On pointed Hopf algebras associated with alternating and dihedral groups

Andruskiewitsch, Nicolás; Fantino, Fernando

Servicios Personalizados

Revista

Articulo

Indicadores

Citado por SciELO

Links relacionados

Similares en SciELO

Otros
Otros

Permalink

Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.3 Bahía Blanca 2007

On pointed Hopf algebras associated with alternating and dihedral groups

Nicolás Andruskiewitsch and Fernando Fantino

Abstract. We classify finite-dimensional complex pointed Hopf algebras with group of group-like elements isomorphic to . We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of π ∈ 𝔸n is infinite-dimensional if the order of is odd except for π = (12 3) in . We also study pointed Hopf algebras over the dihedral groups.

2000 Mathematics Subject Classification. 16W30; 17B37

This work was partially supported by Agencia Córdoba Ciencia, ANPCyT-Foncyt, CONICET and Secyt (UNC)

Dedicado a María Inés Platzeck en sus # años

Introduction

In this article, we continue the work of [AZ, AF] on the classification of finite-dimensional complex pointed Hopf algebras with G = G (H ) non-abelian. We follow the Lifting Method - see [AS2] for a general reference; in particular, we focus on the problem of determining when the dimension of the Nichols algebra associated with conjugacy classes of is infinite. The paper is organized as follows. In Section 1, we review some general facts on Nichols algebras corresponding to finite groups. We discuss the notion of absolutely real element of a finite group in subsection 1.2. We then provide generalizations of [AZ, Lemma 1.3], a basic tool in [AZ, AF], see Lemmata 1.8 and 1.9. Section 2 is devoted to pointed Hopf algebras with coradical ℂ 𝔸 n . We prove that any finite-dimensional complex pointed Hopf algebra with G (H ) ≃ 𝔸5 is isomorphic to the group algebra of ; see Theorem 2.6. This is the first finite non-abelian group such that all pointed Hopf algebras with G (H ) = G are known. We also prove that dim 𝔅 (Oπ, ρ) = ∞ , for any in of odd order, except for π = (1 23 ) or π = (13 2) in 𝔸 4 - see Theorem 2.3. This last case is particularly interesting. It corresponds to a "tetrahedron" rack with constant cocycle ω ∈ 𝔾3 - 1 . The technique in the present paper does not provide information on the corresponding Nichols algebra. We also give partial results on pointed Hopf algebras with groups and , and on Nichols algebras 𝔅 (O π,ρ) , with |π| even. In Section 3, we apply the technique to conjugacy classes in dihedral groups. It turns out that it is possible to decide when the associated Nichols algebra is finite-dimensional in all cases except for M (Ox,sgn ) (if is odd), or M (Ox, sgn ⊗ sgn) or M (Ox, sgn ⊗ɛ) or M (Oxy,sgn ⊗ sgn) or M (Oxy, sgn ⊗ɛ) (if is even). See below for undefined notations. We finally observe in Section 4 that there is no finite-dimensional Hopf algebra with coradical isomorphic to the Hopf algebra J (ℂ 𝔸5 ) discovered in [Ni], except for J (ℂ𝔸5 ) itself.

1. Generalities

For s ∈ G we denote by s G the centralizer of in . If is a subgroup of and s ∈ H we will denote H O s the conjugacy class of in . Sometimes we will write in rack notations x ⊳ y = xyx -1 , , y ∈ G . Also, if (V, c) is a braided vector space, that is c ∈ GL (V ⊗ V ) is a solution of the braid equation, then 𝔅 (V ) denotes its Nichols algebra.

We denote by the group of -th roots of 1 in .

1.1. Preliminaries. Let be a finite group, a conjugacy class of , s ∈ O fixed, an irreducible representation of , M (O,ρ ) the corresponding irreducible Yetter-Drinfeld module. Let t = s 1 , …, t M be a numeration of and let gi ∈ G such that gi ⊳ s = ti for all 1 ≤ i ≤ M . Then M (O,ρ ) = ⊕1 ≤i≤M gi ⊗ V . Let giv := gi ⊗ v ∈ M (O, ρ) , , v ∈ V . If v ∈ V and , then the coaction and the action of g ∈ G are given by

δ(giv) = ti ⊗ giv, g ⋅ (giv) = gj(γ ⋅ v),

where ggi = gjγ , for some 1 ≤ j ≤ M and γ ∈ Gs . The Yetter-Drinfeld module M (O, ρ ) is a braided vector space with braiding given by

c(giv ⊗ gjw) = ti ⋅ (gjw ) ⊗ giv = gh(γ ⋅ v) ⊗ giv

(1)

for any 1 ≤ i,j ≤ M , v,w ∈ V , where tigj = gh γ for unique , 1 ≤ h ≤ M and γ ∈ Gs . Since s ∈ Z (Gs) , the Schur Lemma says that

(2)

Let be a finite non-abelian group. Let be a conjugacy class of and let be an irreducible representation of the centralizer s G of a fixed s ∈ O . Let M (O, ρ) be the irreducible Yetter-Drinfeld module corresponding to (O, ρ) and let 𝔅 (O, ρ) be its Nichols algebra. As explained in [AZ, AF, Gñ], we look for a braided subspace of M (O,ρ ) of diagonal type such that the dimension of the Nichols algebra 𝔅 (U ) is infinite. This implies that the dimension of is infinite too.

