Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VI. Suzuki and Ree groups
aa r X i v : . [ m a t h . QA ] J un FINITE-DIMENSIONAL POINTED HOPF ALGEBRASOVER FINITE SIMPLE GROUPS OF LIE TYPE VI.SUZUKI AND REE GROUPS
GIOVANNA CARNOVALE AND MAURO COSTANTINI
Abstract.
We analyse the rack structure of conjugacy classes in simple Suzuki andRee groups and determine which classes are kthulhu. Combining this results withabelian rack techniques, we show that the only finite-dimensional complex pointedHopf algebras over the simple Suzuki and Ree groups are their group algebras. Introduction
This paper is part of an ongoing project with N. Andruskiewitsch and G. A. Garc´ıa,aimed at understanding finite-dimensional complex pointed Hopf algebras whose groupof grouplikes is a finite simple group of Lie type [1, 2, 3, 4, 5]. We adopt notation andterminology from these papers, and for further details the reader is referred to them.We recall that a finite-dimensional pointed Hopf algebra H has a natural filtrationwhose associated graded contains a graded associative algebra, the so-called Nicholsalgebra B ( V ), whose structure depends on a representation V of the finite group G ofgrouplike elements of H and a compatible G -grading on V (i.e., V is a Yetter-Drinfeldmodule of G ). It is therefore crucial for our purposes to classify finite-dimensionalNichols algebras for Yetter-Drinfeld modules of G .We recall the following folklore conjecture Conjecture 1.1.
Let G be a finite simple non-abelian group. Then, dim B ( V ) = ∞ for every complex Yetter-Drinfeld module V of G . Thus, the only finite-dimensionalcomplex pointed Hopf algebra whose group of grouplikes is G is the group algebra C G . In fact Nichols algebras can be defined in a more general setup, whenever we havean endomorphism c ∈ GL( V ⊗ ) for V a vector space, satisfying the braid equation( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ).If V is a Yetter-Drinfeld module of G , then c is defined by c ( v ⊗ v ′ ) = g · v ′ ⊗ v, v ∈ V g , v ′ ∈ V, V = M g ∈ G V g . (1.1)The presentation of B ( V ) depends only on the braiding c and not on the Yetter-Drinfeldmodule structure itself. Indeed, such a braiding can arise from different actions of Keywords:
Nichols algebra; Hopf algebra; rack; finite group of Lie type; conjugacy class.This work was partially supported by Progetto BIRD179758/17 of the University of Padova. different groups on V and c can be described in terms of a combinatorial object calledrack and a cocycle for it, [8]. Since important properties of racks are preserved by rackinclusions and projections, an approach to Nichols algebras done rack-by-rack might bemore convenient than an approach group-by-group. In particular, the reduction to a(simple) rack is relevant for the problem of classifying finite-dimensional pointed Hopfalgebras whose group of grouplikes is finite but not simple.In our situation, a simple rack will always be a conjugacy class O in G , with rackstructure g ⊲ h = ghg − for g, h ∈ O . Conjugacy classes in different groups can beisomorphic as racks (e.g. unipotent conjugacy classes arising from isogenous algebraicgroups). An important goal is to classify finite-dimensional Nichols algebras for everyconjugacy class in G and every cocycle , and not only those coming from a Yetter-Drinfeld module of G .A series of conditions on racks (called type D, F and C) ensuring that the associatedNichols algebra is infinite-dimensional for any choice of a cocycle were given in [6, 1, 3].In this case we say that the rack collapses . In group theoretic terms these conditionsare easy to state and well-behaved when passing to subgroups and quotients. Theconjugacy classes that are not of type C, D, or F are called kthulhu: they are essentiallythose for which the possible non-empty intersections with a subgroup H ≤ G are eithera single conjugacy class in H or consist of a set of commuting elements. For these classeswe have no general strategy to deal with all cocycles. Yet, one can use the techniquesdeveloped in [6, 7, 10, 14] and the classification in [15] to deal with the Nichols algebrasassociated with a kthulhu class and a cocycle coming from a (simple) Yetter-Drinfeldmodule of G . These techniques are often enough for dealing with Hopf algebras over G but might not propagate when passing to overgroups or groups projecting on G .This paper deals with conjugacy classes in simple Suzuki groups B ( q ), q > F ( q ), q > G ( q ), q >
3. They were firstly presented and studied in[28, 22, 23] but we will use the uniform description in terms of fixed points of a Steinbergendomorphism as in [11, 20]. Working group-by-group we prove the following result:
Theorem 1.2.
Let G be a simple Ree or Suzuki group and let O be the class of x ∈ G .Then O is kthulhu if and only if G = B ( q ) and | x | = 2 or | x | divides q + 1; G = F ( q ) and | x | 6 = 13 and divides q − q + 1; G = G ( q ) and | x | = 3 or | x | 6 = 7 and divides q − q + 1 . In other words, kthulhu classes are either classes of unipotent elements of primeorder, or they are represented by elements in maximal tori whose normaliser is theonly maximal subgroup containing them. Up to some exception of small order, everynon-trivial element in such a torus lies in a kthulhu class. Then we focus on kthulhuclasses in each group and finally we prove:
ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 3
Theorem 1.3.
Conjecture 1.1 holds if G is a simple Suzuki or Ree group. Notation and background
For any automorphism σ of an algebraic structure X , we shall denote by X σ the set ofelements fixed by σ . For G a group, the orbit of an element g ∈ G under the conjugationaction of a subgroup H ≤ G will be denoted by O Hg . The superscript will be omittedif the ambient group is clear from the context. To keep uniformity with the previouspapers in the series, we will denote the conjugation action by: g ⊲ h := ghg − . Thecentraliser of an element x ∈ G will be denoted by C G ( x ), and the set of isomorphismclasses of irreducible representations of a group H will be denoted by Irr( H ).2.1. Preliminaries on racks.
In this section we introduce some preliminary notionson the rack structure specialised to the case of a conjugacy class.
Definition 2.1. ([3, Definition 2.3], [7, Definition 3.5], [1, Definition 2.4]). A conjugacyclass O in a finite group M is said to be of typeC if there are H ≤ M and r, s ∈ O ∩ H such that(a) O Hr = O Hs ,(b) rs = sr ,(c) H = hO Hr , O Hs i ,(d) either min( |O Hr | , |O Hs | ) > |O Hr | , |O Hs | ) > r, s ∈ O such that(a) O h r, s i r = O h r, s i s ,(b) ( rs ) = ( sr ) ;F if there are r i ∈ O , for 1 ≤ i ≤ O h r i , ≤ i ≤ i r i = O h r i , ≤ i ≤ i r j , for i = j ,(b) r i r j = r j r i , for i = j .A conjugacy class is called kthulhu if it is of none of these types.The relevance of the above conditions relies on the following results, that we applyto the special case of conjugacy classes. Proposition 2.2. ( [3, § , [7, § , [1, § ).(1) If a rack X is of type C, D, or F, then dim B ( X, q ) = ∞ for every cocycle q for X , i.e., it collapses.(2) If a rack contains or projects onto a rack of type C, D, or F, then it is of thesame type.Remark . (1) Assume that M is a finite group with M/Z ( M ) simple non-abelian.If m ∈ M \ Z ( M ), then there exists g ∈ M such that [ g ⊲ m, m ] = 1. Otherwise, G. CARNOVALE, M. COSTANTINI N := hO Mm i would be an abelian normal subgroup of M and N/ ( Z ( M ) ∩ N )would be an abelian normal subgroup of M/Z ( M ). Therefore N would becentral, while m Z ( M ), a contradiction.(2) If r ∈ M with | r | odd, and r, s ∈ O Mr satisfy rs = sr , then for any H ≤ M suchthat h r, s i ≤ H we have min( |O Hr | , |O Hs | ) >
2. Indeed, 3 ≤ |O h r i s | ≤ |O Hs | and3 ≤ |O h s i r | ≤ |O Hr | . Lemma 2.4.
Assume M = M × M is a finite group such that M /Z ( M ) is simplenon-abelian and let m i ∈ M i \ Z ( M i ) for i = 1 , . If | m | is odd, | m | 6 = 2 and m isreal in M , then O M ( m ,m ) is of type C.Proof. By Remark 2.3 (1) there is g ∈ M such that [ g ⊲ m , m ] = 1. Let g ∈ M besuch that g ⊲m = m − . We set r = ( m , m ), s := ( g , g ) ⊲ ( m , m ) and H = h r, s i ≤h m , g ⊲ m i × h m i . By construction rs = sr and H = hO Hr , O Hs i . The inequality m = 1 implies O Hr ∩ O Hs = ∅ . In addition |O Hr | = |O h m ,g ⊲m i m | ≥ |O h g ⊲m i m | ≥ | m | = | g ⊲ m | ≥
3, and similarly for |O Hs | . (cid:3) In the following Remark we recall an argument used in [3, Proposition 5.5] in orderto prove that certain classes are kthulhu.
