A ug 2 00 8 NEW TECHNIQUES FOR POINTED HOPF ALGEBRAS

We present techniques that allow to decide that the dimension 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 sexagésimo cumpleaños. Introduction 0.1. Let G be a finite group and let CG CGYD be the category of YetterDrinfeld 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 ∈ CG CGYD, 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 ∈ 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 is a conjugacy class in G and ρ is an irreducible representation of the isotropy group Gs 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 between braided vector spaces and Yetter-Drinfeld modules. Indeed, the same braided vector space could be realized as a Yetter-Drinfeld module over different 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: August 22, 2008. 2000 Mathematics Subject Classification. 16W30; 17B37. This work was partially supported by ANPCyT-Foncyt, CONICET, Ministerio de Ciencia y Tecnoloǵıa (Córdoba) and Secyt (UNC). 1 2 ANDRUSKIEWITSCH AND FANTINO Thus, a comprehensive approach to the question above would be to solve the following: Given a braided vector space (V, c) determined by a rack and a 2-cocycle, decide when dimB(V ) < ∞. But at the present moment and with the exception of the diagonal case mentioned below, we know explicitly very few Nichols algebras of braided vector 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 over some finite abelian group are the diagonal braided vector spaces. This leads to the following question: Given a braided vector space (V, c) of diagonal type, decide when the Nichols algebra B(V ) is finite-dimensional. The full answer to this problem was given in [H2], see [AS2, H1] for braided vector spaces of Cartan type– and [AS4] for applications. These results on Nichols algebras of braided vector spaces of diagonal type were in turn used for more general pointed Hopf algebras. Let us fix a non-abelian finite group G and let V ∈ CG CGYD irreducible. If the underlying braided vector space contains a braided vector subspace of diagonal type, whose Nichols algebra has infinite dimension, then dimB(V ) = ∞. In turns out that, for several finite groups considered so far, many Nichols algebras of irreducible YetterDrinfeld modules have infinite dimension; and there are short lists of those not 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 are irreducible. Then the Nichols algebra of V is studied, under the assumption that the B(Vi) are known and finite-dimensional, 1 ≤ i ≤ θ. Under some circumstances, 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, S3, S4, Dn– the Nichols algebras of some V have infinite dimension. These applications rely on 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 to discard more irreducible Yetter-Drinfeld modules. Namely, let V = V1⊕V2 ∈ CΓ CΓYD, where Γ = S3, S4 or Dn, such that dimB(V ) = ∞ by [AHS, Section 4]. 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 ∈ Ĝs and M(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 braided vector subspace isomorphic to (CX, cq); thus, necessarily, dimB(O, ρ) = ∞. We illustrate these new techniques with several examples; see in particular Example 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 determination to consider families of finite groups, in order to discard those irreducible Yetter-Drinfeld modules over them with infinite-dimensional Nichols algebra by the ‘subrack method’. Natural candidates are the families of simple groups, or closely related; cf. the classification of simple racks in [AG]. The