Lemma 1.1. If is a subspace of such that c(W ⊗ W ) = W ⊗ W and dim 𝔅 (W ) = ∞ , then dim 𝔅 (V) = ∞ .□

Recall that a braided vector space (V,c) is of diagonal type if there exists a basis v1,...,vθ of and non-zero scalars qij , 1 ≤ i,j ≤ θ , such that c(v ⊗ v ) = q v ⊗ v i j ij j i , for all . A braided vector space (V, c) is of Cartan type if it is of diagonal type and there exists aij ∈ ℤ , - |qii| < aij ≤ 0 such that aij qijqji = qii for all 1 ≤ i ⁄= j ≤ θ ; by |qii| we mean if qii is not a root of 1, otherwise it means the order of qii in the multiplicative group of the units in . Set aii = 2 for all 1 ≤ i ≤ θ . Then (aij)1≤i,j≤ θ is a generalized Cartan matrix.

Theorem 1.2. ([H, Th. 4], see also [AS1, Th. 1.1]). Let (V, c) be a braided vector space of Cartan type. Then dim 𝔅 (V ) < ∞ if and only if the Cartan matrix is of finite type. □

We say that s ∈ G is real if it is conjugate to s-1 ; if is real, then the conjugacy class of is also said to be real. We say that is real if any s ∈ G is real.

The next application of Theorem 1.2 was given in [AZ]. Let be a finite group, s ∈ G , the conjugacy class of , s ρ : G → GL (V ) irreducible; × qss ∈ ℂ was defined in (2).

Lemma 1.3. Assume that is real. If dim 𝔅 (O,ρ ) < ∞ then q = - 1 ss and has even order.□

If -1 s ⁄= s , this is [AZ, Lemma 2.2]; if 2 s = id then qss = ±1 but qss = 1 is excluded by Lemma 1.1.

The class of real groups includes finite Coxeter groups. Indeed, all the characters of a finite Coxeter group are real valued, see subsection 1.2 below, and [BG] for H 4 . Therefore, we have:

Theorem 1.4. Let be a finite Coxeter group. If s ∈ G has odd order, then dim 𝔅 (Os, ρ) = ∞ , for any ρ ∈ ^Gs . □

1.2. Absolutely real groups. Let be a finite group. We say that s ∈ G is absolutely real if there exists an involution in such that σsσ = s-1 . If this happens, any element in the conjugacy class of is absolutely real and we will say that the conjugacy class of is absolutely real. We say that is absolutely real if any s ∈ G is so. The finite Coxeter groups are absolutely real. Indeed,

the dihedral groups are absolutely real, by straightforward computations.
the Weyl groups of semisimple finite dimensional Lie algebras are absolutely real, by [C, Th. C (iii), p. 45].
is absolutely real, by Proposition 1.7 below.
is absolutely real, we have checked it using GAP3, [S].

Remark 1.5. Let , be finite groups. We note:

is absolutely real iff both and are absolutely real.
is absolutely real iff both and are absolutely real.
Assume abelian. Then is absolutely real iff has exponent 2, i. e. for some integer .
If is absolutely real and is abelian of exponent 2 then is absolutely real.

We first discuss when an element of is absolutely real. Assume that π ∈ 𝕊n is of type (1m1,2m2, ...,nmn) . Then π ∈ 𝔸n iff ∑ mj j even is even.

Lemma 1.6. (a). If m1 ≥ 2 , then is absolutely real in .

(b). If ∑ (m4h + m4h+3 ) h∈ℕ is even then is absolutely real in .

Proof. Let τj := (1 2 ...j) for some and take

{ (1 j - 1)(2 j - 2 )⋅⋅⋅(k - 1 k + 1), if j = 2k is even, gj = (1 j - 1)(2 j - 2 )⋅⋅⋅(k k + 1), if j = 2k + 1 is odd.

It is easy to see that gjτjgj = τ -1 j

and

{ k-1 sgn(g ) = (- 1) , if j = 2k is even, j (- 1)k, if j = 2k + 1 is odd.

To prove (b), observe that there exists an involution σ ∈ 𝕊n such that σπ σ = π- 1 , which is a product of "translations" of the g j 's. Since the sign of is (- 1)∑h∈ℕ(m4h+m4h+3) , σ ∈ 𝔸 n iff ∑ h∈ℕ(m4h + m4h+3 ) is even; (b) follows. We prove (a). By assumption there are at least two points fixed by , say n - 1 , . By the preceding there exists an involution σ ∈ 𝕊n-2 such that σπ σ = π -1 . If σ ∈ 𝔸n-2 ⊂ 𝔸n we are done, otherwise take ^σ = σ(n - 1 n) ∈ 𝔸n ; is an involution and ^σπ ^σ = π- 1 . □

Proposition 1.7. The groups 𝔸 5 and H 3 are absolutely real.