Remark . Let O be a conjugacy class in a finite group G . Assume that for any H ≤ G the intersection O ∩ H is either empty, a unique conjugacy class in H , orconsists of mutually commuting elements. Then O is kthulhu. We usually deal withintersections with subgroups using the list of maximal subgroups as follows. For anymaximal M < G we analyse
O ∩ M . If O ∩ M = ∅ or consists of commuting elements,it will be again so for any H ≤ M . Then we show that for the remaining subgroups O ∩ M is a single conjugacy class in M and observe that in this case the structure of M and of its (maximal) subgroups are well-understood. In most cases M will be a finitesimple group of Lie type of the same sort as G but over a smaller field, or PSL ( q ).This way we reduce from the pair ( O , G ) to the pair ( O ∩
M, M ). The class
O ∩ M will usually have the same features as O had in G and we proceed inductively.In order to implement the above mentioned analysis, we will make use of the followingstandard observation. Remark . Let M = N ⋊ h a i be a finite group with C N ( a ) = { } .(1) We have the equality O Ma = N a . Indeed, the inclusion ⊂ follows from normalityof N , and C N ( a ) = { } implies that the two sets have the same cardinality.(2) For any H ≤ M such that a ∈ H , then O Ha = ( H ∩ N ) a . Indeed, if a ∈ H , then H = ( N ∩ H ) ⋊ h a i , so (1) applies. ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 5 (3) If G is a finite group containing M and such that O Ga ∩ M ⊂ N a , then forany H ≤ M with a ∈ H we have O Ga ∩ H = O Ha . Indeed, by (2) there holds O Ga ∩ H ⊂ ( N ∩ H ) a = O Ha ⊂ O Ga ∩ H , whence the equality.2.2. Nichols algebras of Yetter-Drinfeld modules and abelian subracks.
Inthis section we provide some necessary ingredients for dealing with Conjecture 1.1.The first key observation is the following.
Remark . ([7, § H be a finite group. If dim B ( V ) = ∞ for every simpleYetter-Drinfeld module V of H , then dim B ( V ′ ) = ∞ for every Yetter-Drinfeld module V ′ of H .Simple Yetter-Drinfeld modules are parametrized by pairs ( O , ρ ) where the O = O Hg is a conjugacy class in H and is the support of the grading, and ρ ∈ Irr( C H ( g )). If ρ : C H ( g ) → GL ( W ), the corresponding simple Yetter-Drinfeld module M ( O , ρ ) has H -module structure and grading defined by: M ( O , ρ ) = Ind HC H ( g ) ρ = C H ⊗ C C H ( g ) W, m ⊗ W ⊂ M ( O , ρ ) m⊲g , m ∈ H. (2.1)If O is of type C, D, or F, then Proposition 2.2 (1) ensures that dim B ( M ( O , ρ )) = ∞ for any choice of ρ ∈ Irr( C H ( g )). For a kthulhu conjugacy class O = O Hh the conclusionof Proposition 2.2 (1) cannot be inferred. We recall a strategy developed in [6, 7, 10, 14]and references therein to estimate the dimension of B ( M ( O , ρ )).Assume A ≤ C H ( g ) is an abelian subgroup containing g . Then, O ∩ A is an abeliansubrack of O and ρ ( A ) stabilises a line C v in W : we call χ its character. We set O ∩ A = { x := g ⊲ g, . . . , x r := g r ⊲ g } , q ij := χ ( g − j g i ⊲ g ) . (2.2)Then M A = span C ( g i ⊗ v, i = 0 , . . . , r ) is a braided subspace of M ( O , ρ ), i.e., c ( M A ⊗ M A ) = M A ⊗ M A , where c is as in (1.1). The restriction of c to M A ⊗ M A is given by c (( g i ⊗ v ) ⊗ ( g j ⊗ v )) = q ij ( g j ⊗ v ) ⊗ ( g i ⊗ v ), i.e., it is of diagonal type with q ii = χ ( g ) forevery i . Now B ( M A ) is a subalgebra of B ( M ( O , ρ )). We can invoke the classificationresults for finite-dimensional Nichols algebras for braided spaces of diagonal type in[15], and if dim B ( M A ) = ∞ we can conclude that dim B ( M ( O , ρ )) = ∞ .2.3. Construction of the groups.
Let p be a prime, h ≥ q = p h +1 and G asimply-connected simple algebraic group over F p . We recall the construction of thegroups B ( q ), F ( q ) and G ( q ) from [11, § F in G . Let T be a fixed maximal torus in G , with correspondingroot system Φ, root subgroups U α for α ∈ Φ, and Weyl group W = N G ( T ) / T . Wefix an isomorphism x α : F p → U α for each α ∈ Φ and a set of simple roots ∆, withcorresponding positive roots Φ + . The group W acts by isometries on E = R ⊗ Z Z Φ.We will focus on the cases in which the pair (Φ , p ) is either ( B , F ,
2) or ( G , x α is as in [11, § G. CARNOVALE, M. COSTANTINI
The non-trivial symmetry of the Coxeter graph of G induces a permutation θ : Φ → Φ, [11, § τ the unique involutory isometry of E such that τ ( α ) ∈ R > θα for all α ∈ Φ: τ ( α ) = ( √ θα if α is short, √ θα if α is long, for Φ of type B or F ,τ ( α ) = ( √ θα if α is short, √ θα if α is long, for Φ of type G . There is a graph automorphism ϑ of G preserving T and such that ϑ ( U α ) = U θα forall α ∈ Φ, [11, § ϑ ( x α ( c )) = ( x θα ( c p ) if α is short, x θα ( c ) if α is long.Let Fr p h be the field automorphism of G induced by the automorphism λ λ p h of F p and let F : G → G be the Steinberg endomorphism F = ϑ ◦ Fr p h = Fr p h ◦ ϑ . Then G := G F = { x ∈ G | F ( x ) = x } are the Suzuki groups for G of type B and the Reegroups for G of type F or G .Note that F : x α ( ξ ) x α ( ξ q )for every α ∈ Φ, so that G is contained in B (2 h +1 ) , F (2 h +1 ) , G (3 h +1 ) respectively.For convenience, we denote G by B ( q ) , F ( q ) , G ( q ) respectively. Notice that thisis not the only notation for the Suzuki or Ree groups, often the convention q = p h +1 is used. We have (cid:12)(cid:12) B ( q ) (cid:12)(cid:12) = q ( q − q + 1) , (cid:12)(cid:12) G ( q ) (cid:12)(cid:12) = q ( q − q + 1) , (cid:12)(cid:12) F ( q ) (cid:12)(cid:12) = q ( q − q + 1)( q − q + 1) . We recall that these groups are simple for h ≥ B ≤ G be the Borel subgroup generated by T and the U α , α ∈ Φ + , let B − ≤ G be the opposite Borel subgroup and let U and U − be their unipotent radicals. Everyunipotent conjugacy class in G is represented by an element in U and, for any fixedordering of Φ + , every element in U can be uniquely written as a product Q γ ∈ Φ + x γ ( c γ )for c γ ∈ F q . Also U := U F and U − := ( U − ) F are Sylow p -subgroups of G , [20, Corollary24.11] so all unipotent conjugacy classes in G intersect U and U − . We recall that every F -stable maximal torus in G is of the form g T g − for some g ∈ G such that ˙ w := g − F ( g ) ∈ N ( T ). Two such tori are G -conjugate if and only if the corresponding Weylgroup elements are F -conjugate, [20, Proposition 25.1]. We denote by T w a maximaltorus whose associated Weyl group element is w = ˙ w T ∈ W and we set T w = T Fw .Every semisimple element is contained in some T w , for some w ∈ W , [20, Proposition26.6]. There is a formula for the order of T w . Let Y = Hom( F p × , T ) be the cocharacter ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 7 group of T . Then W and F act on Y , hence on Y ⊗ R and | T w | = | det Y ⊗ R ( w − ◦ F − | ,see [20, Proposition 25.3 (c)]. An element σ ∈ W has a representative in N G ( T w ) ifand only if σ ∈ C W ( τ w ), and | N G ( T w ) / T w | = | C W ( τ w ) | , [20, Proposition 25.3 (a)]. Remark . When dealing with mixed classes, i.e., classes of elements x ∈ G thatare neither semisimple (i.e. of order coprime with p ) nor unipotent (i.e. of order apower of p ) we adopt the strategy developed in [5, § x = x s x u be the Jordandecomposition of x . We recall that [ C G ( x s ) , C G ( x s )] is a semisimple group whose rootsystem has a base that can be indexed by a set of nodes Σ in the extended Dynkindiagram of G , [20, Remark 14.5]. In addition, Σ must be stable by Ad( ˙ w ) ◦ F forsome w ∈ W . Since W preserves the root lengths and ϑ does not, if it is non-empty,Σ, identified with the corresponding subset of Φ, can only have the same amount ofshort and long roots, providing a strong restriction on the possibilities for Σ. Also,[ C G ( x s ) , C G ( x s )] F ≃ h T , U ± α | α ∈ Σ i Ad( ˙ w ) F and the following natural rack inclusion O [ C G ( x s ) ,C G ( x s )] F x u ≃ x s O [ C G ( x s ) ,C G ( x s )] F x u = O [ C G ( x s ) ,C G ( x s )] F x s x u ⊂ O G F x s x u (2.3)shows that if O [ C G ( x s ) ,C G ( x s )] F x u is not kthulhu, then O G x is again so. Remark . Since in B , F and G the longest element w in W is − id, for any w ∈ W there is always a representative of w in N G ( T w ) and therefore all semisimpleclasses in G are real. Remark . Since X ± X m +1 ±
1, if q = q m +10 and ( d, q k ±
1) = 1, then( d, q k ±
1) = 1. So, if an element of B ( q ), F ( q ) or G ( q ) lies in a torus whose orderis coprime to q k ±
1, and it also lies in a subgroup isomorphic to B ( q ), F ( q ) or G ( q ), respectively, then it will lie in a torus therein whose order is coprime to q k ± Remark . If F ′ is a Steinberg automorphism of G such that G F ′ ≤ G and g ∈ G is semisimple, then O G g ∩ G F ′ ⊂ O G g ∩ G F ′ . Since O G g is semisimple, the right hand sideis either empty or a unique semisimple conjugacy class in G F ′ by [27, § The Suzuki groups B (2 h +1 )In this section p = 2, q = 2 h +1 , h ≥ G = B ( q ) a Suzuki group. We recall somebasic facts from [28].We will need the automorphism δ = Fr h +1 of F q , so that δ = Fr and F δq = F . Remark . (1) Let k ∈ F q be such that kδ ( k ) = 1. Then, 1 = δ ( kδ ( k )) = δ ( k ) k ,so k = k ∈ F . G. CARNOVALE, M. COSTANTINI (2) The group morphism ϕ : F × q → F × q given by ϕ ( k ) = kδ ( k ) is injective, thereforeit is an isomorphism.We realize G = Sp ( F ) as the group of matrices in GL ( F ) preserving the bilinearform associated with the matrix J = (cid:18) (cid:19) . Then T can be chosen to be thesubgroup of diagonal matrices and B the subgroup of upper-triangular matrices. TheSylow 2-subgroup U − of G is given by the matrices of the form U ( a, b ) := a aδ ( a ) + b δ ( a ) 1 a δ ( a ) + ab + δ ( b ) b a , a, b ∈ F q . with multiplication rule U ( a, b ) U ( c, d ) = U ( a + c, aδ ( c ) + b + d ) , a, b, c, d ∈ F q , (3.1)For any k ∈ F × q we have t k := diag( ξ , ξ , ξ − , ξ − ) ∈ T where δ ( ξ ) = kδ ( k ) and δ ( ξ ) = k . There holds: t − k U ( a, b ) t k = U ( ak, bkδ ( k )) , a, b ∈ F q , k ∈ F × q . (3.2)It follows from [28, Proposition 1, 2, 3, 7] that if x is a non-trivial element of a Sylow2-subgroup Q of G , then C G ( x ) ≤ Q . In particular, the order of the elements in G iseither a power of 2 or odd.It follows from (3.1) that all non-trivial involutions are conjugate to an element ofthe form U (0 , b ), so by Remark 3.1 and (3.2) all non-trivial involutions are conjugate.The elements of odd order are semisimple, and therefore their conjugacy classes arerepresented by an element in a maximal torus T w , where w runs through a set ofrepresentatives of F -conjugacy classes in W . Up to conjugacy they are: T , of order q − T s and T s s s whose orders are 2 h +1 ± h +1 + 1 = q ± √ q + 1, so | T | , | T s | and | T s s s | are mutually coprime.The maximal subgroups of G are the conjugates of the following subgroups, [28,Theorems 9, 10]:(1) B − = T ⋉ U − of order q ( q − N G ( T ) of order 2( q − N G ( T s ) and N G ( T s s s ) of order 4( q ± √ q + 1);(4) B (2 m +1 ) for (2 h + 1) / (2 m + 1) a prime number.3.1. Collapsing racks.Lemma 3.2. If O consists of elements of order , then O is kthulhu.Proof. The class O is contained in a unipotent class in Sp ( F q ) and all classes of non-trivial involutions therein are kthulhu [2, Lemma 4.22(2), Lemma 4.26], [3, Lemma2.14]. We conclude by [4, Lemma 2.5]. (cid:3) ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 9
Lemma 3.3.
Assume h > . If O consists elements of order , then it is of type F.Proof. By (3.1) any element of order 4 has a representative r = U ( a, b ) for some a = 0, a, b ∈ F q . Since h >
0, there are distinct k j ∈ F × q for j = 0 , , , r j := t k j ⊲ r = U ( ak j , bk j δ ( k j )). For any c, d ∈ F q we have U ( c, d ) − = U ( c, d ′ ) for some d ′ ∈ F q and U ( c, d ) ⊲ U ( a, b ) = U ( a, b ′ ) for some b ′ ∈ F q . As h r i , i = 0 , , , i ≤ U − ,we deduce that O h r i , i =0 , , , i r i = O h r i , i =0 , , , i r j for i = j . In addition, r i r j = U ( a ( k i + k j ) , aδ ( a ) k i δ ( k j ) + bk i δ ( k i ) + bk j δ ( k j ))so r i r j = r j r i if and only if k i k − j = δ ( k i k − j ) if and only if k i = k j if and only if i = j .Whence, O is of type F. (cid:3) Lemma 3.4.
Let = t ∈ T .Then O G t is of type C.Proof. Recall that t = t − ∈ O G t by Remark 2.9 and that C G ( U (1 , ≤ U − , [28,Proposition 1, 2, 3, 7]. It follows from Remark 2.6 that U − ⊲ t = tU − ⊂ O G t andsimilarly, U − ⊲ t − = t − U − ⊂ O G t . Let H := h t, U − i , r = t , s = t − U (1 , ∈ O G t ∩ H .Then, rs = sr by (3.2). Also, O Hr = O U − t = tU − and O Hs = O Ht − = O U − t − = t − U − , thetwo sets are clearly disjoint and hO Hr , O Hs i = h tU − , t − U − i = h t, U − i = H. In addition |O Hr | = |O Hs | = | U − | = q >
2, so O G t is f type C. (cid:3) Remark . All classes in B (2) are khtulhu. Indeed, | B (2) | = 20 and its elementshave either order 2 , m ( a, x ) = ( x a )where x ∈ F and a ∈ F × . Let g ∈ B (2) with | g | = 5. Then, g = m (0 , x ) for some x ∈ F and all such elements form an abelian normal subgroup of B (2). Hence, allelements in the class of g commute, so classes of elements of order 5 cannot be of typeC, D nor F. Let now | g | = 4. Then g = m ( a, x ), for a = 2 or 3 in F and some x ∈ F and h ∈ O g if and only if h = m ( a, y ) for some y ∈ F . But then h g, h i containselements of order 4 and it is either cyclic of order 4 or the whole group. Hence, O g iskthulhu. For non-trivial involutions we invoke Lemma 3.2. Lemma 3.6.
