, τ ) ⊆ O is oftype O (2) . If σ = σ d and τ = σ e for some d , e ∈ Z , then dim B ( O , ρ ) = ∞ . OINTED HOPF ALGEBRAS 27
Proof.
Note that d and e are odd, since they are relatively prime with | σ | .Hence the hypothesis (H4) and (H5) hold. Now g − σ g = σ e | g |− . Then ρ ( g − σ g ) = − id and (H6) holds. The proof of (H7) is similar. (cid:3) Example 4.13.
Let m ≥ . Let σ ∈ S m of type (1 n , n , n ) , with n ≥ , O the conjugacy class of σ and ρ ∈ c S σm . Then dim B ( O , ρ ) = ∞ .Proof. By Lemma 1.3, we may suppose that q σσ = −
1. If n ≥
3, thendim B ( O , ρ ) = ∞ , from Corollary 3.10. We consider two cases. CASE (I): n = 1. Let A = ( i i i i i i i i ) the 8-cycle appearingin the decomposition of σ as product of disjoint cycles. We set α := σ A − and define σ := σ , σ := σ , τ := σ , τ := σ − , σ := ( i i i i i i i i ) α, σ := ( i i i i i i i i ) α,σ := ( i i i i i i i i ) α, σ := ( i i i i i i i i ) α,τ := ( i i i i i i i i ) α, τ := ( i i i i i i i i ) α,τ := ( i i i i i i i i ) α, τ := ( i i i i i i i i ) α. CASE (II): n = 2. Let A , = ( i i i i i i i i ) and A , = ( i i i i i i i i )the two 8-cycles appearing in the decomposition of σ as product of disjointcycles. We call A = A , A , , α := σ A − and define σ := σ , σ := σ , τ := σ , τ := σ − , σ := ( i i i i i i i i )( i i i i i i i i ) α,σ := ( i i i i i i i i )( i i i i i i i i ) α,σ := ( i i i i i i i i )( i i i i i i i i ) α,σ := ( i i i i i i i i )( i i i i i i i i ) α,τ := ( i i i i i i i i )( i i i i i i i i ) α,τ := ( i i i i i i i i )( i i i i i i i i ) α,τ := ( i i i i i i i i )( i i i i i i i i ) α,τ := ( i i i i i i i i )( i i i i i i i i ) α. In both cases, σ = σ and τ = σ and ( σ, τ ) ⊆ O is of type O (2) . Thenthe result follows from Corollary 4.12. (cid:3) Remarks . (i). The discussion in the preceding example can be adaptedto σ ∈ S m of type (1 n , n , . . . , m n m ) provided that n ≥
1; but then somerequirements on the representation ρ have to be imposed. (ii). Let N = 2 n with n ≥
4. It can be shown that the orbit of the N -cycle in S N contains no family of type O using Lemma 4.6.(iii). The orbit with label j = 4 of the Mathieu group M contains afamily of type O (2) , and therefore this group admits no finite-dimensionalpointed Hopf algebra except the group algebra itself [F1]. Acknowledgement.
The authors are grateful to the referee for carefullyreading the paper and for his/her comments.
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Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional deC´ordoba. CIEM – CONICET.Medina Allende s/n (5000) Ciudad Universitaria, C´ordoba, Argentina
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