Proof. The type of π ∈ 𝔸 5 is either (15) , (12,31) , (1,22) or (51) ; in the first two cases is absolutely real by Lemma 1.6 part (a), in the last two by part (b). Since H3 ≃ 𝔸5 × ℤ2 (see [Hu, Section 2.13]), then the Coxeter group is absolutely real by Remark 1.5. □

1.3. Generalizations of Lemma 1.3. The next two Lemmata are variations of [AZ, Lemma 2.2]. A result in the same spirit appears in [FGV]. We deal with elements having a power in , the conjugacy class of . Clearly, if sj = σs σ-1 is in , then sjl = σlsσ-l is in , for every . So, sj|σ| = s ; this implies that |s| divides |σ| j - 1 . Hence

pict

with N := |qss| , recall (2).

Lemma 1.8. Let be a finite group, s ∈ G , the conjugacy class of and ρ ∈ ^Gs . Assume that there exists an integer such that , and sj2 are distinct elements and is in . If dim 𝔅 (O, ρ) < ∞ , then has even order and qss = - 1 .

Proof. We assume that dim 𝔅 (O, ρ) < ∞ , thus N > 1 . It is easy to see that

pict

for every , . We will call l tl := sj , gl := σl , l = 0 , , ; so tl = glsgl-1 , for l = 0 , , . The other relations between 's and 's are obtained from (4). For v ∈ V - 0 and l = 1 or , we define Wl := ℂ - span of {g0v,glv} . Hence, is a braided vector subspace of M (O,ρ ) of Cartan type with

( j|σ|-l) ( ) Ql = qssl qss , Al = 2 a12(l) , qjss qss a21(l) 2

where

a (l) = a (l) ≡ j |σ|- l + jl mod (N ) 12 21

. Since

, we have that a12(l) = a21(l) = 0

. We consider now two cases.
(i) Let us suppose that a12(1) = a21(1) = 0

. This implies that j|σ|-1 + j ≡ 0 mod (N )

. Since

divides

, we have that

divides

. We consider now two possibilities.

Assume that . Then . Since divides , we have that divides . So, ; hence the result follows.
Assume that . Then . We can see that divides . This implies that divides , a contradiction.

(ii) Let us suppose that a12(1) = a21(1) = - 1 . This implies that j|σ|-1 + j ≡ - 1 mod (N ) . Since divides j|σ| - 1 , we have that divides j2 + j + 1 . We consider now two possibilities.

Assume that . Then . So, divides . It is easy to see that divides . Since and are relatively prime, must be , a contradiction.
Assume that . This means that the subspace $^ W := ℂ - span of {g0v, g1v,g2v}$ of is of Cartan type with
By Theorem 1.2, we have that , a contradiction.

This concludes the proof. □

Lemma 1.9. Let be a finite group, s ∈ G , the conjugacy class of and ^ s ρ = (ρ,V ) ∈ G such that dim 𝔅 (O, ρ) < ∞ . Assume that there exists an integer such that j s ⁄= s and j s is in .

If , then has even order and .
If , then either has even order and , or .

Proof. We will proceed and use the notation as in the proof of Lemma 1.8. If j2 s ⁄= s , then the result follows by Lemma 1.8. Assume that 2 sj = s . This implies that |s| divides j2 - 1 , so divides j2 - 1 .

(a) Let and in linearly independent and let W = ℂ - span of {g0v1, g0v2, g1v1, g1v2} , with g0 := id and g1 := σ . Thus is a braided vector subspace of M (O, ρ) of Cartan type with

( |σ|-1 |σ|-1) ( ) qss qss qjss qjss 2 a12 a13 a14 || qss qss qj|σ|-1 qj|σ|-1|| || a21 2 a23 a24|| Q = ( qj qj ssq ssq ) , A = ( a31 a32 2 a34) , sjs ssj ss ss a a a 2 qss qss qss qss 41 42 43

where

and

a ≡ j|σ|-1 + j mod (N ). 13

, then

divides

and the result follows. Besides, if a13 = 0

, then

; this implies that

divides

, so

divides 2 and the result follows. On the other hand, if aij = - 1

, for all

, we have that the matrix

is not of finite type; hence dim 𝔅 (O, ρ) = ∞

, from Theorem 1.2. This is a contradiction by hypothesis. Therefore, (a) is proved.

(b) For v ∈ V - 0 we define W := ℂ - span of {g0v,g1v} , with g0 := id and g1 := σ . Hence, is a braided vector subspace of M (O,ρ ) of Cartan type with

( |σ|-1) ( ) qss qjss 2 a12 Q = qjss qss , A = a21 2 ,

where

. Since

, we have that a12 = 0

. We consider now two possibilities.
(i) Assume that a12 = 0

. This implies that j|σ|-1 + j ≡ 0 mod (N )

. Since

divides

, we have that

divides

. Thus,

divides

; hence

, and the result follows.
(ii) Assume that a12 = - 1

. This implies that |σ|-1 j + j ≡ - 1 mod (N )

. Since

divides

, we have that

divides

. Hence,

divides

. If

is a prime divisor of

, then

divides

, because

divides

. If

divides

, then

divides

, a contradiction. So,

divides

. Hence,

divides

, i.e.

and the result follows. □

2. On Nichols algebras over

We recall that we will denote or O π the conjugacy class of an element in , and in ^π 𝔸 n , a representative of an isomorphism class of irreducible representations of . We want to determine pairs (O, ρ ) , for which dim 𝔅 (O, ρ) = ∞ , following the strategy given in [AZ, AF]; see also [Gñ].