The non-trivial classes represented by elements in the subgroups T s and T s s s are kthulhu.Proof. We use the inductive argument from Remark 2.5. Let T ′ be one of these tori,let g ∈ T ′ \ { } and let O = O G g . Since | g | divides ( q + √ q + 1)( q − √ q + 1) = q + 1, a maximal subgroup M of G meeting O cannot be conjugate to B − or N G ( T ).Also by order reasons, if it meets N G ( T w ) for w = s or s s s , then T ′ = T w and since N G ( T w ) ≃ T w ⋊ C , all its elements of odd order are contained in T w . Thus, O ∩ N G ( T w ) consists of commuting elements. We finally consider M = B (2 m +1 ) with(2 h + 1) / (2 m + 1) a prime. If M ∩ O 6 = ∅ , then it is a unique semisimple conjugacy classin M by Remark 2.11. By Remark 2.10, this class is represented in a torus of ordercoprime with 2 m +1 −
1, i.e., a torus as in the hypotheses. If m = 0, the intersectionis non-trivial only if | g | = 5 and by Remark 3.5 it consists of commuting elements.Arguing by induction on the number of prime factors of 2 h + 1, we conclude that forany K ≤ G , the intersection K ∩ O is either empty, a conjugacy class in K , or consistsof commuting elements. (cid:3) Remark . Let u ∈ G be a non-trivial involution. Then, there exists v ∈ O G u suchthat ( uv ) = ( vu ) . Indeed, since there is a unique conjugacy class of elements of order2 in G , we may always assume that u ∈ B (2) and a direct computation using therealisation in Remark 3.5 shows that any v ∈ O B (2) u \ { u } has the required property.3.2. Nichols algebras over Suzuki groups.
In this subsection we consider Nicholsalgebras attached to simple Yetter-Drinfeld modules M ( O , ρ ) for O a kthulhu class in G and we prove Theorem 1.3 for simple Suzuki groups for h > Proposition 3.8.
Let g ∈ G with | g | 6 = 1 and odd. Then dim B ( O g , ρ ) = ∞ for everyirreducible representation ρ of C G ( g ) .Proof. Every such element is semisimple, hence real by Remark 2.9. The claim followsfrom [10, Lemma 2.2]. (cid:3)
Lemma 3.9.
Assume h > . Let H = h T , U (0 , b ) , b ∈ F q i = T Z ( U − ) and let O = O HU (0 , . Then dim B ( M ( O HU (0 , , ρ )) = ∞ for every ρ ∈ Irr C H ( U (0 , .Proof. The proof is obtained mutatis mutandis from the proof of [14, Proposition 3.1,case c ]. We sketch it here for completeness’ sake and we use notation and strategy fromSubsection 2.2. For k ∈ F × q we consider the elements t k ⊲ U (0 ,
1) = U (0 , ϕ ( k ) − ). ByRemark 3.1 all non-trivial involutions in H are conjugate to U (0 ,
1) by an element in T and A := C H ( U (0 , Z ( U − ) ≃ ( F q , +) is abelian. We parametrise the elements in O = O ∩ A by elements in H/Z ( U − ) ≃ T ≃ F × q . Let χ be an irreducible representationof Z ( U − ), i.e., a group morphism χ : ( F q , +) → C × . The image is in {± } . Assumethat dim B ( M ( O , χ )) < ∞ . The coefficients of the braiding associated with ( O , χ ) aregiven by q kl = χ ( t − l t k ⊲U (0 , χ ( U (0 , ϕ ( lk − )) so q kl q lk = χ ( U (0 , ϕ ( kl − )+ ϕ ( lk − ))for every k, l ∈ F × q . By [10, Remark 1.1] we necessarily have q kk = χ ( U (0 , − k ∈ F × q . Let k ∈ F q − F and let d be its multiplicative order. Then d ≥ ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 11 if we had q k q k = −
1, then we would have a cycle in the generalised Dynkin diagramassociated with the braiding: q k q k = q kk q k k = · · · = q k d − , q k d − = −
1. This isexcluded by [14, Lemma 2.3]. Thus, q k q k = χ ( U (0 , ϕ ( k ) + ϕ ( k − ))) = 1. However,the additive subgroup of F q generated by the elements of the form ϕ ( k ) + ϕ ( k − ), k ∈ F q − F contains 1, so this would imply χ ( U (0 , (cid:3) Proof of Theorem 1.3 for simple Suzuki groups.
Proposition 2.2 covers the cases ofsimple Yetter-Drinfeld modules associated with classes of elements of odd order or oforder 4 by Lemma 3.3 and Proposition 3.8. We consider the class O of non-trivialinvolutions, represented by U (0 , B ( M ( O HU (0 , , χ )) = ∞ for every χ ∈ Irr C H ( U (0 , B ( M ( O , ρ )) = ∞ for every ρ ∈ Irr C G ( U (0 , (cid:3) The Ree groups F (2 h +1 )In this section p = 2, q = 2 h +1 , h ≥ G = F ( q ) a Ree group of type F .4.1. Collapsing racks.
In this Subsection we list the kthulhu and non-kthulhu con-jugacy classes in G , when it is simple. We consider unipotent, semisimple and mixedclasses separately.4.1.1. Unipotent classes.
We make use of the list of representatives of each conjugacyclass in [24, Table II] and notation from [23, 7, 13] and [16], with the understandingthat our q is q therein. Roots in Φ + are indicated as follows: an index j stands for ε j ; the symbol i ± j stands for ε i ± ε j and 1 ± ± ± ( ε ± ε ± ε ± ε ).For 1 ≤ i ≤
12, and t ∈ F q there are elements α i ( t ) ∈ U such that every u ∈ U can be written uniquely as an ordered product u = Q i =1 α i ( d i ) with d i ∈ F q for each i = 1 , . . . ,
12. Commutation rules between α i ( t ) and α j ( t ) are given in [25]: the case( i, j ) = (2 ,
3) contained a mistake pointed out in [19] and corrected in [16, Table 1].We will also make use of the subsets U i := { α i ( t ) | t ∈ F q } for 1 ≤ i ≤
12 and of thesubgroups U ≥ i := Q j = i U j E U . Lemma 4.1.
Any non-trivial unipotent conjugacy class in G , different from the onerepresented by u in [24, Table II] is of type D.Proof. Each representative of the 19 unipotent conjugacy classes of G in [24, Table II]is defined over F , i.e. it lies in the subgroup F (2). Conjugacy classes in Tits’ group F (2) ′ and in F (2) = Aut( F (2) ′ ) are studied in [7] and [13] respectively. By [7,Table 2] and [13, Table 1] every conjugacy class of F (2), apart from the one labeledby 2 A , is of type D. This class is represented by a non-trivial involution in Z ( U ), henceit is the one represented by u = α (1). (cid:3) It has been shown in [13, Proposition 4.1] that the class of u is not of type D. NextLemma deals with this class provided h > Lemma 4.2.
Let h > . The class O of u is of type F.Proof. Observe that u = α (1) = x (1) x (1). The Weyl group element s − s − liesin C W ( τ ) so it has a representative ˙ σ in F (2) ∩ N G ( T ). Hence ˙ σ ⊲ u = x (1) x (1) = α (1) ∈ O . For 1 ≤ j ≤
4, let ξ j be distinct elements of F q . We consider the involutions α ( ξ j ) = x − − − ( ξ h j ) x − ( ξ j ) ∈ U and we set r j := α ( ξ j ) ⊲ u = α ( ξ j ) ⊲ α (1) ∈ O . Thus, r i r j = α ( ξ i ) α (1) α ( ξ i ) α ( ξ j ) α (1) α ( ξ j ) = α ( ξ i ) α (1) α ( ξ i + ξ j ) α (1) α ( ξ j )where we have used Chevalley’s commutator formula. Hence r i r j = r j r i if and only if α ( ξ i + ξ j ) α (1) α ( ξ i + ξ j ) α (1) = α (1) α ( ξ i + ξ j ) α (1) α ( ξ j + ξ i )and this happens if and only if the commutator of α ( ξ i + ξ j ) and α (1) is an involution.Making use of the commutation relations in [25, 16] we deduce( α (1) α ( η ) α (1) α ( η )) = ( α ( η h +1 ) α ( η ) α ( η ) α ( η h +1 +1 ) α ( η h +1 +1 )) ∈ α ( η ) U ≥ so r i r j = r j r i whenever i = j . By direct computation: r i = α (1) α ( ξ h +1 i ) α ( ξ i ) α ( ξ i ) α ( ξ h +1 +1 i ) α ( u h +1 +1 ) ∈ U U U ≥ and V = U U U ≥ is a subgroup of U with V /U ≥ abelian. Let H = h r , r , r , r i .Then H ≤ V and so O Hr i ⊂ O Vr i ⊂ α (1) α ( ξ h +1 i ) U ≥ . Since ξ h +1 i = ξ h +1 j only if ξ i = ξ j , the classes O Hr i for i = j are disjoint and O is of type F. (cid:3) Mixed classes.
Lemma 4.3.
Let O be the class of an element x ∈ G with Jordan decomposition x = x s x u with x s , x u = 1 . If x u = 1 , then O is not kthulhu.Proof. Using strategy and notation from Remark 2.8 we see that Σ can only be of type A × ˜ A , A × ˜ A , or B . By looking at the action of F and the order of the centralisersgiven in [24, Table IV] we deduce that [ C G ( x s ) , C G ( x s )] F is either PSL ( q ), PSU ( q ),or B ( q ) and x u lies in there. By [4, Proposition 5.1] and Lemma 3.3 we see that if x u = 1, then O [ C G ( x s ) ,C G ( x s )] F x u is not kthulhu. (cid:3) Lemma 4.4.