case of symmetric and alternating groups is treated in [AZ, AF1, AF2, AFZ]; Mathieu groups in [F1]; other sporadic groups in [AFGV]; some finite groups of Lie type with rank one in [FGV, FV]. Particularly, a list of only 9 types of pairs (O, ρ) for Sm whose Nichols algebras might be finite-dimensional is given in [AFZ]; an analogous list of 7 pairs out of 1137 (for all 5 Mathieu simple groups) is given in [F1]; the sporadic groups J1, J2, J3, He and Suz are shown to admit no non-trivial pointed finite-dimensional Hopf algebra in [AFGV]. Our new techniques are crucial for these results. 0.6. If for some finite group G there is at most one irreducible YetterDrinfeld 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 , and investigate 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 (vi)1≤i≤θ, let (qij)1≤i,j≤θ be a matrix of 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– called the multiplication, such that φi : X → X, φi(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−1. In this case, j ⊲i = i whenever i ⊲ j = j and i ⊲ i = i for all i ∈ G. We are mainly interested in subracks of G, e. g. in conjugacy classes in G. Let (X, ⊲) be a rack. A function q : X × X → C× is a 2-cocycle if qi,j⊲k qj,k = qi⊲j,i⊲k qi,k, for all i, j, k ∈ X. Then (CX, cq) is a braided vector space, where CX is the vector space with basis ek, k ∈ X, and the braiding is 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 abelian subrack of X, then CT is a braided vector subspace of (CX, cq) of diagonal type. 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 Ĝ the set of isomorphism classes of irreducible representations of G. We shall often denote a representant of a class in Ĝ with the same symbol as the class itself. Here is an explicit description of the irreducible Yetter-Drinfeld module M(O, ρ). Let t1 = s, . . . , tM be a numeration of O and let gi ∈ G such that gi ⊲ s = ti for all 1 ≤ i ≤ M . Then M(O, ρ) = ⊕1≤i≤M gi ⊗ V , where V is the 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 ∈ G is given by g · (giv) = gj(γ · v), where ggi = gjγ, for some 1 ≤ j ≤ M and γ ∈ Gs, and the coaction is given by δ(giv) = ti ⊗ giv. Then M(O, ρ) is a braided vector space with braiding c(giv ⊗ gjw) = gh(γ · w) ⊗ giv, for any 1 ≤ i, j ≤ M , v,w ∈ V , where tigj = ghγ for unique h, 1 ≤ h ≤ M and γ ∈ 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 all permutations in Sm with the same type as σ; we may use the type as a subscript of a conjugacy class as well. If some emphasis is needed, we add a superscript m to indicate that we are taking conjugacy classes in Sm, like Oj 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 ∈ D̂n be the sign representation (Dn = 〈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 infinite dimension. Note that M(Ox, sgn) ⊕ M(Ox, sgn) is isomorphic as a braided vector space to (CXn, q), where • Xn is the rack with 2n elements si, tj , i, j ∈ Z/n, and with structure si⊲sj = s2i−j, si⊲tj = t2i−j , ti⊲sj = s2i−j, ti⊲tj = t2i−j , i, j ∈ Z/n; • q is the constant cocycle q ≡ −1. If d divides n, then Xd can be identified with a subrack of Xn. Hence, it is enough to consider braided vector spaces (CXp, q), with p an odd prime. We fix a finite group G with the rack structure given by conjugation x ⊲ y = xyx−1, x, y ∈ G. Let O be a conjugacy class in G. Definition 2.2. Let p > 1 be an integer. A family (μi)i∈Z/p of distinct elements of G is of type Dp if (2.1) μi ⊲ μj = μ2i−j, i, j ∈ Z/p. Let (μi)i∈Z/p and (νi)i∈Z/p be two families of type Dp in G, such that μi 6= νj for all i, j ∈ Z/p. Then (μ, ν) := (μi)i∈Z/p ∪ (νi)i∈Z/p is of type D (2) p if (2.2) μi ⊲ νj = ν2i−j, νi ⊲ μj = μ2i−j , i, j ∈ Z/p. It is useful to denote i ⊲ j = 2i− j, for i, j ∈ Z/p. We state some consequences of this definition for further use. Remark 2.3. If (μi)i∈Z/p is of type Dp then μ i ⊲ μj = μ2i−j , μi ⊲ μ −1 j = μ −1 2i−j, μ −1 i ⊲ μ −1 j = μ −1 2i−j , (2.3) μi ⊲ μj = μ2i−j , μi ⊲ μ k j = μ k 2i−j, μ k i ⊲ μ k j = μ k 2i−j , (2.4) for all i, j ∈ Z/p, and for all k odd. Remark 2.4. Assume that p is odd. If (μ, ν) = (μi)i∈Z/p∪ (νi)i∈Z/p is of type D (2) p , then for all i, j, μi = μ 2 j , ν 2 i = ν 2 j , μ 2 i νj = νjμ 2 i , ν 2 i μj = μjν 2 i . (2.5) Indeed, μhμj = μjμ 2 h, hence μ 2 2h−j = μhμ 2 jμ −1 h = μ 2 j . Take now h = i+ j 2 . 6 ANDRUSKIEWITSCH AND FANTINO Lemma 2.5. If (μ, ν) = (μi)i∈Z/p ∪ (νi)i∈Z/p is of type D (2) p , then (i) μkμl = μt(l−k)+k μt(l−k)+l, (ii) μkνl = μ2t(l−k)+k ν2t(l−k)+l, (iii) μkνl = ν(2t+1)(l−k)+k μ(2t+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) are similar. The result is obvious when t = 0. Since μkμl = μl μl⊲k, then the result holds for t = 1. Let us suppose that (i) holds for every s ≤ t. Now, μkμl = μt(l−k)+k μt(l−k)+l = μt(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. Lemma 2.6. Assume that p is odd. If (μ, ν) is of type D (2) p , then for i ∈ Z/p, μiνi = μ0ν0, (2.6) νiμi = ν0μ0. (2.7) Proof. Let i, j ∈ Z/p, with i 6= j. If we write (ii) of Lemma 2.5 with k = i, l = j and t = −1/2 we have that μiνj = μ2i−jνi. Thus, μiνiν 2 j = μiνjνjνi = μ2i−jνiνiν2i−j = μ2i−jν2i−jν 2 i , and, by (2.5), μiνi = μ2i−jν2i−j . Now (2.6) follows taking j = 2i. Now (2.7) follows from (2.6) by (2.2). We now set up some notation that will be used in the rest of this section. Let (μi)i∈Z/p be a family of type Dp in G, with p odd. Set gi = μi/2, (2.8) αij = g −1 i⊲j μi gj = μ −1 i−j/2 μi μj/2, (2.9) for all i, j ∈ Z/p. Then gi ⊲ μ0 = μi, αij ∈ G μ0 , i, j ∈ Z/p. Let now (μ, ν) be of type D (2) p and suppose that there exists g∞ ∈ G such that g∞ ⊲ μ0 = ν0. Set fi = νi/2 g∞, (2.10) βij = f −1 i⊲j μi fj = g −1 ∞ ν −1 i−j/2 μi νj/2 g∞, (2.11) γij = g −1 i⊲j νi gj = μ −1 i−j/2 νi μj/2, (2.12) δij = f −1 i⊲j νi fj = g −1 ∞ ν −1 i−j/2 νi νj/2 g∞. (2.13) POINTED HOPF ALGEBRAS 7 Then fi ⊲ μ0 = νi, βij , γij , δij ∈ G μ0 , i, j ∈ Z/p. We assume from now on that p is an odd prime. This is required in the proof of the next lemma, needed for the main result of the section. Lemma 2.7. Let (μ, ν) = (μi)i∈Z/p ∪ (νi)i∈Z/p be of type D (2) p , and suppose that there exists g∞ ∈ G such that g∞ ⊲μ0 = ν0. Let gi and fi be as in (2.8) and (2.10), respectively. Then, for all i, j ∈ Z/p, (a) αij = δij = μ0, (b) βij = g −1 ∞ μ0g∞, (c) γij = ν0. Proof. Let k, l be in Z/p. Then, for all r ∈ Z/p, we have (2.14) μkμl = μk+rμl+r, μkνl = μk+rνl+r, μkνl = νk+rμl+r. This follows from (2.5) and Lemma 2.6 (when k = l), and Lemma 2.5 (when k 6= l). There are similar equalities interchanging μ’s and ν’s. Now αij = μ −1 i−j/2 μi μj/2 (2.14) = μ0, δij = g −1 ∞ ν −1 i−j/2 νi νj/2 g∞ (2.14) = g ∞ ν0 g∞ = μ0, βij = g −1 ∞ ν −1 i−j/2 μi νj/2 g∞ (2.14) = g ∞ μ0 g∞, γij = μ −1 i−j/2 νi μj/2 (2.14) = μ i−j/2 μi−j/2ν0 = ν0, and the Lemma is proved. We can now prove one of the main results of this paper. Theorem 2.8. Let (μ, ν) = (μi)i∈Z/p ∪ (νi)i∈Z/p be a family of elements in G with μ0 ∈ O. Let (ρ, V ) be an irreducible representation of the centralizer Gμ0 . We assume that (H1) (μ, ν) is of type D (2) p ; (H2) (μ, ν) ⊆ O, with g∞ ∈ G such that g∞ ⊲ μ0 = ν0; (H3) qμ0μ0 = −1; (H4) there exist v,w ∈ V − 0 such that, ρ(g ∞ μ0g∞)w = −w, (2.15)