The following is a helpful criterion to decide when a conjugacy class of an even permutation in splits in .

Proposition 2.1. [JL, Proposition 12.17] Let π ∈ 𝔸n , with n > 1 .

If commutes with some odd permutation in , then and .
If does not commute with any odd permutation in , then splits into two conjugacy classes in of equal size, with representatives and , and .□

Remarks 2.2. (i) Notice that if satisfies (1) of Proposition 2.1, then is real. The reciprocal is not true, e.g. consider τ5 = (12 34 5) in .

(ii) One can see that if in is of type (1m1, 2m2,...,nmn ) , then satisfies (2) of Proposition 2.1 if and only if m1 = 0 or , m = 0 2h and m ≤ 1 2h+1 , for all h ≥ 1 . Thus, if π ∈ 𝔸 n has even order, then is real.

We state the main Theorem of the section.

Theorem 2.3. Let π ∈ 𝔸n and ρ ∈ ^𝔸πn . Assume that is neither (1 23) nor (1 32) in . If dim 𝔅 (O π,ρ) < ∞ , then has even order and qππ = - 1 .

Proof. If |π| is even the result follows by Lemma 1.3 and Remark 2.2 (ii). Let us suppose that |π| ≥ 5 and odd . If -1 π is in , then the result follows by Lemma 1.3. Assume that π -1 ⁄∈ O π . We consider two cases.

(i) If π2 ∈ O π , then is in O π , and π4 ⁄= π2 because |π| ≥ 5 . Hence, the result follows from Lemma 1.8.

(ii) Assume that 2 π ⁄∈ O π . We know that there exist and ′ σ in , necessarily odd permutations, such that - 1 -1 π = σ πσ and π2 = σ′π σ′-1 . Then σ′′ = σ σ′ ∈ 𝔸n and π- 2 = σ′′πσ ′′-1 ; so π-2 is in . This implies that is in , and π4 ⁄= π -2 because 5 ≤ |π| is odd. Now, the result follows from Lemma 1.8.

Finally, let us suppose that |π| = 3 , with type (1a,3b) . If a ≥ 2 or b ≥ 2 , then is real, by Lemma 1.6 (a) and Remark 2.2, respectively. Hence, the result follows by Lemma 1.3. This concludes the proof. □

2.1. Case . Obviously, 𝔸3 ≃ ℤ3 ; thus is not real. This case was considered in [AS1, Theorem 1.3].

2.2. Case . It is straightforward to check that is not real, since (12 3) is not real in 𝔸 4 . Let in 𝔸 4 ; then the type of may be (14) , (22) or (1,3) . If the type of is 4 (1 ) , then dim 𝔅 (O π,ρ) = ∞ , for any in ^𝔸4 , by Lemma 1.1. If the type of is (1,3) , then is not real; moreover we have

O (1 2 3) = {(1 2 3),(1 3 4),(1 4 2),(2 4 3 )},

O (1 3 2) = {(1 3 2),(1 2 4),(1 4 3),(2 3 4 )},

and

. If

is trivial, then dim 𝔅 (O π,ρ) = ∞

; otherwise it is not known.

The following result is a variation of [AZ, Theorem 2.7].

Proposition 2.4. Let in of type 2 (2 ) . Then dim 𝔅 (O π,ρ) = ∞ , for every in ^𝔸 π4 .

Proof. We can assume that π = (1 2)(3 4) . If we call t1 := π , t := (1 3)(24) 2 and t := (1 4)(23) 3 , then O = {t ,t ,t} π 1 2 3 and π 𝔸 4 = ⟨t1⟩ × ⟨t2⟩ ≃ ℤ2 × ℤ2 . If g1 = id , g2 = (1 32) and g3 = (1 23 ) , then - 1 tj = gjπgj , j = 1,2,3 , and pict

Let in ^𝔸π4 and M (O π,ρ) := g1v ⊕ g2v ⊕ g3v , where ⟨v⟩ is the vector space affording . Thus is a braided vector space with braiding given by - see (1)- c(gjv ⊗ gjv) = gjt1 ⋅ v ⊗ gjv and c(g v ⊗ g v) = g t ⋅ v ⊗ g v j 1 1 j j , j = 1,2,3 and pict

Clearly, dim 𝔅 (O π,ɛ ⊗ ɛ) = dim 𝔅 (O π,ɛ ⊗ sgn ) = ∞ , by Lemma 1.1. If we consider ρ = sgn ⊗ ɛ (resp. sgn⊗ sgn ), then M (O π,ρ) is of Cartan type with matrix of coefficients (qij)ij given by

( ) ( ) - 1 - 1 1 - 1 1 - 1 Q = ( 1 - 1 - 1) , ( resp. Q = ( - 1 - 1 1 ) ). - 1 1 - 1 1 - 1 - 1

In both cases the Cartan matrix is ( ) 2 - 1 - 1 A = ( - 1 2 - 1 ) - 1 - 1 2

. Therefore, dim 𝔅 (Oπ, ρ) = ∞

, by Theorem 1.2. □

2.3. Case . Here is the key step in the consideration of this case.

Lemma 2.5. Let π ∈ 𝔸5 . Then dim 𝔅 (O π,ρ) = ∞ , for every in ^𝔸 π 5 .