Let O be the class of an element x ∈ G with Jordan decomposition x = x s x u with x s , x u = 1 . Then O is not kthulhu. ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 13
Proof.
By Lemma 4.3 we may assume that x u = 1. We argue as in the proof of [5,Lemma 3.2] to show that O is of type D. Since w = − id, there is representative˙ w ∈ N G ( ˜ T ) for every F -stable maximal torus ˜ T containing x s . Then, ˙ w ⊲ x = x − s x ′ u , where x s = x − s because x s = 1 and x ′ u is a non-trivial involution in K :=[ C G ( x − s ) , C G ( x − s )] F = [ C G ( x s ) , C G ( x s )] F . The latter is in turn isomorphic to PSL ( q ), PSU ( q ) or B ( q ). All non-trivial involutions are conjugate in these groups, so O Kx ′ u = O Kx u . By [4, Lemma 3.6(a)], [5, Lemma 2.9] and Remark 3.7 there is v ∈ O Kx ′ u such that( x u v ) = ( vx u ) . Thus, s := x − s v ∈ O G x and we have:( xs ) = ( sx ) , O h x,s i x ⊂ x s O Kx u , O h x,s i s ⊂ x − s O Kx u , x s O Kx u ∩ x − s O Kx u = ∅ . The claim follows. (cid:3)
Semisimple classes.
The conjugacy classes of maximal tori in G , the correspond-ing Weyl group elements, their orders and the order of their normalisers are listed in[24, §
3, Table III]. They are represented by T i , for 1 ≤ i ≤
11, with | T i | = d i as follows: d = ( q − , d = ( q − q + 1) ,d , = ( q − q ± p q + 1) , d = ( q + 1) ,d , = ( q ± p q + 1) , d = ( q + 1) ,d = ( q − q + 1) , d = ( q − p q + q − p q + 1) ,d = ( q + p q + q + p q + 1) . We denote by T i , for i ≤
11 the corresponding F -stable tori in G . Observe that( q − p q + 1)( q + p q + 1) = q + 1 , ( q + 1) = ( q − q + 1)( q + 1) , ( q − p q + q − p q + 1)( q + p q + q + p q + 1) = q − q + 1 , ( q − q + 1)( q + 1) = ( q + 1) . According to [19, § G contains subgroups isomorphic to B ( q ) × B ( q ) and SL ( q ) × SL ( q ) and all such isomorphic subgroups are conjugate. By looking at themaximal tori in B ( q ) and SL ( q ) we see that every torus T i for i ≤ M = M × M with either M ≃ M ≃ B ( q ) or M ≃ M ≃ SL ( q ). Thisinclusion induces a decomposition of T i into a product of 2 subtori T i,M j := T i ∩ M j for j = 1 , d i given above. If we write x s = ( x , x ) for an element in T i , we are referring to this decomposition. Also, weshall write T i,M j for the corresponding tori in Sp ( F q ) or SL ( F q ). Lemma 4.5.
Let O be the class of an element of order in G . Then, O is of type D. Proof.
According to [24] there is a unique conjugacy class of elements of order 3 in G .Recall that | F ′ (2) | = 2 · · ·
13, so the class O meets F ′ (2) ≤ F (2) ≤ F ( q ).Since the only non-trivial class of F ′ (2) which is not of type D consists of involutions[7], we have the statement. (cid:3) Lemma 4.6.
Assume h > . Let O = O G x s for x s = ( x , x ) ∈ T i for i ≤ . If x = 1 , x = 1 , then, O is of type C.Proof. We consider the inclusion of T i ≤ M × M with M j ≃ SL ( q ) or M j ≃ B ( q )for j = 1 ,
2. The statement follows from Lemma 2.4 and Remark 2.9. (cid:3)
Lemma 4.7.
Assume h > . Let O = O G x s for x s = ( x , x ) ∈ T i \ for i ≤ . If x = 1 or x = 1 , and | x s | 6 = 3 , then, O is of type C.Proof. We assume that x = 1, x = 1, the other case is treated the same way. If O G x s ∩ T i contains x ′ s = ( x ′ , x ′ ) with x ′ = 1, x ′ = 1 then we invoke Lemma 4.6. Hencewe assume from now on that O G x s ∩ T i = ( O G x s ∩ T i,M ) ∪ ( O G x s ∩ T i,M ). The inclusion x s ∈ M × M imply that x s ∈ T i for i ∈ { , , , } . Also, if x s = ( x , ∈ T , thenit lies in a split torus in SL ( q ) = PSL ( q ) or B ( q ) and the claim follows either from[3, Lemma 3.9] or Lemma 3.4.Thus, for the rest of the proof x s ∈ T i for i = 6 , ,
8, and | x s | 6 = 3. Observe that C G ( x s ) ⊃ T i,M × M , so C G ( x s ) is not abelian. The structure of the centralisers ofsemisimple elements in G is described in [24, Theorem 3.2] and most of them are tori.By inspection we see that x s is necessarily conjugate to some t j from [24, Table IV], with j ∈ { , , } and in these cases C G ( t i ) = T i,M M with | T i,M | ∈ { q + 1 , q ± √ q + 1 } ,so x is regular in M . Observe that M ≃ M ≃ PSL ( q ) when x s ∈ T and x s isconjugate to t and M ≃ M ≃ B ( q ) when x s ∈ T or T and in these cases x s isconjugate to t or t .The inclusion O N G ( T i ) x s ⊆ O G x s ∩ T i yields the inequality |O G x s ∩ T i | ≥ | N G ( T i ) / ( N G ( T i ) ∩ C G ( x s )) | = | N G ( T i ) / T i,M N M ( T i,M ) | = | N G ( T i ) / T i | / | N M ( T i,M ) / T i,M | . By [24, §
3, Table III] the quotient N G ( T i ) / T i has order 96 (for i = 6 ,
7) or 48 (for i = 8). In addition, | N M ( T i,M ) / T i,M | equals either 4 (for i = 6 ,
7) or 2 (for i = 8).In all cases, |O G x s ∩ T i | ≥ |O M × M x s ∩ T i | = |O M x s ∩ T i,M | cannot exceed the order of theWeyl group of Sp ( F q ) for i = 6 , SL ( F q ) for i = 8, so |O M × M x s ∩ T i | ≤ ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 15
This shows that in our situation: O M × M x s ∩ T i,M = O M × M x s ∩ T i ( O G x s ∩ T i = ( O G x s ∩ T i,M ) ∪ ( O G x s ∩ T i,M ) . The estimate |O M × M x s ∩ T i | ≤ x s by any x ′ s in O G x s ∩ T i,M or in O G x s ∩ T i,M . Therefore the elements in O G x s ∩ T i lie in at least 3distinct ( M × M )-orbits and each of these is contained either in M or in M . Withoutloss of generality we may assume that two of them are contained in O G x s ∩ M , say O M s and O M s ′ , with s, s ′ ∈ T i,M and by construction both regular in M . By Remark 2.3(1) there is g ∈ M such that r := g ⊲ s ′ C M ( s ′ ) = T i,M = C M ( s ) so [ r, s ] = 1. ByRemark 2.3 (2) the class is of type C with H = h r, s i . (cid:3) Lemma 4.8.
Assume h > . Let O = O G x s for x s ∈ T \ and | x s | 6 = 3 . Then O is oftype C.Proof. According to [19], G contains a subgroup isomorphic to SU ( q ), which containsa maximal torus of order d , so we may assume x s ∈ T ≤ SU ( q ). This torus consistsof elements in SU ( q ) which are conjugate to diagonal matrices in SL ( F q ) of the formdiag( x, x q , x − q ), for x q − q +1 = 1. An element of this form could be real in SU ( q )only if the set of its eigenvalues would coincide with its inverse set, which is impossiblein our case, so O SU ( q ) x s is not real. However, O is real in G by Remark 2.9. Since | x s | 6 = 3, it is not central in SU ( q ) so by Remark 2.3 (1), there is g ∈ SU ( q ) suchthat [ g ⊲ x − s , x s ] = 1. We take r = g ⊲ x − s , s = x s and H = h r, s i ≤ SU ( q ) so O Hs = O Hr and since | x s | is odd, O is of type C by Remark 2.3 (2). (cid:3) Lemma 4.9.
