aa r X i v : . [ m a t h . QA ] N ov ON NICHOLS ALGEBRAS OF DIAGONAL TYPE
IV ´AN ANGIONO
Abstract.
We give an explicit and essentially minimal list of defining relations of aNichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by An-druskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebraover an algebraically closed field of characteristic zero is generated as an algebra by itsgroup-like and skew-primitive elements. As an application of our main result, we provethe conjecture when the group of group-like elements is abelian.
Introduction Let k be an algebraically closed field of characteristic zero and let θ be a naturalnumber. Let q = ( q ij ) ≤ i,j ≤ θ be a matrix with invertible entries on k and let V be avector space of dimension θ . The Nichols algebra associated to q is a graded connectedHopf algebra B ( V ) = ⊕ n ≥ B n ( V ) with many favorable properties. It plays a fundamentalrole in the classification of finite-dimensional (or finite growth) pointed Hopf algebras.Precisely, a basic question in the classification Program [AS1] is the following: Question 1. [An, Question 5.9] : Given ( V, q ) , determine if the associated Nichols algebra B ( V ) is finite-dimensional. In such case, compute the dimension of B ( V ) and give apresentation by generators and relations. The first part of this question has been answered by Heckenberger [H3], who obtainedthe list of all matrices q whose associated Nichols algebra has a finite root system. Roughly,this list contains three classes of matrices: • Standard matrices [AA]: they are associated with finite Weyl groups. Their rootsystems coincide with root systems of finite Cartan matrices. This family includesproperly the so-called braidings of Cartan type, in particular the matrices relatedwith the positive part of the small quantum groups. • Matrices of super type [AAY], related with the positive part of quantized envelop-ing algebras of contragradient Lie superalgebras. Their root systems become fromthe corresponding Lie superalgebras. • A finite list of exceptional matrices, whose associated diagram has connected com-ponents with at most 7 vertices, and the scalars defining these braidings are rootsof unity of low order.There are several answers to the second part of Question 1 under particular assumptions: ⊲ [L] for the positive part of quantized enveloping algebras of semisimple Lie algebrasand small quantum groups, using the full representation theory of quantum groups; ⊲ [AS2] for braidings of Cartan type; ⊲ [A1] for braidings of standard type; ⊲ [Y] for the positive part of quantized enveloping algebras of contragradient Liesuperalgebras; Mathematics Subject Classification. ⊲ [AAY] for braidings of super type; ⊲ [H1], giving a general form of relations for matrices of rank two; ⊲ [He] for some examples of rank two matrices, but giving explicit relations.In [A2] we gave general formulae for defining relations of Nichols algebras of diagonaltype, see Theorem 1.25 below. The expression of those relations and the proof that theygenerate the defining ideal are independent of Heckenberger’s classification; they rely inKharchenko’s and Rosso’s PBW bases [Kh, R] and a detailed study of convex orders ingeneralized root systems [A2], through the classification of coideal subalgebras [HS]. Inthis paper we refine the main result of [A2] and prove: Theorem 1.
A minimal set of relations of B ( V ) is obtained by considering relations ofthe following type: (1) Quantum Serre relations, and powers of generators x i corresponding to non-Cartanvertices; they are needed to introduce Lusztig’s isomorphisms at the level of doublesof tensor algebras. (2) Relations in the image of the previous ones by the Lusztig’s isomorphisms, andcorrespond to relations (23) in Theorem 1.25. (3)
Relations that guarantee that the ideal generated by the previous relations is abraided biideal: they appear in the coproduct of relations of the item (2) in thetensor algebra T ( V ) . (4) Powers of root vectors (generators of the PBW basis) corresponding to roots in theorbit of Cartan vertices.
See Theorem 3.1 for a complete and explicit set of relations. In this set we distinguishrelations appearing in [A2] for standard braidings, and relations in [Y] related with braid-ings of super type. There exists also a large list of new relations, related with the set ofexceptional braidings or with braidings of super type evaluated in roots of unity of smallorder. The knowledge of the explicit relations of a Nichols algebra has several potential ap-plications to the theory of pointed Hopf algebras, that we discuss now: • One of the basic question in the Lifting Method [AS1, AS3] for the classificationof Hopf algebras is the following:
Conjecture 1. [AS2, Conjecture 1.4]
Let Γ be a finite group and k an algebraically closedfield of characteristic 0. If H is a finite-dimensional pointed Hopf algebra over k such that G ( H ) = Γ , then H is generated as an algebra by Γ and its skew-primitive elements. This question was answered in [AS4] for braidings of Cartan type under some mild con-ditions. This result was extended to the case of standard braidings in [AGI]. In Section 4we obtain as a consequence of Theorem 3.1:
Theorem 2.
Let H be a finite dimensional pointed Hopf algebra over an abelian group Γ .Then H is generated as an algebra by Γ and its skew primitive elements. That is, we answer positively Conjecture 1 in a general context: when G ( H ) is anyabelian group. This Theorem is also applied to the known cases of finite-dimensionalNichols algebras over non-abelian groups • Another crucial step of the Lifting Method is to obtain all deformations of thepointed Hopf algebras B ( V ) k Γ; that is, all the pointed Hopf algebras such thattheir associated coradically graded algebras are isomorphic to B ( V ) k Γ.This problem was solved for Γ abelian in [AS4] – under the restriction that the order isnot divisible by 2,3,5,7. We believe that the explicit presentation in this paper would besubstantial to solve the question for any abelian group.
N NICHOLS ALGEBRAS OF DIAGONAL TYPE 3 • The explicit relations would be also useful in the study of various elements in therepresentation theory of pointed Hopf algebras. In this direction, the theory ofNichols algebras of diagonal type provides an uniform approach to the study ofquantum groups and quantum supergroups. The plan of this paper is the following. We introduce the notion of Nichols algebras inSection 1. We give a PBW basis of any Nichols algebra and some properties of this basisfollowing [Kh, R]. Next we recall the notions of Weyl groupoid and its associated rootsystem following [CH2, HS], and make a connection with the theory of Nichols algebrasof diagonal type. We present the needed material from [A2], in particular Theorem 1.25,a key result for our purposes.Section 2 is devoted to Lusztig’s isomorphisms in the general context of braidings ofdiagonal type [H4], extending analogous isomorphisms from [L].In Section 3 we give the mentioned presentation by generators and relations, based inthe classification of braidings of diagonal type with finite root system [H3]. The strategy ofproof consists first to define Lusztig isomorphisms for the Drinfeld doubles of the braidedHopf algebras U + obtained by quotient by the relations in Theorem 1, except the groupin (4). This quotient is analogous to the algebra U + q ( g ); the Drinfeld double u q ( g ) of theNichols algebra is a quotient of the previous algebra, as it was considered by Lusztig andAndruskiewitsch-Schneider. We denote these two algebras by U + and u + , respectively, so u + = B ( V ). The existence of the Lusztig’s isomorphisms prove that the PBW generatorscorresponding to the algebras U + and their quotients u + are the same, but the heights ofsome generators are not the order of the associated scalar in U + . Therefore we obtain u + after to quotient U + by some powers of root vectors as in (4).Theorem 3.1 extends the presentation obtained in [A1] for standard braidings, and in[AAY] for braidings of super type, and gives a new proof in the case of braidings of Cartantype, in particular quantized enveloping algebras U q ( g ) and small quantum groups u q ( g ).Finally, Section 4 is devoted to the proof of Theorem 2. We prove first that any finitedimensional braided graded Hopf algebra of diagonal type S = ⊕ n ≥ S n , S = k , S ∼ = V, generated as an algebra by V is isomorphic to the Nichols algebra B ( V ); this result extends[AS4, Thm. 5.5], [AGI, Thm. 2.5], but the proof follows the same scheme. Acknowledges.
This work is part of the author’s PhD Thesis. I want to thank speciallyto my advisor Nicol´as Andruskiewitsch for his inspiring guidance, patience and supervisionduring these years. I want to thank also to my family for all their support, and to Antonelafor all her love. 1.
Preliminaries
In this Section we recall results from different works needed in the sequel. First weconsider the existence of PBW bases for Nichols algebras of diagonal type [Kh, R], andthe rich combinatoric related to them. Next we recall the definitions of Weyl groupoid,the associated root systems and some properties thereof [HS, HY]. We close this Sectionstating a general presentation of Nichols algebras coming from [A2].1.1.
Lyndon words and PBW bases for Nichols algebras of diagonal type.
Tobegin with, we recall the definition of a Nichols algebras and show a characterization inthe case of a diagonal braiding.
IV ´AN ANGIONO
Definition 1.1. [AS3] Given V ∈ HH YD , the tensor algebra T ( V ) admits a unique struc-ture of graded braided Hopf algebra in HH YD such that V ⊆ P ( V ). Consider the family S of all the homogeneous Hopf ideals I ⊆ T ( V ) such that • I is generated by homogeneous elements of degree ≥ • I is a Yetter-Drinfeld submodule of T ( V ).The Nichols algebra B ( V ) associated to V is the quotient of T ( V ) by the biggest ideal I ( V ) of S . Let ( V, c ) be a braided vector space of diagonal type such that q ij = q ji for any i, j . LetΓ = Z θ , and α , . . . , α θ be the canonical basis. We set the characters χ , . . . , χ θ of Γ givenby χ j ( α i ) = q ij , 1 ≤ i, j ≤ θ .Consider V as a Yetter-Drinfeld module over k Γ such that x i ∈ V χ i α i . In this context wecan characterize the Nichols algebra as a quotient that admits a certain non-degeneratebilinear form. We use Swedler’s notation for the coproduct in T ( V ): ∆( x ) = x ⊗ x ,where we omit the summation symbol. Proposition 1.2. [L, Prop. 1.2.3] , [AS3, Prop. 2.10] There exists a unique bilinear form ( ·|· ) : T ( V ) × T ( V ) → k such that (1 |
1) = 1 , and: ( x i | x j ) = δ ij , for any i, j ;(1) ( x | yy ′ ) = ( x | y )( x | y ′ ) , for any x, y, y ′ ∈ T ( V );(2) ( xx ′ | y ) = ( x | y )( x ′ | y ) , for any x, x ′ , y ∈ T ( V ) . (3) This is a symmetric form, for which we have: (4) ( x | y ) = 0 , for any x ∈ T ( V ) g , y ∈ T ( V ) h , g, h ∈ Γ , g = h. The radical of this form { x ∈ T ( V ) : ( x | y ) = 0 , ∀ y ∈ T ( V ) } coincides with I ( V ) , so ( ·|· ) induces a non-degenerate bilinear form on B ( V ) = T ( V ) /I ( V ) , denoted also by ( ·|· ) . (cid:3) Therefore I ( V ) is a Z θ -homogeneous ideal, and then B ( V ) is Z θ -graded.Let A be an algebra, P, S ⊂ A and h : S N ∪ {∞} . We fix a linear order < on S . B ( P, S, <, h ) will denote the set (cid:8) p s e . . . s e t t : t ∈ N , s > · · · > s t , s i ∈ S, < e i < h ( s i ) , p ∈ P (cid:9) . If B ( P, S, <, h ) is a k -linear basis, we say that ( P, S, <, h ) is a set of
PBW generators ,whose height is h , and B ( P, S, <, h ) is a
PBW basis of A .We will describe a particular PBW basis for any graded braided Hopf algebra B = ⊕ n ∈ N B n generated by B ∼ = V as an algebra, where V is a braided vector space; we willfollow the results in [Kh].Fix θ ∈ N , and a set X = { x , . . . , x θ } . Let X be the set of words with letters in X and consider the lexicographical order on X . Definition 1.3.
An element u ∈ X , u = 1 is a Lyndon word if for any decomposition u = vw , v, w ∈ X − { } , we have u < w . We will denote the set of all Lyndon words by L . Remark . • Each Lyndon word begin with its smallest letter. • Each u ∈ X − X is a Lyndon word if and only if for each decomposition u = u u with u , u ∈ X \
1, we have u u = u < u u . • If u , u ∈ L and u < u , then u u ∈ L . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 5
A basic Lyndon’s result says that any word u ∈ X admits a unique decomposition asnon-increasing product of Lyndon words:(5) u = l l . . . l r , l i ∈ L, l r ≤ · · · ≤ l . It is called the
Lyndon decomposition of u ∈ X , and the l i ∈ L in (5) are called the Lyndonletters of u .Another characterization of Lyndon words is the following: Lemma 1.5. [Kh, p.6]
Let u ∈ X − X . Then u ∈ L if and only if there exist u , u ∈ L such that u < u and u = u u . (cid:3) Definition 1.6.