Proof. Let π ∈ 𝔸5 . If the type of is either 5 (1 ) , 2 (1 ,3) or (5) , we have that dim 𝔅 (O π,ρ) = ∞ , by Lemma 1.3 and Proposition 1.7. Let us assume that the type of is (22) . For j = 1,2,3 , let and be as in the proof of Proposition 2.4. By Proposition 2.1 and straightforward computations, we have that O 𝔸5 = O 𝕊5 π π and π 𝔸 5 = ⟨t1⟩ × ⟨t2⟩ ≃ ℤ2 × ℤ2 . Notice that 𝔸5 tj ∈ O π , . Let ρ ∈ ^𝔸 π5 and W := g1v ⊕ g2v ⊕ g3v , where ⟨v⟩ is the vector space affording ; then is a braided vector subspace of M (O π,ρ) . Therefore, dim 𝔅 (O π,ρ) = ∞ , by the same argument as in the proof of Proposition 2.4. □

As an immediate consequence of Lemma 2.5 we have the following result.

Theorem 2.6. Any finite-dimensional complex pointed Hopf algebra with G (H ) ≃ 𝔸5 is necessarily isomorphic to the group algebra of .

Proof. Let be a complex pointed Hopf algebra with G(H ) ≃ 𝔸5 . Let ℂ𝔸5 M ∈ ℂ𝔸5YD be the infinitesimal braiding of -see [AS2]. Assume that H ⁄= ℂ 𝔸5 ; thus M ⁄= 0 . Let N ⊂ M be an irreducible submodule. Then dim 𝔅 (N ) = ∞ , by Lemma 2.5. Hence, dim 𝔅 (M ) = ∞ and dim H = ∞ . □

2.4. Case . Let be in . If the type of is (16) , (12,22) , (13,3 ) , (32) or (1,5) , then is absolutely real by Lemma 1.6, and if the type of is (2,4) , then is real because it has even order - see Remark 2.2 (ii). Hence, is a real group. We summarize our results in the following statement.

Theorem 2.7. Let M (O, ρ) be an irreducible Yetter-Drinfeld module over ℂ 𝔸6 , corresponding to a pair (O, ρ) . If dim 𝔅 (O, ρ) < ∞ , then O = O π , with π = (1 2)(34 56) , and ρ = sgn ∈ ^ℤ4 .

Remark 2.8. In this Theorem we do not claim that the condition is sufficient.

Proof. Let be in 𝔸 6 . If the type of is

, then , for any in , by Lemma 1.1.
, or , then , for any in , by Lemma 1.3.

Let us suppose that the type of is 2 2 (1 ,2 ) ; we can assume that π = (1 2)(34) . It is easy to check that

π 𝔸6 = ⟨a := (34 )(5 6), b := (13 24)(5 6)⟩ ≃ 𝔻4.

Notice that π = b2 . It is known that $^𝔻4 = {ρ1, ρ2, ρ3, ρ4, ρ5}$ , where , j = 1 , , and , are the following characters

pict

and is the -dimensional representation given by

pict

It is clear that ρj(π) = 1 , j = 1 , , and . Then dim 𝔅 (Oπ, ρ) = ∞ , by Lemma 1.1. Let us consider now that ρ = ρ5 . We define t1 := (12 )(3 4) , t2 := (13 )(2 4) , t3 := (14)(2 3) , g1 := id , g2 := (13 2) and g3 := (12 3) . It is clear that

pict

If ( ) i v1 := 1 we have that ρ(t1)(v1) = - v1 and ρ(t2)(v1) = v1 = - ρ(t3)(v1) . We define W := ℂ - span of {g v ,g v ,g v } 1 1 2 1 3 1 . Then is a braiding subspace of M (O π,ρ) of Cartan type with

pict

Since is not of finite type we have that dim 𝔅 (Oπ, ρ) = ∞ , by Theorem 1.2.

Finally, let us assume that the type of is (2,4) . Then O π has elements and 𝔸π = ⟨π ⟩ ≃ ℤ 6 4 . We call $^ℤ = { χ ,χ ,χ ,χ } 4 0 1 2 3$ , where l χl(π) = i , l = 0 , , , . It is clear that if ρ = χl , with l = 0 , or , then ρ(π) ⁄= - 1 . This implies that dim 𝔅 (O π,ρ) = ∞ , by Lemma 1.3. □

Remark 2.9. We can see that every maximal abelian subrack of O (12)(3456) has two elements. Hence, M (O (12)(3456),ρ) is a negative braided space in the sense of [AF].