Let O = O G x s for x s ∈ T i \ and i = 10 , . If | x s | 6 = 13 , then O iskthulhu.Proof. The list of conjugacy classes of maximal subgroups in G is the main result in[19] (note that q there is our q ). We use Remark 2.5 and consider the intersectionof O with all maximal subgroups of G . Using coprimality of q − q + 1 with q ,( q ±
1) and ( q − q + 1), we verify that O can have non-empty intersection only with N G ( T i ) = T i ⋊ C , for i = 10 ,
11 and F ( q ) where q = 2 m +1 and (2 h + 1) / (2 m + 1)is prime. Since q ≡ q − q + 1 ,
12) = 1, whence
O ∩ N G ( T i ) ⊂ T i is a commuting set. Assume O ∩ F ( q ) = ∅ . Then, it is a unique semisimple classin F ( q ) by Remark 2.11 and by Remark 2.10, it has empty intersection with the F -stable maximal tori T ′ i of F ( q ) for i = 10 ,
11. Hence, we are in the same hypothesesas above, with q < q . We proceed inductively on the number of prime factors of 2 h +1.When 2 h +1 = 1 we have | F (2) | = 2 · · ·
13. Since 2 , +1 are coprimewith d and d and | x s | 6 = 13, our assumptions imply that O ∩ F (2) = ∅ . (cid:3) Lemma 4.10.
Let O = O G x s for | x s | = 13 . Then, O is not kthulhu.Proof. If x s lies in T i for i ≤
9, the result follows from Lemmata 4.6, 4.7 and 4.8.Assume x s ∈ T j for j = 10 ,
11. The torus T j is cyclic and has empty intersection withall maximal tori of different order, so h x s i is the only subgroup of order 13 in T j andall subgroups of this order are conjugate to it. Therefore O ∩ F (2) ′ = ∅ , as 13 dividesthe order of F (2) ′ By [7, Theorem II] the classes contained in
O ∩ F (2) ′ are of typeD, whence so is O . (cid:3) Nichols algebras over the Ree groups of type F . We are now in a positionto prove Theorem 1.3 for G . Proof of Theorem 1.3 for simple Ree groups of type F . Proposition 2.2 covers thecases of simple Yetter-Drinfeld modules associated with unipotent classes by Lemmata4.1 and 4.2; mixed classes by Lemmata 4.3 and 4.4 and semisimple classes representedin T i , for i ≤ T or T are covered by the combination ofRemark 2.9 and [10, Lemma 2.2]. We conclude by Remark 2.7. (cid:3) The Ree groups G (3 h +1 )In this section p = 3, q = 3 h +1 , h ≥ G = G ( q ) a Ree group of type G . We fixa basis { α, β } of Φ with α short. We recall the list of maximal subgroups of G up toconjugation from [17, Theorem C]:(1) B F ;(2) the centraliser of a non-trivial involution σ , isomorphic to h σ i × PSL ( q ) (for h > C × C (for h > N G ( T s α s β s α ) ≃ T s α s β s α ⋊ C , of order 6( q − √ q + 1) (for h > N G ( T s α s β s α s β s α ) ≃ T s α s β s α s β s α ⋊ C , of order 6( q + √ q + 1);(6) G (3 f +1 ) for (2 h + 1) / (2 f + 1) a prime.If h = 0, then G (3) ≃ PSL (8) ⋊ h ϕ i where ϕ acts as Fr on PSL (8) and, up toconjugation, we have the additional maximal subgroups PSL (8) and the normaliser ofthe Sylow 2-subgroup of upper triangular matrices in PSL (8), whose order is 2 · · Remark . We collect some properties of maximal subgroups of G and fix somenotation:(1) σ will denote the involution σ = α ∨ ( − β ∨ ( − C G ( σ ) ≃ h σ i × PSL ( q ). There is only one class of non-trivialinvolutions in G so by Remark 2.11 there is only one class of non-trivial invo-lutions in G . ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 17 (2) Assume h >
0. We provide a description of the normaliser of a subgroupisomorphic to C × C alternative to the one in [17]. There is a unique conju-gacy class of such subgroups, [30, p. 69]. A representative is given by h σ, σ ′ i where σ ′ the unique non-trivial involution in the (cyclic) maximal torus T ′ of PSL ( q ) ≤ C G ( σ ) of order q +12 . The subgroup h σ i × T ′ is a maximal torusof G of order q + 1 and there is only one class of tori of this order in G ,namely the one represented by T s α . We set T s α = h σ i × T ′ . By construc-tion T s α ≤ C G ( h σ, σ ′ i ) ≤ N G ( h σ, σ ′ i ). By [17, List C] we have N G ( h σ, σ ′ i ) ≃ ( C × D ( q +1) / ) ⋊ C , so T s α is normal of index 6 in N G ( h σ, σ ′ i ). Simplic-ity of G and maximality of N G ( h σ, σ ′ i ) give N G ( h σ, σ ′ i ) = N G ( T s α ). Also N G ( T s α ) ≤ N G ( T s α ) and since s α τ is a rotation on E we have | C W ( s α τ ) | = 6.Hence, by order reason, N G ( T s α ) = N G ( T s α ) and there exists an element ̺ of order 6 in N G ( h σ, σ ′ i ) such that ̺ ⊲ t = t − for every t ∈ T s α . Therefore h ̺ i ∩ T s α = 1 and comparing orders we have N G ( h σ, σ ′ i ) = T s α ⋊ h ̺ i .(3) Let ξ ∈ F × with ξ + ξ + 1 = 0. Then H := h (cid:0) ξ (cid:1) , (cid:0) ξ (cid:1) i ≤ PSL (8) ≤ G (3)is a representative of the conjugacy class of subgroups isomorphic to C × C .Its normaliser in G (3) is generated by ϕ and the subgroup of upper triangularunipotent matrices in PSL (8). Observe that ϕ permutes cyclically the threenon-trivial elements in H . Since all subgroups isomorphic to H are conjugatein G , it follows that also ̺ as in (2) permutes σ, σ ′ and σσ ′ cyclically.5.1. Collapsing racks.
In this Subsection we list the kthulhu and non-kthulhu con-jugacy classes in G , when it is simple. We consider unipotent, semisimple and mixedclasses separately.5.1.1. Unipotent classes.
We see from [18, Table 22.1.5] that G has 5 non-trivial unipo-tent classes: the regular one O reg , the subregular one O subreg , represented by x β (1) x α + β (1),the class labeled by ( ˜ A ) , represented by x α + β (1) x α +2 β (1), and the classes ˜ A and A , represented by x α (1) and x β (1), [26, Table 2, Example 4.3]. The last two classesare interchanged by F , hence they do not intersect G . The dimensions of the remain-ing ones are all distinct, hence they are all F -stable. Regular unipotent elements haveorder 9, all others have order 3. By [18, Table 22.1.5] the component group of C G ( u )is cyclic of order 2 if u ∈ O subreg and trivial if u ∈ O ( ˜ A ) . In the first case C G ( u )is parted into two F -conjugacy classes, so O subreg ∩ G is the union of two G -classes,whereas O ( ˜ A ) ∩ G is a single unipotent conjugacy class. In addition O ( ˜ A ) is theunique unipotent class of dimension 8 in G , hence it is real, and therefore O ( ˜ A ) ∩ G isagain so. The classes of ϕ ± in G (3) are not real, hence these elements lie in O subreg and represent the two unipotent classes in G contained therein. Lemma 5.2.
Let h > . If O ⊂ O reg , then O is of type D. Proof.