For each u ∈ L − X , the Shirshov decomposition of u is the decomposition u = u u , u , u ∈ L , such that u is the smallest end of u between all the possibledecompositions with these conditions.Given a finite-dimensional vector space V , fix a basis X = { x , . . . , x θ } of V ; we canidentify k X with T ( V ). In what follows we consider two graduations for the algebra T ( V ):the usual N -graduation T ( V ) = ⊕ n ≥ T n ( V ), and Z θ -graduation of T ( V ), determined bythe condition deg x i = α i , 1 ≤ i ≤ θ , where { α , . . . , α θ } is the canonical basis of Z θ .Consider a braiding c for V . The braided bracket of x, y ∈ T ( V ) is defined by(6) [ x, y ] c := multiplication ◦ (id − c ) ( x ⊗ y ) . Assume that (
V, c ) is of diagonal type, and let χ : Z θ × Z θ → k × be the bicharacterdetermined by the condition(7) χ ( α i , α j ) = q ij , for each pair 1 ≤ i, j ≤ θ. Then, for each pair of Z θ -homogeneous elements u, v ∈ X ,(8) c ( u ⊗ v ) = q u,v v ⊗ u, q u,v = χ (deg u, deg v ) ∈ k × . In such case, the braided bracket satisfies a “braided Jacobi identity” and determinesskew-derivations as follows:[[ u, v ] c , w ] c = [ u, [ v, w ] c ] c − χ ( α, β ) v [ u, w ] c + χ ( β, γ ) [ u, w ] c v, (9) [ u, v w ] c = [ u, v ] c w + χ ( α, β ) v [ u, w ] c , (10) [ u v, w ] c = χ ( β, γ ) [ u, w ] c v + u [ v, w ] c , (11)where u, v, w ∈ T ( V ) are homogeneous of degree α, β, γ ∈ N θ , respectively.Using the previous decompositions, we can define the k -linear endomorphism [ − ] c of k X as follows:[ u ] c := u, if u = 1 or u ∈ X ;[[ v ] c , [ w ] c ] c , if u ∈ L, ℓ ( u ) > , u = vw is the Shirshov decomposition;[ u ] c . . . [ u t ] c , if u ∈ X − L and its Lyndon decomposition is u = u . . . u t . Definition 1.7.
The hyperletter corresponding to l ∈ L is [ l ] c . An hyperword is aword whose letters are hyperletters, and a monotone hyperword is an hyperword W =[ u ] k c . . . [ u m ] k m c such that u > · · · > u m . Remark . For any u ∈ L , [ u ] c is a Z [ q ij ]-linear combination of words with the same Z θ -graduation than u , such that [ u ] c ∈ u + k X ℓ ( u ) >u . Theorem 1.9. [R, Thm. 10]
Let u, v ∈ L , u < v . Then [[ u ] c , [ v ] c ] c is a Z [ q ij ] -linearcombination of monotone hyperwords [ l ] c . . . [ l r ] c , l i ∈ L , such that the correspondinghyperletters satisfy v > l i ≥ uv . Moreover, [ uv ] c appears in such combination with non-zero coefficient and each hyperword has the same Z θ -graduation than uv . (cid:3) IV ´AN ANGIONO
We consider the polynomials in Z [ t ]:( n ) t := 1 + t + · · · + t n − , ( n ) t ! = (1) t (2) t · · · ( n ) t , n ∈ N . The q-combinatorial numbers are defined as the following quotient: (cid:18) ni (cid:19) t := ( n ) t !( n − i ) t !( i ) t ! , ≤ i ≤ n. It follows inductively that (cid:0) ni (cid:1) t ∈ Z [ t ], for all n and all i ∈ { , , . . . , n } . Therefore (cid:0) ni (cid:1) q will denote the evaluation of (cid:0) ni (cid:1) t for t = q , where q ∈ k .The comultiplication of hyperwords in T ( V ) has a nice expression, as we can see in thefollowing result. Lemma 1.10. [R, Thm.13]
Let u , . . . , u r , v ∈ L , with v < u r ≤ · · · ≤ u . Then, ∆ ([ u ] c · · · [ u r ] c [ v ] mc ) = 1 ⊗ [ u ] c · · · [ u r ] c [ v ] mc + m X i =0 (cid:18) mi (cid:19) q v,v [ u ] c . . . [ u r ] c [ v ] ic ⊗ [ v ] m − ic + X l ≥···≥ l p >v, l i ∈ L ≤ j ≤ m x ( j ) l ,...,l p ⊗ [ l ] c · · · [ l p ] c [ v ] jc ; Where x ( j ) l ,...,l p is Z θ -homogeneous, and deg( x ( j ) l ,...,l p ) + deg( l . . . l p v j ) = deg( u . . . u r v m ) . (cid:3) Another useful result from [R] is the following one.
Lemma 1.11.
For each l ∈ L let W l be the subspace of T ( V ) generated by (12) [ l ] c [ l ] c · · · [ l k ] c , k ∈ N , l i ∈ L, l ≥ . . . ≥ l k ≥ l. Then W l is a left coideal subalgebra of T ( V ) . (cid:3) We consider another order in X as in [U]; it was implicitly used in [Kh]. Let u, v ∈ X .We say that u ≻ v if ℓ ( u ) < ℓ ( v ), or ℓ ( u ) = ℓ ( v ) and u > v for the lexicographical order.This order ≻ is total, and it is called the deg-lex order .The empty word 1 is the maximal element for ≻ , and this order is invariant by left andright multiplication.Let I be an ideal of T ( V ), and R = T ( V ) /I . Let π : T ( V ) → R be the canonicalprojection. We set: G I := { u ∈ X : u / ∈ k X ≻ u + I } . Note that if u ∈ G I and u = vw , then v, w ∈ G I . Therefore each u ∈ G I is a non-increasingproduct of Lyndon words of G I . Proposition 1.12. [Kh, R]
The set π ( G I ) is a basis of R . (cid:3) In what follows I will denote a Hopf ideal. Consider the set S I := G I ∩ L . Define h I : S I → { , , . . . } ∪ {∞} according to the following condition:(13) h I ( u ) := min (cid:8) t ∈ N : u t ∈ k X ≻ u t + I (cid:9) . We recall the following result and its corollaries following [Kh].
Theorem 1.13. B ′ I := B ( { I } , [ S I ] c + I, <, h I ) is a PBW basis of H = T ( V ) /I . (cid:3) Corollary 1.14.
A word u belongs to G I if and only if the corresponding hyperletter [ u ] c isnot a linear combination, modulo I , of greater hyperwords [ w ] c , w ≻ u , whose hyperlettersare in S I . (cid:3) N NICHOLS ALGEBRAS OF DIAGONAL TYPE 7
Corollary 1.15. If v ∈ S I is such that h I ( v ) < ∞ , then q v,v is a root of unity. Moreover,if ord q v,v = h , then h I ( v ) = h , and [ v ] h is a linear combination of hyperwords [ w ] c , w ≻ v h . (cid:3) Weyl groupoids and root systems.
We follow the notation in [CH1]. Fix a non-empty set X , and a finite set I . For each i ∈ I we fix a bijective function r i : X → X , andfor each X ∈ X a generalized Cartan matrix A X = ( a Xij ) i,j ∈ I . Let ( α i ) i ∈ I be the canonicalbasis of Z I . Definition 1.16. [HY, CH1] The 4-uple C := C ( I, X , ( r i ) i ∈ I , ( A X ) X ∈ X ) is a Cartan scheme if it holds: • for any i ∈ I , r i = id , and • for any X ∈ X and any pair i, j ∈ I : a Xij = a r i ( X ) ij .For each i ∈ I and each X ∈ X we denote by s Xi the automorphism of Z I given by s Xi ( α j ) = α j − a Xij α i , j ∈ I. The
Weyl groupoid of C is the groupoid W ( C ) whose set of objects is X and whose mor-phisms are generated by s Xi , considered as elements s Xi ∈ Hom(
X, r i ( X )), i ∈ I , X ∈ X .In general we denote W ( C ) simply by W , and for each X ∈ X :Hom( W , X ) := ∪ Y ∈ X Hom(
Y, X ) , (14) ∆ X re := { w ( α i ) : i ∈ I, w ∈ Hom( W , X ) } . (15)∆ X re is the set of real roots of X . Each w ∈ Hom( W , X ) is written as a product s X i s X i · · · s X m +1 i m , where X j = r i j − · · · r i ( X ), i ≥
2. We denote it by w = id X s i · · · s i m :it means that w ∈ Hom( W , X ), because each X j ∈ X is univocally determined by thiscondition. The length of w is defined by ℓ ( w ) = min { n ∈ N : ∃ i , . . . , i n ∈ I such that w = id X s i · · · s i n } . We assume that W is connected : that is, Hom( Y, X ) = ∅ , for any pair X, Y ∈ X . Definition 1.17. [HY, CH1] Given a Cartan scheme C , consider for each X ∈ X a set∆ X ⊂ Z I . We say that R := R ( C , (∆ X ) X ∈ X ) is a root system of type C if(1) for any X ∈ X , ∆ X = (∆ X ∩ N I ) ∪ − (∆ X ∩ N I ),(2) for any i ∈ I and any X ∈ X , ∆ X ∩ Z α i = {± α i } ,(3) for any i ∈ I and any X ∈ X , s Xi (∆ X ) = ∆ r i ( X ) ,(4) if m Xij := | ∆ X ∩ ( N α i + N α j ) | , then ( r i r j ) m Xij ( X ) = X for any pair i = j ∈ I andany X ∈ X .∆ X + := ∆ X ∩ N I is called the set of positive roots , and ∆ X − := − ∆ X + is the set of negativeroots . Remark . From (2) and (3) we deduce that ∆
X re ⊂ ∆ X , for any X ∈ X .For each positive root α = P i n i α i , the support of α is the setsupp α := { i : 1 ≤ i ≤ θ, n i = 0 } . By (3) we have that w (∆ X ) = ∆ Y for any w ∈ Hom(
Y, X ). We say that R is finite if∆ X is finite for some X ∈ X . By [CH1, Lemma 2.11], it is equivalent to the fact that allthe sets ∆ X are finite, X ∈ X , and also that W is finite. Moreover, for any pair i = j ∈ I and any X ∈ X , we have that kα i + α j ∈ ∆ X if and only if 0 ≤ k ≤ − a Xij . Therefore,(16) a Xij = − max { k ∈ N : kα i + α j ∈ ∆ X } . IV ´AN ANGIONO
A fundamental result involving root systems is the following one:
Theorem 1.19. [CH2, Thm. 2.10]
For every α ∈ ∆ X + \ { α i : i = 1 , . . . θ } , there exist β, γ ∈ ∆ X + such that α = β + γ . (cid:3) We give now some results about real roots and the length of elements.