2.5. Case , m ≥ 7 . Let π ∈ 𝔸m , with |π | even. We now investigate the Nichols algebras associated with by reduction to the analogous study for the orbit of in 𝕊 m , [AF]. By Remark 2.2 (ii), O 𝔸m = O 𝕊m π π and [𝕊π : 𝔸 π] = 2 m m . So, we can determinate the irreducible representations of π 𝔸m from those of π 𝕊 m . We know that if the type of is b1 b2 bm (1 ,2 ,...,m ) , then 𝕊 πm = T1 ⋅⋅⋅Tm, with b Ti ≃ ℤii⋊ 𝕊bi , 1 ≤ i ≤ m .

Some generalities and notation. Let be a finite group, a subgroup of of index two, and a representation of . It is easy to see that

${ ′ η(g) ,if g ∈ H, η (g) := - η(g) ,if g ∈ G \ H,$

defines a new representation of

. Notice that ResGH η = ResGH η′

. On the other hand, any representation

defines a representation

, call it the conjugate representation of

, given by ρ(h ) := ρ(ghg -1)

, for every h ∈ H

, where

is an arbitrary fixed element in $G \ H$ . Since

is unique up to multiplication by an element of

, the conjugate representation is unique up to isomorphism.

Let s ∈ H such that OHs = OGs ; thus [Gs : Hs ] = 2 . Let in G^s . Then we have two cases:

. If , then , and .
. We have that and .

Moreover, if is an irreducible representation of , then is a restriction of some ^ s η ∈ G or is a direct summand of Gs Res Hs η as in (ii), see [FH, Ch. 5].

Remark 2.10. If η ∈ ^Gs and ρ := ResGss η H , it is easy to check that

pict

as braided vector spaces.

We apply these observations to the case G = 𝕊m and H = 𝔸m . We use some notations given in [AF, Section II.D].

Lemma 2.11. Assume that the type of is ((2r)n) , with r ≥ 1 and even. Let in ^π 𝔸 m , with m = 2rn .

If , then .
If and , then
- if , then .
- Assume that . If , then . Assume that . If , with even or odd, or if , with even and or , where or , then the braiding is negative; otherwise, .

Proof. (a) follows by Remark 2.2 (ii) and Lemma 1.3. (b). Since ρ ≃ ρ- , 𝕊πm ρ = Res 𝔸πm(η ) , with η ∈ ^𝕊πm , ′ η ⁄≃ η and 𝕊πm ′ Ind 𝔸πm ρ ≃ η ⊕ η . Notice that η (π ) = - Id because ρ(π) = - Id . Now, as the racks are the same, i.e. O 𝔸πm = O𝕊πm , we can apply [AF, Theorem 1]. □

Remark 2.12. Keep the notation of the Lemma. If is not isomorphic to its conjugate representation , then there exists η ∈ ^𝕊π m such that 𝕊πm -- Res 𝔸πm(η) = ρ ⊕ ρ and 𝕊πm ′ 𝕊πm -- Ind𝔸πm ρ ≃ η ≃ η ≃ Ind 𝔸πm ρ . Clearly, η(π) and -- ρ(π) act by scalar , and we have that M (O𝕊πm ,η) ≃ M (O 𝔸πm ,ρ) ⊕ M (O𝔸πm ,ρ) as braided vector spaces. We do not get new information with the techniques available today.

3. On Nichols algebras over

We fix the notation: the dihedral group of order is generated by and with defining relations x2 = e = yn and xyx = y-1 . Let be a primitive -th root of 1 and let be the character of ⟨y⟩ , χ (y) = ω . If s ∈ 𝔻 n then we denote the conjugacy class by On s or simply .

Theorem 3.1. Let M (O, ρ) be the irreducible Yetter-Drinfeld module over ℂ 𝔻n corresponding to a pair (O, ρ) . Assume that its Nichols algebra 𝔅 (O, ρ) is finite-dimensional.

If is odd, then , where , .
If is even, then is one of the following:
1. where satisfies .
2. where , and .
3. or , where , , .
4. or , where , , .

In the cases (i) and (ii) the dimension is finite. In the cases (iii) and (iv), the braiding is negative in the sense of [AF].

Remark 3.2. There are isomorphisms of braided vector spaces

pict

Remark 3.3. Assume for simplicity that is odd and that n = de , where , are integers ≥ 2 . Then the (indecomposable) rack On x is a disjoint union of racks isomorphic to Od x ; in other words, On x is an extension of Oe x by Od x (and vice versa), see [AG, Section 2]. Thus, there is an epimorphism of braided vector spaces n e M (O x,sgn ) → M (O x,sgn) , as well as an inclusion M (Odx,sgn ) → M (Onx,sgn ) . The techniques available today do not allow to compute the Nichols algebra 𝔅 (Onx ,sgn ) from the knowledge of the Nichols algebra 𝔅 (Oe ,sgn) x .

Remark 3.4. In Theorem 3.1 we do not claim that the conditions are sufficient. See Tables 1, 2. For instance, it is known that n dim 𝔅 (Ox ,sgn ) < ∞ when n = 3 - see [MS]; for other odd , this is open.

Table 1. Nichols algebras of irreducible Yetter-Drinfeld modules over , odd.

Let us now proceed with the proof of Theorem 3.1.

Proof. If s = id , then q = 1 ss and dim 𝔅 (O, ρ) = ∞ , from Lemma 1.1.

We consider now two cases.

CASE 1: odd.