This is [2, Proposition 3.7]. (cid:3)
Lemma 5.3. If O ⊂ O subreg ∪ O ( ˜ A ) , then O is kthulhu.Proof. We apply the strategy described in Remark 2.5 and consider the intersection of O with a maximal subgroup M of G . Notice that if O subreg ∩ H = ∅ for some H ≤ G ,then |O G ϕ ∩ H | = |O G ϕ − ∩ H | 6 = 0. Let M = B F . The inclusion x α ( F q × ) x β ( F q × ) U α + β U α + β U α + β U α +2 β ⊂ O reg and F -invariance imply that O ∩ M ⊂ U α + β U α + β U α + β U α +2 β and all elements thereincommute, [12, III.6].Let M = C G ( σ ). It follows from [21, Lemmata 3.2 and 3.3, Corollary 3.4(ii)], that O ∩ M ⊂ PSL ( q ) is empty if O ⊂ O ( ˜ A ) and a single conjugacy class in PSL ( q ) if O is one of the two G -classes contained in O subreg . Furthermore, when it is non-empty,the intersection of O with a subgroup of PSL ( q ) is either a single conjugacy class orconsists of commuting elements [1, Lemma 3.5].Let M = N G ( h σ, σ ′ i ) = T s α ⋊ h ̺ i . Observe that ̺ ∈ C G ( ̺ ), the centraliser of aninvolution, and all elements of order 3 therein are not real. Hence, ̺ ∈ O subreg ∩ M =( O G ϕ ∩ M ) ∪ ( O G ϕ − ∩ M ). All elements of order 3 in M lie either in T s α ̺ or in T s α ̺ ,so ( O subreg ∪ O ( ˜ A ) ) ∩ M ⊂ T s α ̺ ∪ T s α ̺ . Setting M := T s α ⋊ h ̺ i we have O subreg ∩ M = O subreg ∩ M = ( O G ϕ ∩ M ) ∪ ( O G ϕ − ∩ M ) . Observe that ( q + 1) / T s α ≃ C × C × C ( q +1) / and h σ, σ ′ i and C ( q +1) / are characteristic in T s α . We claim that C T sα ( ̺ ) = 1. Indeed, if ̺ t = t̺ for some t ∈ T s α , then ̺ would commute with the components of t in C × C and C ( q +1) / .The first component is trivial by Remark 5.1(3), whereas the second one is trivial by[30, p. 75]. By Remark 2.6 (2), up to interchanging ̺ and ̺ − , we have O subreg ∩ M = O subreg ∩ M = T s α ̺ ∪ T s α ̺ = O M ̺ ∪ O M ̺ = O M̺ ∪ O M̺ . Hence O G ϕ ± ∩ M = O M̺ ± and O ( ˜ A ) ∩ M = ∅ . Let now H ≤ M be such that O G ϕ ± ∩ H = ∅ . Replacing if needed H by an M -conjugate of H containing ̺ we apply Remark 2.6(3) to deduce that O G ϕ ± ∩ H = O H̺ ± .Let w = s α s β s α or w = s α s β s α s β s α and let M be N G ( T w ) = T w ⋊ h g i for some g ∈ G with | g | = 6. This case is similar to the case of M = C G ( h σ, σ ′ i ), but simpler.Here we use [30, Theorem, part (4)] to show that C T w ( g ) = 1 and proceed as before.Let M = G ( q f +1 ). There are three conjugacy classes of elements of order 3 in M : the real one, which is M ∩ O ( ˜ A ) , and the two non-real ones, that are M ∩ O G ϕ ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 19 and M ∩ O G ϕ , so each intersection is a single conjugacy class in M and we proceedinductively on the number of prime factors of 2 h + 1.Finally, assume q = 3. Let M = PSL (8). The only class of elements of order 3 in M is real, hence M ∩ O ( ˜ A ) is this class in M and M ∩ O subreg = ∅ . The intersectionof O ( ˜ A ) with any subgroup of PSL (8) is either empty, a single conjugacy class, orconsists of commuting elements, [3, Proposition 4.2, Case 2].Let M be the normaliser of a Sylow 2-subgroup in G (3) ≃ PSL (8) ⋊ h ϕ i . Setting B := { ( α x α − ) | x ∈ F , α ∈ F × } we take M = B ⋊ h ϕ i . Clearly, O subreg ∩ M = ∅ and since ( | B | ,
3) = 1, we have the inclusion ( O subreg ∪ O ( ˜ A ) ) ∩ M ⊂ B ϕ ∪ B ϕ − .Now, C B ( ϕ ) = h ( ) i and O Mϕ ± ⊂ B ϕ ± , so |O Mϕ ± | = |O B ϕ ± | = | B | /
2. The sameargument shows that the orbits of the elements ( ) ϕ ± , whose order is 6, have | B | / B ϕ ± . Hence, O Mϕ ± = O G ϕ ± ∩ M and O ( ˜ A ) ∩ M = ∅ . Let now H ≤ M such that H ∩ O subreg = ∅ . Conjugating in M we can always make surethat ϕ ∈ H , so H = ( B ∩ H ) ⋊ h ϕ i . If ( ) H , then Remark 2.6 (2) shows that O Hϕ ± = ( B ∩ H ) ϕ ± = O G ϕ ± ∩ H . If, instead ( ) ∈ H , then we use a countingargument as above to see that O Hϕ ± = O G ϕ ± ∩ H . (cid:3) Mixed classes.
Lemma 5.4.
Let h > and let O = O G x where x has Jordan decomposition x = x s x u with x s , x u = 1 . Then O is of type D.Proof. Arguing as in Remark 2.8 we see that [ C G ( x s ) , C G ( x s )] is necessarily of type A × ˜ A . In this case, x s is a non-trivial involution and we take x s = σ = α ∨ ( − β ∨ ( − C G ( σ ) = h T , U ± ( α + β ) , U ± (3 α + β ) i , the roots α + β and 3 α + β are interchanged by θ and C G ( σ ) ≃ h σ i × PSL ( q ). Then x u can be chosen to be x α + β ( ǫ ) x α + β ( ǫ ) with ǫ = ±
1, and the two choices represent two distinct conjugacy classes of elements oforder 6 in G . By construction, there are no others. Let F × q = h ζ i . We consider theelements v, t, r, s and the subgroup H as follows: v := x α + β (1) x α +2 β (1) ∈ G ∩ C G ( U α + β U α + β ) ,t := α ∨ ( ζ h ) β ∨ ( ζ ) ∈ G ,r := v ⊲ x s x u = α ∨ ( − β ∨ ( − x α + β ( ǫ ) x α + β ( ǫ ) x α + β (1) x α +2 β (1) ∈ O ,s := t ⊲ x s x u = α ∨ ( − β ∨ ( − x α + β ( ǫζ − h ) x α + β ( ǫζ h +1 − ) ∈ O ,H := h r, s i ⊂ h U α + β U α + β U α + β U α +2 β , α ∨ ( − β ∨ ( − i . where we have used that in characteristic 3 there hold: [ x s , s ] = 1, [ U α + β U α +2 β , x u ] =1 and that x s ⊲ x α + β ( ξ ) x α +2 β ( η ) = x α + β ( − ξ ) x α +2 β ( − η ) for every ξ, η ∈ F . Hence, O Hs ⊂ α ∨ ( − β ∨ ( − x α + β ( ǫζ − h ) x α + β ( ǫζ h +1 − ) U α + β U α +2 β , O Hr ⊂ α ∨ ( − β ∨ ( − x α + β ( ǫ ) x α + β ( ǫ ) U α + β U α +2 β so O Hs ∩ O Hr = ∅ . In addition( rs ) = (cid:16) x α + β ( ǫ (1 + ζ − h )) x α + β ( ǫ (1 + ζ h +1 − )) x α + β ( − x α +2 β ( − (cid:17) = x α + β (2 ǫ (1 + ζ − h )) x α + β (2 ǫ (1 + ζ h +1 − )) x α + β (1) x α +2 β (1) , ( sr ) = (cid:16) x α + β ( ǫ (1 + ζ − h )) x α + β ( ǫ (1 + ζ h +1 − )) x α + β (1) x α +2 β (1) (cid:17) = x α + β (2 ǫ (1 + ζ − h )) x α + β (2 ǫ (1 + ζ h +1 − )) x α + β (2) x α +2 β (2) , so ( rs ) = ( sr ) and O is of type D. (cid:3) Semisimple classes.
There are four G -conjugacy classes of maximal tori repre-sented by T , T s α , T s α s β s α and T s α s β s α s β s α of order q ∓ q ∓ √ q + 1, respectively.Their orders are mutually coprime in all cases except from ( | T | , | T s α | ) = 2. We realise T and T s α in C G ( σ ) as the direct product of h σ i and a maximal torus in PSL ( q ), so T ∩ T s α is a cyclic group of order 2. The tori T , T s α s β s α and T s α s β s α s β s α are cyclic. Remark . Let t ∈ G be semisimple, and such that t = 1. Then [ C G ( t ) , C G ( t )] is notof type A × ˜ A , so it is trivial. In other words, t is regular in G and it lies in a uniquemaximal torus of G . Hence, if T w = T Fw is a maximal torus in G and t , t ∈ T w satisfy g ⊲ t = t for some g ∈ G and t = 1, then g ⊲ T w = g ⊲ C G ( t ) = C G ( t ) = T w . Hence, g ∈ N G ( T w ) and (cid:12)(cid:12) O G t ∩ T w (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) O N G ( T w ) t (cid:12)(cid:12)(cid:12) = | N G ( T w ) / T w | = | C W ( wτ ) | . Lemma 5.6.