Lemma 1.20. [HY, Cor. 3]
Let m ∈ N , X, Y ∈ X , i , . . . , i m , j ∈ I , w = id X s i · · · s i m ∈ Hom(
Y, X ) , where ℓ ( w ) = m . Then, • ℓ ( ws j ) = m + 1 if and only if w ( α j ) ∈ ∆ X + , • ℓ ( ws j ) = m − if and only if w ( α j ) ∈ ∆ X − . (cid:3) Proposition 1.21. [CH1, Prop. 2.12]
For any w = id X s i · · · s i m ∈ W such that ℓ ( w ) = m , the roots β j = s i · · · s i j − ( α i j ) ∈ ∆ X are positive and all different. If R is finite and w is an element of maximal length, say N , then ∆ X + = { β j | ≤ j ≤ N } . The roots β , ,..., β N are pairwise different, and hence for each α ∈ ∆ X + there exist i , . . . , i k , j ∈ I such that α = s i k · · · s i ( α j ) . (cid:3) Call ∆ V + the set of degrees of a PBW basis of B ( V ), counted with their multiplicities,as in [H2]. It does not depend on the PBW basis, see [H2, AA]. We can attach a Cartanscheme C , a Weyl groupoid W and a root system R , see [HS, Thms. 6.2, 6.9]. To do this,define for each 1 ≤ i = j ≤ θ ,(17) − a ij := min { n ∈ N : ( n + 1) q ii (1 − q nii q ij q ji ) = 0 } , and set a ii = 2, s i ∈ Aut( Z θ ) such that s i ( α j ) = α j − a ij α i .Set q rs = χ ( s i ( α r ) , s i ( α s )). Let V i be another vector space of dimension θ , and attachto it the matrix q = ( q rs ). By [H2], ∆ V i + = s i (cid:0) ∆ V + \ { α i } (cid:1) ∪ { α i } . If we consider ∆ V =∆ V + ∪ ( − ∆ V + ), last equation lets us to define the Weyl groupoid of V , whose root systemis defined by the sets ∆ V ′ , V ′ obtained after to apply some reflections to the matrix of V .1.3. Defining relations of Nichols algebras of diagonal type.Proposition 1.22. [A2, Prop. 3.1]
Assume that the braiding matrix is symmetric. Thena PBW basis of Lyndon hyperwords of B ( V ) is orthogonal with respect to the bilinear formin Proposition 1.2. (cid:3) Corollary 1.23. [A2, Cor. 3.2] If u = x n M β M · · · x n β , where ≤ n j < N β j , then (18) c u := ( u | u ) = M Y j =1 n j ! q βj c n j x βj = 0 . (cid:3) Remark . Notice that:( x β i x β j | u ) = ( x β i | u (1) )( x β j | u (2) ) = d i,j c x βi c x βj , where d i,j is the coefficient x β i ⊗ x β j for the expression of ∆( u ) in terms of the PBW basis(both factors of the tensor product).For each pair 1 ≤ i ≤ j ≤ θ , we denote B ij := n x n j β j · · · x n i β i : 0 ≤ n k < N β k o ;that is, the set of hyperwords whose hyperletters are between x β i and x β j N NICHOLS ALGEBRAS OF DIAGONAL TYPE 9
Let (
W, d ) be a braided vector space of diagonal type that admits a basis b x , . . . , b x θ such that, for some b q ij ∈ k × , d ( b x i ⊗ b x j ) = b q ij b x j ⊗ b x i , where b q ij = b q ji , and ( V, c ), (
W, d )are twist equivalent: q ij q ji = b q ij b q ji , q ii = b q ii , ≤ i = j ≤ θ. Call b x β = [ l β ] d , the hyperletter corresponding to l β for the braiding d . If u = x n M β M · · · x n β ,call also b u = b x n M β M · · · b x n β . Let σ : Z θ × Z θ → k × be the bicharacter determined by the condition(19) σ ( α i , α j ) = (cid:26) b q ij q − ij , i ≤ j , i > j Define t α i = 1 for any 1 ≤ i ≤ θ , and inductively, t β = σ ( β , β ) t β t β , Sh( l β ) = ( l β , l β ) . For each u = x n M β M · · · x n β call(20) f ( u ) := Y ≤ i Let ( V, c ) be a finite-dimensional braided vector space ofdiagonal type such that ∆ V + is finite. Let x , · · · , x θ be a basis of V such that c ( x i ⊗ x j ) = q ij x j ⊗ x i , where ( q ij ) ∈ ( k × ) θ × θ is the braiding matrix, and let { x β k } β k ∈ ∆ V + be the set ofhyperletters corresponding to the fixed order of the basis of V .Then B ( V ) is presented by generators x , . . . , x θ , and relations x N β β = 0 , β ∈ ∆ V + , ord( q β ) = N β < ∞ , (22) (cid:2) x β i , x β j (cid:3) c = X u ∈ B ij −{ x βj x βi } : deg u = β i + β j c ui,j u, (23) 1 ≤ i < j ≤ M, Sh( l β i l β j ) = ( l β i , l β j ) , l β i l β j = l β k , ∀ k, where c ui,j are as in (21) . Moreover, { x n M β M · · · x n β : 0 ≤ n j < N β j } is a basis of B ( V ) . (cid:3) Lusztig Isomorphisms of Nichols algebras of diagonal type In this Section we recall the Lusztig isomorphisms [H4] of Nichols algebras of diagonaltype, which are a generalization of the isomorphisms of quantized enveloping algebras in[L]. We shall consider different quotients of the tensor algebra of a braided vector spaceof diagonal type and the Drinfeld doubles of their bosonizations by a free abelian group. Notation: Let χ : Z θ × Z θ → k × be a bicharacter, q ij = χ ( α i , α j ). Then χ op and χ − will denote the bicharacters: χ op ( α, β ) := χ ( β, α ) , χ − ( α, β ) := χ ( α, β ) − , α, β ∈ Z θ . Also, for any automorphism s : Z θ → Z θ , s ∗ χ will denote the bicharacter defined by(24) ( s ∗ χ )( α, β ) := χ (cid:0) s − ( α ) , s − ( β ) (cid:1) , α, β ∈ Z θ . Let ( V, c ) a braided vector space of diagonal type, whose braiding matrix is ( q ij ). Weconsider T ( V ) as an algebra in the category of Yetter-Drinfeld modules over k Z θ as above.We follow the results in [H4, Section 4.1]. Definition 2.1. The Drinfeld double U ( χ ) of the Hopf algebra T ( V ) k Z θ is the algebragenerated by elements E i , F i , K i , K − i , L i , L − i , 1 ≤ i ≤ θ , and relations XY = Y X, X, Y ∈ { K ± i , L ± i : 1 ≤ i ≤ θ } ,K i K − i = L i L − i = 1 ,K i E j K − i = q ij E j , L i E j L − i = q − ji E j ,K i F j K − i = q − ij F j , L i F j L − i = q ji F j ,E i F j − F j E i = δ i,j ( K i − L i ) . It admits a Hopf algebra structure, where the comultiplication satisfies∆( K i ) = K i ⊗ K i , ∆( E i ) = E i ⊗ K i ⊗ E i , ∆( L i ) = L i ⊗ L i , ∆( F i ) = F i ⊗ L i + 1 ⊗ F i , and then ε ( K i ) = ε ( L i ) = 1, ε ( E i ) = ε ( F i ) = 0.Notice that U ( χ ) is a Z θ -graded Hopf algebra, where the graduation is characterizedby the following conditions:deg( K i ) = deg( L i ) = 0 , deg( E i ) = α i , deg( F i ) = − α i . U + ( χ ) (respectively, U − ( χ )) denotes the subalgebra generated by E i (respectively, F i ), 1 ≤ i ≤ θ , U +0 ( χ ) (respectively, U − ( χ )) is the subalgebra generated by K i , K − i (respectively, L i , L − i ), 1 ≤ i ≤ θ , and finally U ( χ ) is the subalgebra generated by K i , K − i , L i and L − i . Note that U ( χ ) is isomorphic to k Z θ as Hopf algebras. Moreover, thesubalgebra generated by U + ( χ ), K i and K − i , 1 ≤ i ≤ θ , is isomorphic to T ( V ) k Z θ , so U + ( χ ) is isomorphic to T ( V ) as braided graded Hopf algebras in the category of Yetter-Drinfeld modules over k Z θ , where we consider the actions and coactions: K i · E j = q ij E j , δ ( E i ) = K i ⊗ E i . We will consider a family of useful isomorphisms as in [H4, Section 4.1]. Proposition 2.2. (a) For any a = ( a , . . . , a θ ) ∈ ( k × ) θ there exists a unique algebraautomorphism ϕ a of U ( χ ) such that (25) ϕ a ( K i ) = K i , ϕ a ( L i ) = L i , ϕ a ( E i ) = a i E i , ϕ a ( F i ) = a − i F i . (b) There exists a unique algebra automorphism φ of U ( χ ) such that (26) φ ( K i ) = K − i , φ ( L i ) = L − i , φ ( E i ) = F i L − i , φ ( F i ) = K − i E i . (c) There exists a unique algebra isomorphism φ : U ( χ ) → U ( χ − ) such that (27) φ ( K i ) = K i , φ ( L i ) = L i , φ ( E i ) = F i , φ ( F i ) = − E i . (d) There exists a unique Hopf algebra isomorphism φ : U ( χ ) → U ( χ op ) cop such that (28) φ ( K i ) = L i , φ ( L i ) = K i , φ ( E i ) = F i , φ ( F i ) = E i . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 11 (e) There exists a unique algebra antiautomorphism φ of U ( χ ) such that (29) φ ( K i ) = K i , φ ( L i ) = L i , φ ( E i ) = F i , φ ( F i ) = E i . (f) Let a = ( − , · · · , − . The antipode S of U ( χ ) is given by the composition S = φ φ ϕ a . Also, φ = id . (cid:3) As in [H4] we will consider some skew-derivations. ∆ will denote the braided comul-tiplication of U + ( χ ), which is N -graded: if E ∈ U + ( χ ) is homogeneous of degree n , and k ∈ { , , . . . , n } , ∆ n − k,k ( E ) will denote the component of ∆( E ) in U + ( χ ) n − k ⊗ U + ( χ ) k . Proposition 2.3. For any i ∈ { , . . . , θ } there exist linear endomorphisms ∂ Ki , ∂ Li of U + ( χ ) such that EF i − F i E = ∂ Ki ( E ) K i − L i ∂ Li ( E ) for all E ∈ U + ( χ ) . Such endomorphisms are given by: ∆ n − , ( E ) = θ X i =1 ∂ Ki ( E ) ⊗ E i , ∆ ,n − ( E ) = θ X i =1 E i ⊗ ∂ Li ( E ) , E ∈ U + ( χ ) n , and satisfy the following conditions: ∂ Ki (1) = ∂ Li (1) = 0 ,∂ Ki ( E j ) = ∂ Li ( E j ) = δ i,j ,∂ Ki ( EE ′ ) = ∂ Ki ( E )( K i · E ′ ) + E∂ Ki ( E ′ ) ,∂ Li ( EE ′ ) = ∂ Li ( E ) E ′ + ( L − i · E ) ∂ Li ( E ′ ) , for all j ∈ { , . . . , θ } , and all pair of elements E, E ′ ∈ U + ( χ ) . (cid:3) We recall now a characterization of quotients of the algebra U ( χ ) with a triangulardecomposition [H4, Section 4.1]. According to [H4, Prop. 4.14], the multiplication(30) m : U + ( χ ) ⊗ U ( χ ) ⊗ U − ( χ ) → U ( χ )is an isomorphism of Z θ -graded vector spaces. Proposition 2.4. Let I + ⊂ U + ∩ ker ε (respectively, I − ⊂ U − ∩ ker ε ) be an ideal of U + ( χ ) (respectively, U − ( χ ) ). The following conditions are equivalent: • The multiplication (30) induces an isomorphism m : U + ( χ ) / I + ⊗ U ( χ ) ⊗ U − ( χ ) / I − → U ( χ ) / ( I + + I − ) . • The vector spaces I + U ( χ ) U − ( χ ) and U + ( χ ) U ( χ ) I − are ideals of U ( χ ) . • For all X ∈ U ( χ ) and all i ∈ { , . . . , θ } we have X · I + ⊂ I + , ∂ Ki ( I + ) ⊂ I + , ∂ Ki ( φ ( I − )) ⊂ φ ( I − ) ,X · I − ⊂ I − , ∂ Li ( I + ) ⊂ I + , ∂ Li ( φ ( I − )) ⊂ φ ( I − ) . (cid:3) Lemma 2.5. [H4, Cor. 4.20] Let I + be an braided biideal of U + ( χ ) , which is also aYetter-Drinfeld U ( χ ) -submodule and satisfies I + ⊂ ⊕ n ≥ U + ( χ ) n . Then I + U ( χ ) U − ( χ ) is a Hopf ideal of U ( χ ) . (cid:3) We will assume that all the integers m ij := − a ij of (17) associated with the matrices( q ij ) are defined. Then we consider the automorphisms s p,χ : Z θ → Z θ . We define also thescalars(31) λ i ( χ ) := ( − a pi ) q pp − a pi − Y s =0 ( q spp q pi q ip − , i = p. Denote by E i , F i , K i , L i the generators corresponding to U ( s ∗ p χ ), and by q ij = s ∗ p χ ( α i , α j )the coefficients of the braiding matrix corresponding to s ∗ p χ . Definition 2.6. We say that p ∈ { , . . . , θ } is a Cartan vertex if it satisfies q − a pj pp q pj q jp = 1 , for every j = p. In such case, note that the existence of the integers − a pj implies that ord q pp ≥ − a pj + 1.We denote by O ( χ ) the union of the orbits of the simple roots α p by the action of theWeyl groupoid, where p is a Cartan vertex.Fix p ∈ { , . . . , θ } . For any i = p we define as in [H4], E + i, p ) , E − i, p ) := E i , F + i, p ) , F − i, p ) := F i , and recursively, E + i,m +1( p ) := E p E + i,m ( p ) − ( K p · E + i,m ( p ) ) E p = (ad c E p ) m +1 E i , (32) E − i,m +1( p ) := E p E − i,m ( p ) − ( L p · E − i,m ( p ) ) E p , (33) F + i,m +1( p ) := F p F + i,m ( p ) − ( L p · F + i,m ( p ) ) F p , (34) F − i,m +1( p ) := F p F − i,m ( p ) − ( K p · F − i,m ( p ) ) F p . (35)When p is explicit, we simply denote E ± i,m ( p ) by E ± i,m . By [H4, Cor. 5.4] the followingidentity holds for any m ∈ N :(36) E + i,m F i − F i E + i,m = ( m ) q pp ( q m − pp q pi q ip − L p E + i,m − . Fix a braided graded Hopf algebra B ∼ = T ( V ) /I , where I is a graded Hopf idealgenerated by homogeneous elements of degree ≥ 2. For each 1 ≤ j ≤ θ , p = j , we define(37) M ± p,j ( B ) := n E ± j,m : m ∈ N o . In what follows we consider ord(1) = 1. Remark . If E Ni = 0 in B , with N minimal (it is called the nilpotency order of x i ),then q ii is a primitive root of unity of order N . Moreover, (ad c E i ) N E j = 0.Also, the nilpotency order of E i is infinite if E ni = 0 for all n ∈ N .We recall a result from [A1] extending [H2, Prop. 1, Eqn. (18)]. Lemma 2.8. For each p ∈ { , . . . , θ } , let B ± p be the subalgebra of B generated by ∪ j = p M ± p,j ( B ) ,and denote N p = ord q pp . There exist isomorphisms of graded vector spaces: • ker( ∂ Kp ) ∼ = B + p ⊗ k h E N p p i , ker( ∂ Lp ) ∼ = B − p ⊗ k h E N p p i , if < ord( q pp ) < ∞ but E p is not nilpotent, or • ker( ∂ Kp ) ∼ = B + p , ker( ∂ Lp ) ∼ = B − p , if ord( q pp ) is the nilpotency order of E p or q pp = 1 .Moreover, the multiplication induces an isomorphism of graded vector spaces B ∼ = B ± p ⊗ k [ E p ] . (cid:3) Set N p = ord q pp . We call, following [H4], I + p ( χ ) (respectively, I − p ( χ )) to the ideal of U + ( χ ) (respectively, U − ( χ )) generated by(a) E N p p (respectively, F N p p ), if p is not a Cartan vertex,(b) E + i,m pi +1 (respectively, F + i,m pi +1 ), for each i such that N p > m pi + 1. N NICHOLS ALGEBRAS OF DIAGONAL TYPE 13 Notice that E + i,m pi +1 ∈ I + p ( χ ) for any i such that N p = m pi + 1. We denote: U p ( χ ) := U ( χ ) / (cid:0) I + p ( χ ) + I − p ( χ ) (cid:1) , U + p ( χ ) := U + ( χ ) / I + p ( χ ) , U − p ( χ ) := U − ( χ ) / I − p ( χ ) .I + ( χ ) will denote the ideal of U + ( χ ) such that the quotient U + ( χ ) /I + ( χ ) is isomorphicto the Nichols algebra of V ; that is, the greatest braided Hopf ideal of U + ( χ ) generatedby elements of degree ≥ 2. Call I − ( χ ) = φ ( I + ( χ )), where φ is defined by (29), and u ± ( χ ) := U ± ( χ ) /I ± ( χ ) , u ( χ ) := U ( χ ) / ( I − ( χ ) + I + ( χ )) . In such case, u ( χ ) is the Drinfeld double of the algebra u + ( χ ) k Z θ , where k Z θ = U +0 ( χ ).The Lusztig isomorphisms can be defined in this general context. Theorem 2.9. [H4, Lemma 6.5, Theorem 6.12] There exist algebra morphisms (38) T p , T − p : U p ( χ ) → U p ( s ∗ p χ ) univocally determined by the following conditions: T p ( K p ) = T − p ( K p ) = K − p , T p ( K i ) = T − p ( K i ) = K m pi p K i ,T p ( L p ) = T − p ( L p ) = L − p , T p ( L i ) = T − p ( L i ) = L m pi p L i ,T p ( E p ) = F p L − p , T p ( E i ) = E + i,m pi ,T p ( F p ) = K − p E p , T p ( F i ) = λ p ( s ∗ p χ ) − F + i,m pi ,T − p ( E p ) = K − p F p , T − p ( E i ) = λ p ( s ∗ p χ − ) − E − i,m pi ,T − p ( F p ) = E p L − p , T − p ( F i ) = F − i,m pi . for every i = p . Both are isomorphisms satisfying T p T − p = T − p T p = id , T p ( U ++ p ( χ )) = U + − p ( s ∗ p χ ) . Moreover, there exists µ ∈ ( k × ) θ such that (39) T p ◦ φ = φ ◦ T − p ◦ ϕ µ . Such isomorphisms induce algebra isomorphisms (denoted by the same name): T p , T − p : u ( χ ) → u ( s ∗ p χ ) . (cid:3) Remark . If the homogeneous elements X, Y ∈ U + p ( χ ) are such that T p ( X ) , T p ( Y ) ∈U + p ( s ∗ p χ ), as deg T p ( X ) = s p (deg X ), it follows that T p ([ X, Y ] c ) = [ T p ( X ) , T p ( y )] c . An explicit presentation by generators and relations of Nicholsalgebras of diagonal type We shall obtain a family of isomorphisms induced by the ones in the previous Section.In this case we shall consider a quotient of U ( χ ) by an ideal which is smaller than ( I − ( χ ) + I + ( χ )). Such ideal will be generated by some of the relations in Theorem 1.25, and willbe the smallest one such that it is possible to define all the family of isomorphisms overthe Weyl groupoid. It will give us a relation between the Hilbert series of these algebras,and new sets of roots. We shall use at the end the uniqueness of the root system, whenthe Weyl groupoid is finite.We introduce some notation. We denote f q ij = q ij q ji . Also, x i i ··· i k = (ad c x i ) · · · (ad c x i k − ) x i k , i j ∈ { , . . . , θ } . For each m ∈ N , we define the elements x ( m +1) α i + mα j ∈ U ( χ ) recursively: • if m = 1, x α i + α j := (ad c x i ) x j = x iij , • x ( m +2) α i +( m +1) α j := [ x ( m +1) α i + mα j , x ij ] c .We give now the main result of this section, which is Theorem 3.1: it gives an explicitpresentation by generators and relations of any Nichols algebra of diagonal type withfinite root system. We begin by proving several Lemmata to show the existence of Lusztigisomorphisms for some Hopf algebras. These Hopf algebras are intermediate between thetensor algebra and the Nichols algebras of a given braided vector space. Finally we usethose Lusztig isomorphisms to prove the Theorem. Theorem 3.1. Let ( V, c ) be a finite-dimensional braided vector space of diagonal type,with braiding matrix ( q ij ) ≤ i,j ≤ θ , θ = dim V , and fix a basis x , . . . , x θ of V such that c ( x i ⊗ x j ) = q ij x j ⊗ x i . Let χ be the bicharacter associated to ( q ij ) . Assume that the rootsystem ∆ χ is finite. Then B ( V ) is presented by generators x , . . . , x θ and relations: x N α α , α ∈ O ( χ );(40) (ad c x i ) m ij +1 x j , q m ij +1 ii = 1;(41) x N i i , i is not a Cartan vertex ;(42) ⋄ if i, j ∈ { , . . . , θ } satisfy q ii = f q ij = q jj = − , and there exists k = i, j such that f q ik = 1 or f q jk = 1 , (43) x ij ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q jj = − , f q ik = f q ij f q kj = 1 , f q ij = − , (44) [ x ijk , x j ] c ; ⋄ if i, j ∈ { , . . . , θ } satisfy q jj = − , q ii f q ij ∈ G , f q ij = − , and also q ii ∈ G or m ij ≥ , (45) [ x iij , x ij ] c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q ii = ± f q ij ∈ G , f q ik = 1 , and also − q jj = f q ij f q jk = 1 or q − jj = f q ij = f q jk = − , (46) [ x iijk , x ij ] c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy f q ik , f q ij , f q jk = 1 , (47) x ijk − − f q jk q kj (1 − f q ik ) [ x ik , x j ] c − q ij (1 − f q jk ) x j x ik ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy one of the following situations ◦ q ii = q jj = − , f q ij = f q jk − , f q ik = 1 , or ◦ f q ij = q jj = − , q ii = − f q jk ∈ G , f q ik = 1 , or ◦ q kk = f q jk = q jj = − , q ii = − f q ij ∈ G , f q ik = 1 , or ◦ q jj = − , f q ij = q − ii , f q jk = − q − ii , f q ik = 1 , or ◦ q ii = q jj = q kk = − , ± f q ij = f q jk ∈ G , f q ik = 1 , (48) (cid:2) [ x ij , x ijk ] c , x j (cid:3) c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q ii = q jj = − , f q ij = f q jk − , f q ik = 1 , (49) (cid:2)(cid:2) x ij , [ x ij , x ijk ] c (cid:3) c , x j (cid:3) c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q jj = f q ij = f q jk ∈ G , f q ik = 1 , (50) (cid:2) [ x ijk , x j ] c x j (cid:3) c ; N NICHOLS ALGEBRAS OF DIAGONAL TYPE 15 ⋄ if i, j, k ∈ { , . . . , θ } satisfy q kk = q jj = f q ij − = f q jk − ∈ G , f q ik = 1 , q ii = q kk (51) (cid:2) [ x iij , x iijk ] c , x ij (cid:3) c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q ii = f q ij − ∈ G , q jj = f q jk − = q ii , f q ik = 1 , q kk = q ii (52) [[ x ijk , x j ] c , x k ] c − (1 + f q jk ) − q jk (cid:2) [ x ijk , x k ] c , x j (cid:3) c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q jj = f q ij = f q jk ∈ G , f q ik = 1 , (53) (cid:2)(cid:2) [ x ijk , x j ] c , x j (cid:3) c , x j (cid:3) c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q ii = f q ij = − , q jj = f q jk − = − , f q ik = 1 , (54) [ x ij , x ijk ] c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q ii = q kk = − , f q ik = 1 , f q ij ∈ G , q jj = − f q jk = ± f q ij , (55) [ x i , x jjk ] c − (1 + q jj ) q − kj [ x ijk , x j ] c − (1 + q jj )(1 + q jj ) q ij x j x ijk ; ⋄ if i, j, k, l ∈ { , . . . , θ } satisfy q jj f q ij = q jj f q jk = 1 , q kk = − , f q ik = e q il = f q jl = 1 , f q jk = f q lk − = q ll , (56) (cid:2)(cid:2) [ x ijkl , x k ] c , x j (cid:3) c , x k (cid:3) c ; ⋄ if i, j, k, l ∈ { , . . . , θ } satisfy f q jk = f q ij = q − jj ∈ G ′ ∪ G ′ , q ii = q kk = − , f q ik = e q il = f q jl = 1 , f q jk = f q lk , (57) (cid:2)(cid:2) x ijk , [ x ijkl , x k ] c (cid:3) c , x jk (cid:3) c ; ⋄ if i, j, k, l ∈ { , . . . , θ } satisfy q ll = f q lk − = q kk = f q jk − = q , f q ij = q − ii = q for some q ∈ k × , q jj = − , f q ik = e q il = f q jl = 1 , (58) (cid:2)(cid:2) [ x ijk , x j ] c , [ x ijkl , x j ] c (cid:3) c , x jk (cid:3) c ; ⋄ if i, j, k, l ∈ { , . . . , θ } satisfy one of the following situations ◦ q kk = − , q ii = f q ij − = q jj , f q kl = q − ll = q jj , f q jk = q − jj , f q ik = e q il = f q jl = 1 , or ◦ q ii = f q ij − = − q − ll = − f q kl , q jj = f q jk = q kk = − , f q ik = e q il = f q jl = 1 , (59) (cid:2) [ x ijkl , x j ] c , x k (cid:3) c − q jk ( f q ij − − q jj ) (cid:2) [ x ijkl , x k ] c , x j (cid:3) c ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy f q jk = 1 , q ii = f q ij = − f q ik ∈ G , (60) (cid:2) x i , [ x ij , x ik ] c (cid:3) c + q jk q ik q ji [ x iik , x ij ] c + q ij x ij x iik ; ⋄ if i, j, k ∈ { , . . . , θ } satisfy q jj = q kk = f q jk = − , q ii = − f q ij ∈ G , f q ik = 1 , (61) [ x iijk , x ijk ] c ; ⋄ if i, j ∈ { , . . . , θ } satisfy − q ii , − q jj , q ii f q ij , q jj f q ij = 1 , (62) (1 − f q ij ) q jj q ji (cid:2) x i , [ x ij , x j ] c (cid:3) c − (1 + q jj )(1 − q jj f q ij ) x ij ; ⋄ if i, j ∈ { , . . . , θ } satisfy that m ij ∈ { , } , or q jj = − , m ij = 3 , q ii ∈ G , (63) (cid:2) x i , x α i +2 α j (cid:3) c − − q ii f q ij − q ii f q ij q jj (1 − q ii f q ij ) q ji x iij ; ⋄ if i, j ∈ { , . . . , θ } satisfy α i + 3 α j / ∈ ∆ χ + , q jj = − or m ji ≥ , and also m ij ≥ , or m ij = 2 , q ii ∈ G , (64) x α i +3 α j = [ x α i +2 α j , x ij ] c ; ⋄ if i, j ∈ { , . . . , θ } satisfy α i + 2 α j ∈ ∆ χ + , α i + 3 α j / ∈ ∆ χ + , and q ii f q ij , q ii f q ij = 1 , (65) [ x iij , x α i +2 α j ] c ; ⋄ if i, j ∈ { , . . . , θ } satisfy α i + 3 α j ∈ ∆ χ + , α i + 4 α j / ∈ ∆ χ + , (66) x α i +4 α j = [ x α i +3 α j , x ij ] c ; ⋄ if i, j ∈ { , . . . , θ } satisfy α i + 2 α j ∈ ∆ χ + , α i + 3 α j / ∈ ∆ χ + , (67) [[ x iiij , x iij ] , x iij ] c ; ⋄ if i, j ∈ { , . . . , θ } satisfy q jj = − , α i + 4 α j ∈ ∆ χ + , (68) [ x iij , x α i +3 α j ] c − b − (1 + q ii )(1 − q ii ζ )(1 + ζ + q ii ζ ) q ii ζ a q ii q ij q ji x α i +2 α j , where ζ = f q ij , a = (1 − ζ )(1 − q ii ζ ) − (1 − q ii ζ )(1 + q ii ) q ii ζ , b = (1 − ζ )(1 − q ii ζ ) − a q ii ζ . In what follows we will use implicitly the isomorphism B ( V ) ∼ = u + ( χ ) determined by x i E i ; in this way, we identify B ( V ) as a subalgebra of u ( χ ).For any bicharacter χ whose root system is finite, J + ( χ ) denotes the ideal of U + ( χ )generated by all the relations in Theorem 3.1, except (40), plus the quantum Serre relations(ad c x i ) − a ij x j for those vertices such that q a ij ii = q ij q ji = q ii . The last ingredient is toobtain a quotient of all the algebras U p ( χ ), 1 ≤ p ≤ θ .Call also J − ( χ ) := φ ( J + ( χ )), J ( χ ) := ( J + ( χ ) + J − ( χ )), U ( χ ) := U ( χ ) / J ( χ ) , U ± ( χ ) := U ± ( χ ) / J ± ( χ ) . We prove first that J + ( χ ) is contained in the ideal defining the corresponding Nicholsalgebra. The following Lemma was proved with Agust´ın Garc´ıa Iglesias and is implicit inother papers. Lemma 3.2. Let I ⊂ T ( V ) be a braided homogeneous biideal of T ( V ) , so there existsa surjective morphism of braided graded Hopf algebras π : R := T ( V ) /I → B ( V ) . Let x ∈ ker π , x = 0 of minimal degree k ≥ . Then x is primitive.Proof. As π is a morphism of graded braided bialgebras, ker π is a graded biideal:∆( x ) = x ⊗ ⊗ x + n X j =1 b j ⊗ c j ∈ ker π ⊗ R + R ⊗ ker π, for some homogeneous elements b j , c j ∈ k − L i =1 R i , such that deg( b j )+ deg( c j ) = k . For each j we may assume either b j ∈ ker π or c j ∈ ker π . If b j ∈ ker π , then b j = 0 by the minimalitycondition on k . Similarly, if c j ∈ ker π , then c j = 0. Hence x is primitive in R . (cid:3) We will work with N θ -graded ideals, so the following notation will be useful: given β = P i b i α i , γ = P i c i α i , for some b i , c i ∈ N , we say that β ≥ γ (respectively, β > γ ) if b i ≥ c i (respectively, b i > c i ) for all i ∈ { , . . . , θ } . Proposition 3.3. J + ( χ ) is a braided biideal of U + ( χ ) , and there exist a canonical pro-jection of Hopf algebras π χ : U ( χ ) ։ u ( χ ) such that π ( U ± ( χ )) = u ± ( χ ) .Moreover, the multiplication m : U + ( χ ) ⊗ U ( χ ) ⊗ U − ( χ ) → U ( χ ) is an isomorphismof graded vector spaces.Proof. We can order the relations according to their N -graduation. When we quotient bythe relations of degree at most n − 1, the relations of degree n are primitive by Lemma3.2, because for any of them we can see that the relations in Theorem 1.25 of degree atmost n − α ∈ N θ , it is enough to verifythat the relations of N θ -degree lower than α hold in this partial quotient. For example, N NICHOLS ALGEBRAS OF DIAGONAL TYPE 17 each quantum Serre relation is primitive, and the same holds for x N i i ; therefore, when wequotient by these relation we have that x = (cid:2) (ad c x i ) x j , (ad c x i ) x j (cid:3) c is primitive underthe conditions for (45), because we have quotiented by x i , x j , so it also holds that(ad c x i ) x j = (ad c x j ) x i = 0 . We work in a similar way with the other relations so each partial quotient is a braidedbialgebra (and then a Hopf algebra with the induced antipode); finally, U + ( χ ) is a braidedbialgebra, because J + ( χ ) is a braided biideal.By the definition of Nichols algebra we conclude that J + ( χ ) ⊆ I + ( χ ). By Lemma2.5, J + ( χ ) U ( χ ) U − ( χ ) is a Hopf ideal of U ( χ ), and then the equivalent conditions inProposition 2.4 hold. Therefore there exists a projection of Hopf algebras and a triangulardecomposition as in the Proposition. (cid:3) Now we prove that the isomorphisms at the beginning of Theorem 2.9 induce iso-morphisms between the corresponding algebras U ( χ ). The first step is to prove that T p ( J ( χ )) ⊂ J ( s ∗ p χ ), which will be proved considering each relation generating the ideal.The following two Lemmata help us to reduce the number of explicit computations. Lemma 3.4. Let l be a Lyndon word such that [ l ] c ∼ = P w ∈ S I +( χ ) ,w ≻ l a w w ( mod I + ( χ )) , forsome a w ∈ k . Let I be a braided Hopf ideal N θ -graded of U + ( χ ) such that the set of goodwords S I + ( χ ) , S I coincide for those terms w ≻ l , and assume that l is written as a linearcombination of words greater than l modulo I . Then [ l ] c ∼ = P w ∈ S I +( χ ) ,w ≻ l a w w ( mod I ) .Proof. It is a direct consequence of Corollary 1.14: by this result, [ l ] c is written as a linearcombination of good hyperwords greater than [ l ] c modulo I . Such hyperwords coincidewith the corresponding good hyperwords for I + ( χ ) by hypothesis, and also I ⊆ I + ( χ ).Hence the linear combination should be the same, because the good hyperwords generatea linear complement of the ideal in U + ( χ ). (cid:3) Lemma 3.5. Let I be an N θ -homogeneous ideal of T ( V ) , θ = dim V . Let S , T be twominimal sets of homogeneous generators of I . Assume that for each α ∈ N θ there exists atmost one generator in S (respectively in T ) of degree α , and denote by I ( S, α ) (respectively, I ( T, α ) ) the ideal generated by the elements of S (respectively, T ) of degree β < α .For each s ∈ S of degree α ∈ N θ , there exists t ∈ T of the same degree, and c ∈ k × such that s ∼ = c t ( mod ( I ( S, α )) , and then I ( S, α ) = I ( T, α ) .Proof. We prove it by induction on the degree of the generators. Let s be of degree α ,minimal for the partial order defined on N θ . Therefore dim I α = 1, so there exists anelement of T which belongs to this subspace of I of dimension 1.If the degree of s is not minimal, we apply inductive hypothesis for all the generatorsin lower degree, so for each s ′ of lower degree there exists a corresponding t ′ ∈ T of thesame degree which satisfies the conditions above, and I ( S, α ) = I ( T, α ). Thereforedim I ( S, α ) α = dim I ( T, α ) α = dim I α − , because S is a minimal set of generators, and by hypothesis there exists a unique generatorof degree α . As T is also a set of generators of I , there exists t ∈ I − I ( T, α ) = I − I ( S, α ),of degree α , so the statement follows. (cid:3) Remark . This Lemma lets us to identify relations of the same degree for two sets ofminimal generators of an ideal, up to relations of lower degree and scalars. In this waywe can consider relations from Theorem 1.25 for a fixed order on the letters, and considerrelations for another order. If we have a minimal set and this set contains relations all in different degrees (as we will have for the set of relations of the Nichols algebra or somepartial quotients), then we can find a correspondence as above between the relations ofthe same Z θ -degree.For example, if q m ij +1 ii = 1 for some pair of vertices i, j , then the quantum Serre relation(ad c x i ) m ij +1 x j is a generator for the minimal set of generators corresponding to the order x i < x j , so for the order x i > x j we have:[ x j x m ij +1 i ] c = (cid:2) · · · (cid:2) [(ad c x j ) x i , x i ] c , · · · , (cid:3) c , x i (cid:3) c = a (ad c x i ) m ij +1 x j , for some scalar a ∈ k × .Also, if q ii ∈ G , f q ij ∈ {± q ii , − } , q jj = − 1, we notice that (cid:2) (ad c x i ) x j , (ad c x i ) x j (cid:3) c ∼ = b (cid:2) (ad c x j ) x i , [(ad c x j ) x i , x i ] c (cid:3) c (mod I ) , for some b ∈ k × , where I is the ideal generated by x i and x j , because such relationscorrespond to different minimal sets of generators of the ideal of relations of the Nicholsalgebra, and these are the generators of degree 3 α i + 2 α j for each set. Lemma 3.7. Let I be a Z θ -graded ideal of U + p ( χ ) . Let Y, Z ∈ U + p ( χ ) /I be homogeneouselements of degree β, γ ∈ N θ , respectively, such that (ad c E p ) Y = 0 . Then, (69) [(ad c E p ) Z, Y ] c = (ad c E p ) [ Z, Y ] c . If also χ ( α p , β ) χ ( β, α p ) = 1 , then (70) χ ( α p , β ) [ Y, (ad c E p ) Z ] c = (ad c E p ) [ Y, Z ] c . Proof. Both identities follow from (9). For example, for the second one,(ad c E p ) [ Y, Z ] c = [ E p , [ Y, Z ] c ] c = (cid:2) [ E p , Y ] c , Z (cid:3) c + χ ( α p , β ) Y [ E p , Z ] c − χ ( β, γ ) [ E p , Z ] c Y = χ ( α p , β ) ( Y (ad c E p ) Z − χ ( β, γ ) χ ( β, α p )(ad c E p ) ZY )= χ ( α p , β ) [ Y, (ad c E p ) Z ] c , where we use the condition χ ( α p , β ) χ ( β, α p ) = 1. (cid:3) Lemma 3.8. Let i, p ∈ { , . . . , θ } be such that m pi ≥ and m ip = 1 . Then, in U ( s ∗ p χ ) , h E + i,m pi , E + i,m pi − i c = (cid:2) (ad c E p ) m pi E i , (ad c E p ) m pi − E i (cid:3) c = 0 . Remark . Such relation belongs to the ideal I + ( s ∗ p χ ). In fact, as 2 α i + α p / ∈ ∆ χ + , wehave s p (2 α i + α p ) = 2 α i + (2 m pi − α p / ∈ ∆ χ + , so such relation holds in the correspondingNichols algebra u + ( s ∗ p χ ).On the other hand, some of these relations are generators of the ideal J ( s ∗ p χ ) bydefinition, for example (45). We prove here that the other relations not in the definitionof this ideal are redundant; that is, they are generated by relations of lower degree. Proof. We consider the different possible values of m pi ; we begin with m pi = 2. Therefore q pp ∈ G or q pp q ip q pi = 1, and also q ii = − q ii q ip q pi = 1. If m ip = 1 for s ∗ p χ , then p , i determine a subdiagram of standard type. If q pp = 1 or q ii = − E p E i E p E i iswritten as a linear combination of words greater than E p E i E p E i , modulo J ( s ∗ p χ ), usingthe quantum Serre relations, because in the first case E p E i E p appears with non-zerocoefficient in (ad c E p ) E i , so E p E i E p E i is a linear combination of greater words and E p E i ,but this last word is in the ideal if q ii = − 1, or E p E i appears in (ad c E i ) E p with non-zero coefficient, so in both cases we obtain E p E i E p E i as a combination of greater words. N NICHOLS ALGEBRAS OF DIAGONAL TYPE 19 Using Lemma 3.4, we conclude that h E + i, , E + i, i c = 0. A similar proof in the case q pp = 1, q ii = − q pp = 1, q ii = − 1, the relation corresponds to(45), which is by definition a generator of J ( s ∗ p χ ).If m pi = 2 and m ip > s ∗ p χ , then (62) is a generator of J ( s ∗ p χ ), and then E p E i E p E i is a linear combination of E p E i and greater words. Therefore h E + i, , E + i, i c ∈ J ( s ∗ p χ ), bya similar argument.If m pi = 3, then m ip = 1 for s ∗ p χ , or there exists ζ ∈ G such that q pp = ζ , q − ii = q pi q ip = ζ . For the first case, we notice that (63) holds also if q pp / ∈ G , because E p E i E p can be written as a linear combination of other words from the quantum Serre relation(ad c E p ) E i = 0, and then E p E i E p E i is a linear combination of greater words multiplyingby E i , so we apply Lemma 3.4; from this relation we work as above, so we write E p E i E p E i as a linear combination of other words and deduce that E p E i E p E i is a linear combinationof greater words, and we can apply Lemma 3.4 again. For the second case, we write E p E i E p E i as a linear combination of greater words using the quantum Serre relations orthe relation (63), with the same conclusion.If m pi = 4 , 5, then m ip = 1 for s ∗ p χ . Therefore we work as before and we obtain thedesired relation from (63) or (45), according to 3 α p + 2 α i belongs to ∆ s ∗ p χ + or not. In bothsituations, we can write E p E i E p E i or E p E i E p E i as a linear combination of greater words,so we apply Lemma 3.4 again. (cid:3) We will prove now that T p ( x ) ∈ J ( s ∗ p χ ) for any generator x of the ideal J + ( χ ) so wewill have a family of morphisms between the algebras U ( χ ). Lemma 3.10. Let i be a non-Cartan vertex. Then T p ( E N i i ) ∈ J ( s ∗ p χ ) .If i, j satisfy q ii = f q ij = q jj = − , and there exists k such that f q ik = − or f q jk = − ,then T p (cid:16) E ij (cid:17) ∈ J ( s ∗ p χ ) .Proof. Consider the first relation. If i = p , then p is not Cartan for χ , so p is not Cartanfor s ∗ p χ too. Therefore, by the definition of the ideal J ( s ∗ p χ ), T p ( E N p p ) = F N p p = φ ( E N p p ) ∈ J ( s ∗ p χ ) . We consider then i = p . In such case, T p ( E N i i ) = (cid:16) E + i,m pi (cid:17) N i .If m pi = 0, then E + i, = E i and q ip q pi = 1, so for each j = p we have q ij q ji = f q ij .Therefore i is not Cartan for s ∗ p χ , and T p ( E N i i ) = E N i i ∈ J ( s ∗ p χ ).Consider m pi = 0. As i is not Cartan, there exists j = i such that q m ij ii f q ij = 1.Assume first that m ip + 1 = N i . If m ip = 1, that is q ii = − 1, there are two possibil-ities. If q ip q pi = − 1, using Lemma 3.8, the identity (9) and the quantum Serre relation(ad c E p ) m pi +1 E i = 0, we compute in U ( s ∗ p χ ),0 = h E p , h E + i,m pi , E + i,m pi − i c i c = (cid:0) s ∗ p χ ( α p , m pi α p + α i ) − s ∗ p χ ( m pi α p + α i , ( m pi − α p + α i ) (cid:1) (cid:16) E + i,m pi (cid:17) = ( χ ( − α p , α i ) − χ ( α i , α p + α i )) (cid:16) E + i,m pi (cid:17) = q − pi (1 + q ip q pi ) (cid:16) E + i,m pi (cid:17) , so T p ( E i ) = (cid:16) E + i,m pi (cid:17) ∈ J ( s ∗ p χ ). If q pi q ip = − 1, there are 3 possible subdiagramsdetermined by i, p : it is standard with q = − 1, or it is Cartan of type B with q ∈ G , or it is Cartan of type G with q ∈ G . For the first case, if the diagram is of type A associated to q = − 1, it follows by definition of the ideal J ( s ∗ p χ ); for the other cases, wewrite E m pi p E i E m pi p E i as a linear combination of greater words using (45) or the quantumSerre relations, and also the previous Lemmata.If m ip > m pi = 1, we compute, using (9) and the relation (ad c E p ) E i = 0,ad c E p h E + i, , E i i c = (cid:0) s ∗ p χ ( α p , α i + α p ) − s ∗ p χ ( α i + α p , α i ) (cid:1) (cid:16) E + i, (cid:17) = q − pi (1 − q ii q ip q pi ) (cid:16) E + i, (cid:17) . From this relation and (9) again, we calculatead c E p (cid:20) E + i, , h E + i, , E i i c (cid:21) c = (cid:0) s ∗ p χ ( α p , α i + α p ) − s ∗ p χ ( α i + α p , α i + α p ) (cid:1) ( q − pi − q ii q ip ) (cid:16) E + i, (cid:17) = q − pi (1 − q ii q ip q pi )(1 − q ii q ip q pi ) (cid:16) E + i, (cid:17) . So if m ip = 2 and m pi = 1, it follows that α p + 3 α i / ∈ ∆ χ + , and s p ( α p + 3 α i ) = 3 α i + 2 α p / ∈ ∆ s ∗ p χ + . Using the previous Lemma, (cid:2) (ad c E i ) E p , (ad c E i ) E p (cid:3) c ∈ J + ( s ∗ p χ ), so (cid:20) E + i, , h E + i, , E i i c (cid:21) c ∈ J ( s ∗ p χ ) , because we apply Lemma 3.5 if the relation belongs to a minimal set of generators( q ii q ip q pi = 1, so q ii ∈ G ), or we compute it directly for the cases in which it is Car-tan of type B or standard with q pp = − 1. Then, by a similar argument, T p ( E i ) = (cid:16) E + i, (cid:17) ∈ J ( s ∗ p χ ) . If m ip = 3, m pi = 1, we have that s p ( α p + 4 α i ) = 4 α i + 3 α p / ∈ ∆ s ∗ p χ + , so " E + i, , (cid:20) E + i, , h E + i, , E i i c (cid:21) c c ∈ J ( s ∗ p χ ) , by a similar argument, using (64). In this case we deduce that (cid:16) E + i, (cid:17) ∈ J ( s ∗ p χ ).If m ip = 4, then q ii q ip q pi = 1, so (cid:16) E + i, (cid:17) ∈ J ( s ∗ p χ ) in a similar way, using (66). Wenotice that there are no diagrams such that q m ip +1 ii = 1 and m ip ≥ m ip , m pi > 1, so there are 3 possibilities: • m ip = m pi = 2, so (64) is a generator of J ( s ∗ p χ ), and q pp ∈ G . Therefore wewrite E i E p E i E p E i as a linear combination of other words, which begin with E p or they contain E p as a factor. If we multiply by E p on the left, E p E i E p E i E p E i is a linear combination of greater words modulo J ( s ∗ p χ ), because E p ∈ J ( s ∗ p χ ), so T p ( E i ) = ((ad c E p ) E i ) ∈ J ( s ∗ p χ ). N NICHOLS ALGEBRAS OF DIAGONAL TYPE 21 • m ip = 3, m pi = 2, so q ii = ζ , q pp = ζ , q ip q pi = ζ for some ζ ∈ G , and (68) isa generator of the ideal J ( s ∗ p χ ). Using this relation we write E i E p E i E p E i E p E i as a linear combination of words beginning with E p or words containing E p asa factor. Multiplying by E p on the left, we write ( E p E i ) as a linear com-bination of greater words modulo J ( s ∗ p χ ), because E p ∈ J ( s ∗ p χ ), so T p ( E i ) =((ad c E p ) E i ) ∈ J ( s ∗ p χ ). • m ip = 2, m pi = 3, so there are two possible diagrams; in both cases (67) is agenerator of J ( s ∗ p χ ). From this relation we write E i E p E i E p E i as a combina-tion of words beginning with E p o containing E p . Multiplying on the left by E p , E p E i E p E i E p E i can be written as a linear combination of greater words, mod-ulo J ( s ∗ p χ ), because E p ∈ J ( s ∗ p χ ) or (ad c E p ) E i ∈ J ( s ∗ p χ ), so, in both cases, T p ( E i ) = ((ad c E p ) E i ) ∈ J ( s ∗ p χ ).Finally we consider q m ip ii q ip q pi = 1, m ip < N i − 1, so there exists j = p such that1 ≤ m ip < m ij = N i − 1. In this case, i , j , p determine a connected diagram, where i isnot Cartan, connected with j and p , and also q ii is a root of unity of order N i > 2. Wehave the following possible diagrams under the previous conditions: • q ii ∈ G , q pp ∈ { q ii , − } , m ip = m pi = 1, m ij = 2, m pj = m jp = 0, or • q ii ∈ G , q pp = − q ip q pi q ii = 1, f q ij = q ii , m pj = m jp = 0, m ij = 3 (a diagram oftype super G (3), with q ∈ G ).Both possibilities follow in a similar way to the case m pi = 1.We analyze now the second relation. As f q ij = − c E j ) E i ) = q ji ((ad c E i ) E j ) + 2 q ji ( E i E j E i + E j E i E j ) . By the first part T p ( E i ) , T p ( E j ) ∈ J ( s ∗ p χ ), and as T p is an algebra morphism, it is enoughto prove that T p (cid:16) ((ad c E i ) E j ) (cid:17) ∈ J ( s ∗ p χ ), to conclude that also T p (cid:16) ((ad c E j ) E i ) (cid:17) ∈J ( s ∗ p χ ), and vice versa. Moreover we need just one of these two relations in order to havea minimal set of relations.If p = j , we have T p (cid:16) ((ad c E i ) E p ) (cid:17) = (cid:16) E + i, F p L − p − q ip F p L − p E + i, (cid:17) = (cid:16)(cid:16) F p E − i, L p E i (cid:17) L − p − q ip q pp q ip F p E + i, L − p (cid:17) = (cid:16) − q − ip E i (cid:17) = 4 q ip E i ∈ J ( s ∗ p χ ) , because q pp = q pi q ip = q ii = − p = i, j . If m pi , m pj = 0, we have two possibilities: • q pp = − q ip q pi q jp q pj = − D (2 , α )), or • q pp = q − ip q − pi = − q jp q pj ∈ G ∪ G ∪ G .For the first case, or the second when q pp ∈ G , T p (cid:16) ((ad c E i ) E j ) (cid:17) = (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c . Using (47) and E p if q pp = − 1, or the quantum Serre relations(ad c E p ) E i = (ad c E p ) E j = 0 if q pp ∈ G , E i E p E j E p is a linear combination of greater words, so ( E p E i E p E j ) is alsoa linear combination of greater words. Then, (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c ∈ J + ( s ∗ p χ ) , by an analogous statement to Lemma 3.4 but for powers of hyperwords, and such relationis in I + ( s ∗ p χ ).For the remaining cases, q pp ∈ G ∪ G and T p (cid:16) ((ad c E i ) E j ) (cid:17) = (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c . We write ( E p E i E p E j ) as a linear combination of greater words using the quantum Serrerelations or (47), so T p (cid:16) ((ad c E i ) E j ) (cid:17) ∈ J + ( s ∗ p χ )by an analogous argument.If m pi = 1, m pj = 0, we have two possibilities. If q pp = − 1, then f q pi = − q ii = f q pi = f q pi − , so (54) is a generator of J ( s ∗ p χ ), as well as E p and E pj . By (9), we havethat q pi q pj (1 + f q pi ) E pij = [ E ppij , E ij ] c − [ E p , [ E pij , E ij ] c ] c ∈ J ( s ∗ p χ ) , so T p ( E ij ) = E pij ∈ J ( s ∗ p χ ). If not, then q − pp = f q pi = − 1, and we write E p E i E j E p E i E j asa linear combination of greater words modulo J ( s ∗ p χ ), using the quantum Serre relations(observe that ( q ij ) is twist equivalent to the original braiding). Therefore T p ( E ij ) = E pij ∈J ( s ∗ p χ ).If m pi > m pj = 0, then q pp = − f q pi ∈ G , and the proof follows in a similar way tothe case q pp = − 1, but using the relation (61).If m pi = m pj = 0, the proof follows easily, because q ii = q ij q ji = q jj = − 1, and T p (cid:16) E ij (cid:17) = E ij ∈ J ( s ∗ p χ ) by definition of the ideal. (cid:3) Lemma 3.11. Let i, j ∈ { , . . . , θ } be such that q m ij ii f q ij = 1 . Then T p (cid:0) (ad c E i ) m ij +1 E j (cid:1) ∈ J ( s ∗ p χ ) . Proof. ( i ) The case p = i was considered in the first part of Theorem 2.9.( ii ) Let p = j : we analyze all the possible values of m ip . If m ip = 0, then q ip q pi = 1, and T p ((ad c E i ) E p ) = E i F p L − p − q ip F p L − p E i = ( E i F p − F p E i ) L − p ∈ J ( s ∗ p χ ) . Consider m ip = 1; by (36) we have T p ((ad c E i ) E p ) = E + i,m pi F p L − p − q ip F p L − p E + i,m pi = (cid:16) F p E + i,m pi + ( m pi ) q pp ( q m pi − pp q pi q ip − L p E + i,m pi − (cid:17) L − p − q ip s + p χ ( m pi α p + α i , α p ) F p E + i,m pi L − p =( m pi ) q pp ( q m pi − pp q pi q ip − s ∗ p χ (( m pi − α p + α i , α p ) − E + i,m pi − + F p E + i,m pi L − p − q ip χ ( α i , − α p ) F p E + i,m pi L − p =( m pi ) q pp ( q − − m pi pp q − pi q − ip − q ip q pp E + i,m pi − . (71)If m pi = 1, we have by this identity and Remark 2.10: T p (cid:0) (ad c E i ) E p (cid:1) = T p (cid:0) [ E i , (ad c E i ) E p ] c (cid:1) = (cid:2) (ad c E p ) E i , a E i (cid:3) c , N NICHOLS ALGEBRAS OF DIAGONAL TYPE 23 where a m pi := ( m pi ) q pp ( q − − m pi pp q − pi q − ip − q ip q pp ∈ k × . This element is in J ( s ∗ p χ ) because m ip = 1, so (ad c E i ) E p ∈ J ( s ∗ p χ ). We consider now m pi ≥ 2; by Lemma 3.8, T p (cid:0) (ad c E i ) E p (cid:1) = h E + i,m pi , a m pi E + i,m pi − i c ∈ J ( s ∗ p χ ) . Consider now m ip = 2. We look at all the possible diagrams with two vertices andnote that m pi = 1, or there exists ζ ∈ G such that q ii = − ζ , q ip q pi = ζ , q pp = ζ . Inthe first case, q pp ∈ {− , q ii } , so this diagram is standard of type B , and (45) belongs to J ( s ∗ p χ ) by Lemma 3.8. Therefore T p (cid:0) (ad c E i ) E p (cid:1) = a h (ad c E p ) E i , (cid:2) (ad c E p ) E i , E i (cid:3) c i c ∈ J ( s ∗ p χ ) . For the second case, the braiding matrix of s ∗ p χ is q ii = − q ip q pi = ζ , q pp = ζ . Then T p (cid:0) (ad c E i ) E p (cid:1) = h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p ) E i (cid:3) c i c ∈ J ( s ∗ p χ ) , because (65) is a generator of this ideal. It remains to consider m ip ∈ { , , } . The unique diagram with m pi > ◦ − ζ − ζ ◦ ζ , where ζ ∈ G , q ii = − ζ and m ip = 3, m pi = 2; applying s p we obtain ◦ − − ζ ◦ ζ . By (66) we write E i E p E i E p E i E p E i as a linear combination of words beginning with E p ,or containing E p as a factor, or greater than this word for the order p < i . Multiplyingon the left by E p and using that E p ∈ J ( s ∗ p χ ), ( E p E i ) E p E i can be written as a linearcombination of greater words, modulo J ( s ∗ p χ ). By Lemma 3.4 we conclude that T p (cid:0) (ad c E i ) E p (cid:1) = " E + i, , (cid:20) E + i, , h E + i, , E + i, i c (cid:21) c c = [( E p E i ) E p E i ] c ∈ J ( s ∗ p χ ) . Finally we consider m pi = 1, so we have s p ( m ip α i + α p ) = m ip α i + ( m pi − α p ∈ ∆ s ∗ p χ + ,s p (( m ip + 1) α i + α p ) = ( m ip + 1) α i + m pi α p / ∈ ∆ s ∗ p χ + . If q pp = q pp = − m pi = 3 and ( E p E i ) E p E i is a linear combination of greater wordsmodulo J ( s ∗ p χ ), where we use first the quantum Serre relation (ad c E p ) E i = 0 to write E p E i E p as a combination of the words E p E i , E i E p and then (63), which also holds in U ( s ∗ p χ ). By this relation, T p (cid:0) (ad c E i ) E p (cid:1) = " E + i, , (cid:20) E + i, , h E + i, , E i i c (cid:21) c c ∈ J ( s ∗ p χ ) . In other case, q pp = − m ip ∈ { , , } , so we also have that T p (cid:0) (ad c E i ) m ip +1 E p (cid:1) = [ E m pi α i +( m pi − α p , (ad c E p ) E i ] c ∈ J ( s ∗ p χ ) , by (64), (66) or (68), depending on the value of m ip . ( iii ) Let p = j : if m pi = 0 (i.e. f q ip = 1), then q ii = q ii , q ij q ji = f q ij , so m ij = m ij , and(ad c E i ) m ij +1 E j = 0 holds in U ( s ∗ p χ ). Moreover, in U ( s ∗ p χ ) we have E p E i = q pi E p E i , so(ad c E i )(ad c E p ) X = q ip (ad c E p )(ad c E i ) X, for any X ∈ U ( s ∗ p χ ), by the second part of Lemma 3.7. By Remark 2.10 and the previousresults, in U ( s ∗ p χ ) we have T p (cid:0) (ad c E i ) m ij +1 E j (cid:1) = (ad c E i ) m ij +1 (ad c E p ) m pj E j = q m pj ( m ij +1) ip (ad c E p ) m pj (ad c E i ) m ij +1 E j = 0 . Consider now m pi = 0. If m ij = m pj = 0, we apply Lemma 3.7 to obtain T p ((ad c E i ) E j ) = (cid:2) (ad c E p ) m pi E i , E j (cid:3) c = (ad c E p ) m pi (cid:16)(cid:2) E i , E j (cid:3) c (cid:17) = 0 . where we use that q ij q ji = f q ij = 1, so in U ( s ∗ p χ ) it holds that (cid:2) E i , E j (cid:3) c = 0. It remains toconsider the case in which i , j and p determine a connected diagram, and m pi = 0. First we analyze the case m ij = 0 , m pj = 0. If q pp = − m pi = m pj = 1. Then q ij q ji = q ip q pi q jp q pj , and E p E i E p E j is a linear combination of greaterwords for the order p < i < j , modulo J ( s ∗ p χ ): • if q ij q ji = 1, it follows from (44), • if q ij q ji = 1, we write E i E p E j as a linear combination of other words by (47),where those words are greater than E i E p E j or begin with E p , so we multiply onthe left by E p and use that E p ∈ J ( s ∗ p χ ).In this way, T p ((ad c E i ) E j ) = (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c ∈ J ( s ∗ p χ ). If m pi = m pj = 1 and q pp = − 1, then (ad c E p ) E i , (ad c E p ) E j ∈ J ( s ∗ p χ ); by these relations and (ad c E i ) E j , E p E i E p E j can be written as a linear combination of greater words for the order p < i < j ,modulo J ( s ∗ p χ ), and also T p ((ad c E i ) E j ) ∈ J ( s ∗ p χ ) in this case.If m pj = 1 and m pi > m pj > m pi = 1), then q pp q pj q jp = 1, and q pp = − 1. Note that f q ij = s ∗ p χ ( α i , α j ) s ∗ p χ ( α j , α i ) = q m pi pp q pi q ip . If q m pi pp q pi q ip = 1, then (47) holds in U ( s ∗ p χ ), so we can write E i E p E j as a linear combi-nation of other words, greater than E i E p E j for the order p < i < j , or beginning with E p . Multiplying on the left by E m pi p , we express E m ij p E i E p E j as a linear combination ofgreater words, using that E m pi +1 p ∈ J ( s ∗ p χ ), or (ad c E p ) m pi +1 E i ∈ J ( s ∗ p χ ), so T p ((ad c E i ) E j ) = (cid:2) (ad c E p ) m pi E i , (ad c E p ) E j (cid:3) c ∈ J ( s ∗ p χ ) . If q m pi pp q pi q ip = 1 and q m pi +1 pp = 1, E m ij p E i E p E j is written as a linear combination of greaterwords for the same order using (ad c E p ) m pi +1 E i , (ad c E p ) E j and (ad c E i ) E j , so we obtainthe same conclusion. If q m pi pp q pi q ip = 1 and q m pi +1 pp = 1, then m pi = 2 or m pi = 3, and theconclusion follows from (50) or (53), respectively.If m pi , m pj > 1, there is only one possibility: m pi = m pj = 2. The proof is as above,expressing E p E i E p E j as a linear combination of greater words in the two possible cases:if q pp / ∈ G , using the quantum Serre relations; if q pp ∈ G , by (60) and E p . We consider now m pj = 0 , m ij = 0. Note that m ij ≤ 3, because we have aconnected diagram with three vertices and q ii = − q m ij +1 ii = 1. If m ij = 3, it corresponds N NICHOLS ALGEBRAS OF DIAGONAL TYPE 25 to a diagram of type super G (3): χ : ◦ − q − ◦ q q − ◦ q ! s p s ∗ p χ : ◦ − q ◦ − q − ◦ q . Using (49), E i ( E p E i ) E j can be written as a linear combination of other words modulo J ( s ∗ p χ ), which are greater than this word for the order p < i < j , or begin with E p (recallthat E i ∈ J ( s ∗ p χ )). Multiplying on the left by E p , ( E p E i ) E j is expressed as a linearcombination of greater words, modulo J ( s ∗ p χ ), using that E p ∈ J ( s ∗ p χ ). By Lemma 3.4, T p (cid:0) (ad c E i ) E j (cid:1) = (cid:20) (ad c E p ) E i , h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p )(ad c E i ) E j (cid:3) c i c (cid:21) c =[( E p E i ) E j ] c ∈ J ( s ∗ p χ ) . If m ij = 2, then m pi = 2 for the diagram χ : ◦ ζ ζ ◦ ζ ζ ◦ ζ ! s p s ∗ p χ : ◦ ζ ζ ◦ ζ ζ ◦ ζ , where ζ ∈ G , or m pi = 1. In the first case, we use (51), E p ∈ J ( s ∗ p χ ) and the quantumSerre relations, and call c = s ∗ p χ (4 α p + 2 α i + α j , α p + α i ) = χ (2 α i + α j , α p + α i ), so T p (cid:0) (ad c E