(I) If s = yh , with 1 ≤ h ≤ n , it is easy to see that O h = {yh,y- h} y and 𝔻yh = ⟨y ⟩ ≃ ℤ n n . Then $ℤ^ = {χ }n n ll=1$ , where l χl(y) = ω , with i2π- ω = exp (n ) ∈ 𝔾n a primitive -th root of . Let us consider M (Oyh,χl) ; it is a braided vector space of diagonal type. If qss ⁄= - 1 , then dim 𝔅 (Oyh, χl) = ∞ , from Lemma 1.3. Assume qss = - 1 ; so we have - 1 = χl(s) = χl(yh) = ωlh. This is a contradiction because is odd.

(II) If s = x , then n- 1 Ox = {x, xy,...,xy } and x 𝔻 n = ⟨x⟩ ≃ ℤ2 . Clearly, dim 𝔅 (Ox, ɛ) = ∞ . On the other hand, M (Ox,sgn ) is a negative braided vector space, since every abelian subrack of has one element; indeed xyjxyk = xykxyj , 0 ≤ j,k ≤ n - 1 , if and only if j = k .

Therefore, the part (a) of the Theorem is proved.

$|------------------------------|------------|------------|----------| |Orbit |Isotropy |Rep. |dim 픅 (V )| |------------------------------|group-------|------------|----------| |e-----------------------------|픻n----------|any---------|∞---------| |Oym = {ym }, | Oym |= 1 |픻n | |∞ | | | |(V, ρ) ∈ | | | | |픻^n, | | | | |ρ(ym ) = 1 | | | | |------------|----------| | | |(V, ρ) ∈ |2dim V | | | | ^ | | | | |픻n, | | | | |ρ(ym ) = | | |------------------------------|------------|--1---------|----------| |Oyh = {y±h}, h ⁄= 0,m, |ℤn ≃ ⟨y⟩ |χj, |4 | || O h |= 2 | |ωhj = - 1 | | | y | |--j---------|----------| | | |χ h,j |∞ | |------------------------------|------------|ω---⁄=--- 1--|----------| |Ox = {xy2h : 0 ≤ h ≤ m - 1} |ℤ2 × ℤ2 ≃ |ɛ ⊗ ɛ, |∞ | || Ox |= m |⟨x⟩ ⊕ ⟨ym⟩ |ɛ-⊗-sgn-----|----------| | | |sgn ⊗ sgn, |negative | | | |sgn ⊗ ɛ |braiding | |----------2h+1----------------|------------|------------|----------| |Oxy = {xy : 0 ≤ h ≤ m - |ℤ2 × ℤ2 ≃m |ɛ ⊗ ɛ, |∞ | |1} |⟨xy⟩ ⊕ ⟨y ⟩ |ɛ ⊗ sgn | | || Ox |= m | |------------|----------| | | |sgn ⊗ sgn, |negative | | | |sgn ⊗ ɛ |braiding | |------------------------------|------------------------------------- | |$
Table 2. Nichols algebras of irreducible Yetter-Drinfeld modules over , n = 2m even.

CASE 2: even. Let us say n = 2m .

(I) If s = ym , then O m = {ym } y and 𝔻ym = 𝔻 n n . Clearly, dim 𝔅 (Oym, ρ) = ∞ , for every ^ ρ ∈ 𝔻n with ρ(s) = Id . On the other hand, if (ρ,V ) ∈ ^𝔻n is such that ρ(s) = - Id , then it is straightforward to prove that 𝔅 (Oym, ρ) = ∧ (V ) , the exterior algebra of ; hence dim 𝔅 (O m, ρ) = 2dim V y .

(II) If h s = y , h ⁄= 0,m ; then h - h Oyh = {y ,y } and yh 𝔻 n = ⟨y⟩ ≃ ℤn . From Lemma 1.3, it is clear that dim 𝔅 (Oyh, χl) = ∞ , for every such that χl(yh) ⁄= - 1 , i.e. ωhl ⁄= - 1 . On the other hand, it is easy to see that 𝔅 (O h,χ ) = ∧ (M (O h,χ )) y l y l , hence dim 𝔅 (O h,χ ) = 4 y l , for every χ l with h χl(y ) = - 1 .

(III) If s = x , then 2h Ox = {xy : 0 ≤ h ≤ m - 1} and x m 𝔻n = ⟨x ⟩ ⊕ ⟨y ⟩ ≃ ℤ2 × ℤ2 . From Lemma 1.1, dim 𝔅 (Ox,ɛ ⊗ ɛ) = dim 𝔅 (Ox, ɛ ⊗ sgn) = ∞ .

For the cases ρ = sgn ⊗ ɛ or sgn ⊗ sgn , we note the following fact.

If is odd and , we have that
If is even and , we have that

The cases (i) and (ii) say that every maximal abelian subrack of has one and two elements, respectively. Hence, in both cases the braiding is negative. Indeed, the result is obvious for the case (i), while in the case (ii) we have that if 2j j j -1 tj := xy = xy x(xy ) and 2k k k -1 tk := xy = xy x (xy ) commute in , then qjj = - 1 = qkk and qjkqkj = 1 ; thus the braiding is negative.