Let h > . If x s ∈ T \ { } , then O G x s is of type D.Proof. There is only one class of non-trivial involutions in G so if | x s | = 2 its classis represented by σ ′ ∈ PSL ( q ). If, instead, | x s | >
2, we may assume that σ ∈ T so T ≤ C G ( σ ) ≃ h σ i × PSL ( q ) and x s is conjugate either to y or to σy for some y = 1 ina maximal torus for PSL ( q ). By [3, Corollary 3.5, Lemma 3.9] the racks O PSL ( q ) σ ′ and O h σ i× PSL ( q ) σy ≃ O PSL ( q ) y are of type D. We conclude by using Proposition 2.2 (2). (cid:3) Lemma 5.7.
Let h > . If | x s | = 7 , then O G x s is of type D.Proof. There is precisely one maximal torus up to conjugacy containing elements oforder 7 and by the structure of the tori, it contains exactly one subgroup of order 7.Thus, all subgroups of order 7 are conjugate in G . We recall that there is a subgroupisomorphic to PSL (8) in G (3) ≤ G . It contains a subgroup of order 7, namely ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 21 its split torus, which intersects O G x s . We conclude by invoking [3, Lemma 3.9] andProposition 2.2 (2). (cid:3) Lemma 5.8.
Let w = s α s β s α or s α s β s α s β s α and let x s ∈ T w \ { } . If | x s | 6 = 7 , then O G x s is kthulhu.Proof. We use the strategy from Remark 2.5. The order of x s divides q − q + 1 so it isodd and coprime with q ± q . Hence, the only maximal subgroups intersecting O G x s are N G ( T w ) and G (3 f +1 ) with (2 h + 1) / (2 f + 1) a prime number. In the first case O G x s ∩ M consists of commuting elements; in the second case it is a single conjugacyclass ( s is semisimple). In addition, Remark 2.10 shows that s cannot lie in a torusof order 3 f +1 ±
1. Working inductively on the number of prime factors of 2 h + 1, wehave the statement. (cid:3) Lemma 5.9.
Let h > . If x s ∈ T s α \ { } and | x s | 6 = 2 , then O G x s is of type C.Proof. We recall from Remarks 5.1 (2) and 5.5 that T s α = h σ i × T ′ ≤ h σ i × PSL ( q )and that x s is regular. Observe that s α τ is a rotation on E so C W ( s α τ ) is cyclic oforder 6. By Remark 5.5 we have (cid:12)(cid:12) O G x s ∩ T s α (cid:12)(cid:12) = 6 > (cid:12)(cid:12) { x s , x − s } (cid:12)(cid:12) = (cid:12)(cid:12) O h σ i× PSL ( q ) x s ∩ T s α (cid:12)(cid:12) . Hence, there is s ∈ ( O G x s ∩ T s α ) \ O h σ i× PSL ( q ) x s . By Remark 2.3 (1) there is g ∈h σ i × PSL ( q ) such that r := g ⊲ x s C G ( x s ) = T s α = C G ( s ). If we set H := h r, s i ≤h σ i × PSL ( q ), then O Hs = O Hr . If | x s | is odd the class is of type C by Remark 2.3(2). If | x s | is even, then we decompose s = s e s o into its 2-part and 2-regular part, so s o ∈ H , all prime factors of its order are ≥ s o is regular in G by Remark 5.5.We thus have (cid:12)(cid:12) O Hr (cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) O h s o i r (cid:12)(cid:12)(cid:12) ≥ O G x s is of type C. (cid:3) Nichols algebras over the Ree groups of type G . In this subsection weconsider Nichols algebras attached to simple Yetter-Drinfeld modules M ( O , ρ ) for O a kthulhu class in G and we prove Theorem 1.3 for simple Ree groups of type G for h > Proposition 5.10.
Assume h > . Let g ∈ G be a non-trivial unipotent element.Then dim B ( O G g , ρ ) = ∞ for every irreducible representation ρ of C G ( g ) .Proof. If g ∈ O reg , this follows from Proposition 2.2 (1), Lemma 5.2 and [6, Theorem3.6]. If g ∈ O ( ˜ A ) , then O G g is real and of odd order and the claim follows from [10,Lemma 2.2]. Assume now g = ϕ ∈ O G ϕ , the case of g = ϕ − is treated similarly.We will show that dim B ( O G (3) ϕ , ρ ) = ∞ for every irreducible representation ρ of C G (3) ( ϕ ) and apply [6, Lemma 3.2]. Recall that G (3) = PSL (8) ⋊ h ϕ i , so setting N := PSL (8) we see that O G (3) ϕ ± ⊂ N ϕ ± . The elements of order 3 in N are real,so the real class of elements of order 3 in G (3) is all contained therein. Thus, anelement of order 3 lies in O G (3) ϕ ± if and only if it lies in N ϕ ± . We proceed as outlinedin Subsection 2.2, from which we adopt notation. We consider C G (3) ( ϕ ) ≃ SL (2) × h ϕ i ≃ S × h ϕ i , A := A × h ϕ i ≤ C G (3) ( ϕ ) . The intersection O G (3) ϕ ∩ C G (3) ( ϕ ) = { ϕ, (123) ϕ, (132) ϕ } = O G (3) ϕ ∩ A is a commutingset and we put x = ϕ , x = (123) ϕ , x = (132) ϕ , so x x x = 1. We claim that thereis g ∈ G (3) such that g j ⊲ x i = x i + j mod 3 for j ∈ Z , i = 0 , ,
2. Let z ∈ G (3) be suchthat z⊲x = x , so [ z⊲x , x ] = 1, whence z − ⊲x ∈ O G (3) ϕ ∩ C G (3) ( ϕ ) and z − ⊲x = x .If z − ⊲ x = x , then z ⊲ x = x and so z ⊲ x = z ⊲ ( x x ) − = ( x x ) − = x and weput g = z . If, instead, z − ⊲ x = x , then z ⊲ x = x and we consider y ∈ G (3) suchthat y ⊲ x = x and repeat the argument. Then either g = y will do, or g = yz will do.Let ρ be an irreducible representation of C G (3) ( x ) and let C v be any line stabilisedby A . Let ρ ( x i ) v = ω i v for i = 0 , ,
2. We have ω i = 1 and ω ω ω = 1. By (2.2) thebraided vector subspace M A = span C { g i ⊗ v, i = 0 , , } of M ( O G (3) ϕ , ρ ) has braiding c (( g i ⊗ v ) ⊗ ( g j ⊗ v )) = q ij ( g j ⊗ v ) ⊗ ( g i ⊗ v ) , q ij v = ρ ( g − j + i ⊲ x ) v = ω i − j mod3 v. This gives q q = q q = q q = ω ω = ω , ω ii = ω , i = 0 , , . If ω = 1, then dim B ( M ( O G (3) ϕ , ρ )) = ∞ by [10, Remark 1.1]. If, instead, ω is aprimitive third root of 1, then the generalized Dynkin diagram of M A is connectedand does not occur in [15, Table 2]. This implies dim B ( M A ) = ∞ , whence againdim B ( M ( O G (3) ϕ , ρ )) = ∞ . (cid:3) Proof of Theorem 1.3 for simple Ree groups of type G . Proposition 2.2 covers the casesof simple Yetter-Drinfeld modules associated with unipotent classes by Proposition5.10; mixed classes by Lemma 5.4 and semisimple classes represented in T w for w = 1or s α by Lemmata 5.6, 5.7 and 5.9. The simple Yetter-Drinfeld modules associated withclasses represented in T w for w = s α s β s α or s α s β s α s β s α are covered by the combinationof Remark 2.9 and [10, Lemma 2.2]. We conclude by Remark 2.7. (cid:3) Remark . The non-simple groups F (2) and G (3) have simple derived subgroup G ′ . The group F (2) ′ has been dealt with in [7, 13], whereas G (3) ′ ≃ PSL (8)has been treated in [14]. In both cases, dim B ( V ) = ∞ for every V ∈ G ′ G ′ Y D . Thegroup B (2) has order 20 and all its classes are kthulhu, see Remark 3.5. Its derived ICHOLS ALGEBRAS OVER SUZUKI AND REE GROUPS 23 subgroup is cyclic of order 5, so it is simple and abelian, and there do exist finite-dimensional pointed Hopf algebras over C , for example the Taft algebras in [29], seealso [9, Theorem 1.3]. 6. Acknowledgements
The authors could benefit from several discussions with N. Andruskiewitsch and G.A. Garc´ıa, and we are grateful to them. In particular, the content of Remark 2.3 (1)was pointed out to G.C. by N. Andruskiewitsch.
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G. C.: Dipartimento di Matematica Tullio Levi-Civita, Universit`a degli Studi di Padova,via Trieste 63, 35121 Padova, Italia
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