(IV) If s = xy , then Oxy = {xy2h+1 : 0 ≤ h ≤ m - 1} and 𝔻xy = ⟨xy⟩ ⊕ ⟨ym⟩ ≃ ℤ2 × ℤ2 n . The result follows as in (III) using the isomorphism 𝔻 → 𝔻 n n , x ↦→ xy , y ↦→ y . □

4. On Nichols algebras over semisimple Hopf algebras

Let be a Hopf algebra. Let J ∈ A ⊗ A be a twist and let J A be the corresponding twisted Hopf algebra. If is a Hopf subalgebra of a Hopf algebra , then is a twist for and is a Hopf subalgebra of . Now, if is semisimple, then this induces a bijection

{isoclasses of Hopf algebras with coradical ≃ A } -∽→ {isoclasses of Hopf algebras with coradical ≃ AJ },

(7)

that preserves standard invariants like dimension, Gelfand-Kirillov dimension, etc. Let now A = ℂ 𝔸5 and let J ∈ A ⊗ A be the non-trivial twist defined in [Ni]. By (7), we conclude immediately from Theorem 2.6.

Theorem 4.1. Let be a finite-dimensional Hopf algebra with coradical isomorphic to (ℂ𝔸5 )J . Then H ≃ (ℂ𝔸5 )J . □

Again, this is the first classification result we are aware of, for finite-dimensional Hopf algebras with coradical isomorphic to a fixed non-trivial semisimple Hopf algebra. Recently, a semisimple Hopf algebra B ≃ (ℂ 𝔻 × 𝔻 )J′ 3 3 was discovered in [GN]. This Hopf algebra is simple, that is it has no non-trivial normal Hopf subalgebra. Since there are finite-dimensional non-semisimple pointed Hopf algebras with group 𝔻3 ≃ 𝕊3 , there is a finite-dimensional non-semisimple Hopf algebra with coradical isomorphic to .

Acknowledgement. We are grateful to Professor John Stembridge for information on Coxeter groups, in particular reference [BG]. We thank Matías Graña, Sebastián Freyre and Leandro Vendramín for interesting discussions.

References

[AF] N. Andruskiewitsch and F. Fantino, On pointed Hopf algebras associated with unmixed conjugacy classes in , J. Math. Phys. 48 (2007), 033502-1 - 033502-26. [ Links ]

[AG] N. Andruskiewitsch and M. Graña, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), 177 - 243. [ Links ]

[AS1] N. Andruskiewitsch and H.-J. Schneider, Finite quantum groups and Cartan matrices, Adv. Math. 154 (2000), 1-45. [ Links ]

[AS2] N. Andruskiewitsch, Pointed Hopf Algebras, in "New directions in Hopf algebras", 1-68, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, Cambridge, 2002. [ Links ]

[AZ] N. Andruskiewitsch and S. Zhang, On pointed Hopf algebras associated to some conjugacy classes in , Proc. Amer. Math. Soc. 135 (2007), 2723-2731. [ Links ]

[BG] C. T. Benson and L. C. Grove, The Schur indices of the reflection group , J. Algebra, 27 (1973), 574-578. [ Links ]

[C] R. W. Carter, Conjugacy classes in the Weyl group, Compos. Math. 25 (1972), 1-59. [ Links ]

[FGV] S. Freyre, M. Graña and L. Vendramin, On Nichols algebras over SL (2,𝔽q) and GL(2,𝔽q) , preprint math.QA/0703498. [ Links ]

[FH] W. Fulton and J. Harris, Representation theory, Springer-Verlag, New York 1991. [ Links ]

[GN] C. Galindo and S. Natale, Simple Hopf algebras and deformations of finite groups, Math. Res. Lett., to appear. [ Links ]

[Gñ] M. Graña, On Nichols algebras of low dimension, Contemp. Math. 267 (2000), 111-134. [ Links ]

[H] I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), 175-188. [ Links ]

[Hu] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge 1990. [ Links ]

[JL] A. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, Cambridge 2001. [ Links ]

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

[Ni] D. Nikshych, -rings and twisting of finite dimensional semisimple Hopf algebras, Commun. Algebra 26 (1998), 321-342. (Corrigendum: Commun. Algebra 26 (1998), 2019.) [ Links ]

[S] Martin Schönert et al. "GAP - Groups, Algorithms, and Programming - version 3 release 4 patchlevel 4". Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1997. [ Links ]

Nicolás Andruskiewitsch
Facultad de Matemática, Astronomía y Física,
Universidad Nacional de Córdoba. CIEM - CONICET.
Medina Allende s/n
(5000) Ciudad Universitaria, Córdoba, Argentina
andrus@mate.uncor.edu

Fernando Fantino
Facultad de Matemática, Astronomía y Física,
Universidad Nacional de Córdoba. CIEM - CONICET.
Medina Allende s/n
(5000) Ciudad Universitaria, Córdoba, Argentina
fantino@mate.uncor.edu

Recibido: 16 de febrero de 2007
Aceptado: 2 de agosto de 2007

Servicios Personalizados

Revista

Articulo

Indicadores

Links relacionados

Compartir

Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.3 Bahía Blanca 2007