i ) E j (cid:1) = h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p ) (ad c E i ) E j (cid:3) c i c = (ad c E p ) (cid:18)h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p ) (ad c E i ) E j (cid:3) c i c (cid:19) = − c (ad c E p ) (cid:16)(cid:2) [ E ppi , E ppij ] , E pi (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) . If m pi = 1 and q pp = − 1, using (ad c E p ) E i we write E p E i E p E i E j E i as a linear com-bination of greater words and E p E i E j E i , for the order induced by p < i < j , modulo J ( s ∗ p χ ). Using now (ad c E p ) E j , and (ad c E i ) E j when f q ip = 1, or (50) in other case, E p E i E p E i E j E i is expressed as a linear combination of greater words modulo J ( s ∗ p χ ). If m pi = 1 and q pp = − 1, then (48) is a generator of J ( s ∗ p χ ), so h(cid:2) (ad c E p ) E i , (ad c E p )(ad c E i ) E j (cid:3) c , E i i c ∈ J ( s ∗ p χ ) , or (45) (considered for the pair p, i ) is a generator of the ideal. In any case we have that T p (cid:0) (ad c E i ) E j (cid:1) = h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p )(ad c E i ) E j (cid:3) c i c = (ad c E p ) h E i , (cid:2) (ad c E p ) E i , (ad c E p )(ad c E i ) E j (cid:3) c i c ∈ J ( s ∗ p χ ) . Now we fix m ij = 1. If m pi = 1, we analyze each different possible diagram. • If q pp = − 1, then s ∗ p χ is twist equivalent to χ (restricted to the vertices p, i, j ), and E p E i E p E i E j can be expressed as a linear combination of greater words modulo J ( s ∗ p χ ), using the quantum Serre relations(ad c E p ) E i = (ad c E i ) E j = (ad c E p ) E j = 0 . • If q pp = − q ii q ip q pi = 1, then q ii = − q ip q pi q ij q ji = 1. In this way (44)is a generator of the ideal, and by Lemma 3.7, T p (cid:0) (ad c E i ) E j (cid:1) = (cid:2) (ad c E p ) E i , (ad c E p )(ad c E i ) E j (cid:3) c = (ad c E p ) (cid:16)(cid:2) E i , (ad c E p )(ad c E i ) E j (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) . • If q pp = − q ii q ip q pi = 1, then q pj q jp = q pj q jp = 1, q ij q ji = f q ij = − q − ii = q − pp q − ip q − pi q − ii = − q ip q pi q ij q ji = − , so (54) or (55) are generators of J ( s ∗ p χ ), and then T p (cid:0) (ad c E i ) E j (cid:1) ∈ J ( s ∗ p χ ).Now we fix m pi = 1. The possible connected diagrams of rank three with these condi-tions must verify m pi = 2, m ip = 1. Using the quantum Serre relations (ad c E p ) E j =(ad c E p ) E i = 0 if q pp = − 1, or (46), (52) depending on the case, E p E i E j E p E i can beexpressed as a linear combination of greater words for the order p < i < j , and by Lemma3.4, (cid:2) (ad c E p ) (ad c E i ) E j , (ad c E p ) E i (cid:3) c ∈ J ( s ∗ p χ ) . By Lemma (3.7) we conclude that T p (cid:0) (ad c E i ) E j (cid:1) = (cid:2) (ad c E p ) E i , (ad c E p ) (ad c E i ) E j (cid:3) c = (ad c E p ) (cid:16)(cid:2) (ad c E p ) E i , (ad c E p ) (ad c E i ) E j (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) . Finally we consider m ij , m pj = 0, so each pair of vertices is connected. If m ij = 2,there is just one possibility, χ : ◦ − q ❉❉❉❉❉❉❉❉ ◦ q q − ⑤⑤⑤⑤⑤⑤⑤⑤ q − ◦ − ! s p s ∗ p χ : ◦ − q ◦ − q − ◦ q which is a diagram of type super G (3). By (49) and Lemma 3.5 we have that T p (cid:0) (ad c E i ) E j (cid:1) = (cid:20) (ad c E p ) E i , h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c i c (cid:21) c ∈ J ( s ∗ p χ ) . The remaining case is m ij = 1. If m pi = m pj = 1, there are two possible cases: • q pp = − 1; in this case (48) is a generator of the ideal J ( s ∗ p χ ) by definition, and byLemma 3.5 we have that T p (cid:0) (ad c E i ) E j (cid:1) = h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c i c ∈ J ( s ∗ p χ ) . • q pp = − q pp q pi q ip = q pp q pj q jp = 1; s ∗ p χ is twist equivalent to χ , so (47) is a gener-ator of J ( s ∗ p χ ). Using also the quantum Serre relations (ad c E p ) E i , (ad c E p ) E j ,(ad c E i ) E j , E p E i E p E i E p E j is written as a linear combination of greater words,modulo J ( s ∗ p χ ), so as before T p (cid:0) (ad c E i ) E j (cid:1) ∈ J ( s ∗ p χ ).It remains to consider the following braiding: ◦ ζ ζ − ❇❇❇❇❇❇❇❇ ◦ − ζ − ζ − − ζ − ⑤⑤⑤⑤⑤⑤⑤⑤ ◦ − , q pp = ζ ∈ G , m ij = m pj = 1 , m pi = 2 . The diagram of s ∗ p χ is ◦ − − ◦ ζ ζ − ◦ − . Then (45) holds for p , i , and so E i E p E i E p is expressed as a linear combination of other words of the same Z θ -degree. Multiplying onthe left by E p , on the right by E j , and using that E p = 0, E p E i E p E i E p E j can be writtenas a linear combination of greater words, so T p (cid:0) (ad c E i ) E j (cid:1) = h (ad c E p ) E i , (cid:2) (ad c E p ) E i , (ad c E p ) E j (cid:3) c i c ∈ J ( s ∗ p χ ) . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 27 Therefore we analyze all the cases and the proof is completed. (cid:3) Lemma 3.12. Let i, j, k ∈ { , . . . , θ } be such that q jj = − , f q ik = f q ij f q jk = 1 . Then, T p (cid:0) [ E ijk , E j ] c (cid:1) ∈ J ( s ∗ p χ ) . Proof. Note that in U ( χ ) we have the following identity, using the condition on the scalars,(9) and (ad c E i ) E k = E j = 0:[ E ijk , E j ] c = q ij q kj [ E j , E ijk ] c = q ij q kj [ E ji , E jk ] c , so it is enough to prove that one of these relations is applied in J ( s ∗ p χ ) by T p for eachpossible diagram. Let p = j . Note that q pp = − q ip q pi q pk q kp = q ik q ki = 1, so (ad c E i ) E k and (44) are generators of J ( s ∗ p χ ). By (36) we have: T p (cid:0) [ E ipk , E p ] c (cid:1) = (cid:2) a E i , (ad c E p ) E k (cid:3) c F p L − p + q ip q kp F p L − p (cid:2) a E i , (ad c E p ) E k (cid:3) c = a (cid:16) q − ip q − pp q − kp + q ip q kp (cid:17) F p L − p (cid:2) E i , (ad c E p ) E k (cid:3) c + a q − pk (1 − q pk q kp ) ( E i E k + q ip q pk q ik E k E i )= a q − pk (1 − q pk q kp ) (cid:0) ad c E i (cid:1) E k ∈ J ( s ∗ p χ ) . Let p = i , which is analogous to the case p = k . By (69) and (71), T p (cid:0) [ E jp , E jk ] c (cid:1) = (cid:2) a m pj (ad c E p ) m pj − E j , (ad c E p ) m pj (ad c E j ) E k (cid:3) c . Note that m pj = 1 , 2. If m pj = 1, q pp = − m pj = 2, q pp / ∈ G , then q m pj pp q pj q jp = 1. Inthis case, − q jj = q pk q kp = q pj q jp q kj q jk = 1, so (ad c E p ) m pj +1 E j = 0 = (ad c E p ) E k . Notealso that (44) is a generator of J ( s ∗ p χ ), hence T p (cid:0) [(ad c E j ) E i , (ad c E j ) E k ] c (cid:1) ∈ J ( s ∗ p χ ). If q pp = − 1, then q jj q jp q pj = q jj q jk q kj = 1 , q pk q kp = 1 , hence in U ( s ∗ p χ ), (ad c E j ) E p = (ad c E j ) E k = 0 if q jj = − 1, or (44) if q jj = − 1: in thisway, T p (cid:0) [ E jp , E jk ] c (cid:1) = a (cid:2) E j , (ad c E p )(ad c E j ) E k (cid:3) c ∈ J ( s ∗ p χ ) . The remaining case is q pp ∈ G : by (46) we obtain that T p (cid:0) [ E jp , E jk ] c (cid:1) = a (cid:2) (ad c E p ) E j , (ad c E p ) (ad c E j ) E k (cid:3) c ∈ J ( s ∗ p χ ) . Finally take p = i, j, k . First, the proof is trivial if p is not connected with i , j , k ,because in such case s ∗ p χ is twist equivalent to χ , and then T p (cid:0) [ E ijk , E j ] c (cid:1) = (cid:2) E ijk , E j (cid:3) c ∈ J ( s ∗ p χ ) . Now, if p is connected just with i (or analogously, just with k ), we have q jj = − , q ji q ij q jk q kj = 1 , q ik q ki = 1 , and by Lemma 3.7: T p (cid:0) [ E ijk , E j ] c (cid:1) = (cid:2) (ad c E p ) m pi ( E ijk ) , E j (cid:3) c = (ad c E p ) m pi (cid:16)(cid:2) E ijk , E j (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) . If p is connected just with j , then f q pj ∈ { f q ij , f q jk } , and m pj = 1. We assume that f q pj = f q ij = f q kj − . If q pp f q ip = 1, s ∗ p χ is twist equivalent to χ ; in other case, q pp = − 1, and then q jj = q pj q jp , q pj q jp = q − pj q − jp = q kj q jk = q kj q jk , so in both cases (cid:2) (ad c E p )(ad c E j ) E k , E j (cid:3) c ∈ J ( s ∗ p χ ). Therefore E p E j E k E p E j E i is alinear combination of greater words (for the order p < j < k < i ), so we have that T p (cid:0) [(ad c E j ) E k , (ad c E j ) E i ] c (cid:1) = (cid:2) (ad c E p )(ad c E j ) E k , (ad c E p )(ad c E j ) E i (cid:3) c ∈ J ( s ∗ p χ ) . The remaining case is that p is connected with two consecutive vertices. We canassume that p is connected with i and j . There exist six possible diagrams satisfying theseconditions. For two of them, the diagram of s ∗ p χ is the same. ◦ − − ζ − ❊❊❊❊❊❊❊❊❊ ◦ − − ζ ζ ◦ − ζ ◦ q , ζ ∈ G , ◦ − q q − ❊❊❊❊❊❊❊❊❊ ◦ − q − q − ◦ q kk ◦ q , q = − . In the second case, m pi = m pj = 1. Consider the order p < j < i < k : T p (cid:0) [ E ji , E jk ] c (cid:1) = h(cid:2) E pj , E pi (cid:3) c , (cid:2) E pj , E k (cid:3) c i c = [ E p E j E p E i E p E j E k ] . Note that E j E p E i is a linear combination of other words of the same degree by (47), wherethose words are greater than E j E p E i or begin with E p . In all the cases we conclude that T p (cid:0) [ E ji , E jk ] c (cid:1) ∈ J ( s ∗ p χ ). The proof for the first case is analogous.For the remaining four diagrams, we write also the diagram corresponding to s ∗ p χ : ◦ − q − q ❉❉❉❉❉❉❉❉ ◦ − qq − ◦ q kk s p / / /o/o/o/o/o/o/o ◦ q q − ❉❉❉❉❉❉❉❉ ◦ q − qq ◦ q kk ◦ − ◦ − , ◦ − q − q − ❉❉❉❉❉❉❉❉ ◦ − q − − ◦ q kk s p / / /o/o/o/o/o/o/o ◦ − q − − q ❋❋❋❋❋❋❋❋❋ ◦ − q − − ◦ q kk ◦ − ◦ − , ◦ − q q − ❉❉❉❉❉❉❉❉ ◦ − q − q ◦ q s p / / /o/o/o/o/o/o/o ◦ q − q ❊❊❊❊❊❊❊❊ ◦ q q − q − ◦ q ◦ − ◦ − , ◦ − − ζ − ζ ❉❉❉❉❉❉❉❉ ◦ − − ζ ζ ◦ − ζ s p / / /o/o/o/o/o/o/o ◦ − ζ − ζ ❉❉❉❉❉❉❉❉ ◦ ζ − ζ ζ ◦ − ζ ◦ − ◦ − . Note that m pi = m pj = 1. If we fix the order p < j < i < k , and obtain that T p (cid:0) [ E ji , E jk ] c (cid:1) = [ E p E j E p E i E p E j E k ]. We can write E p E j E p E i E p E j E k as a linear com-bination of greater words modulo J ( s ∗ p χ ): in the first case, using (48); for the remainingcases, we use (59). (cid:3) Lemma 3.13. Let i, j ∈ { , . . . , θ } be such that q jj = − , and also q ii = ± f q ij ∈ G , or q ii f q ij ∈ G . Then, T p (cid:0) [( E iij , E ij ] c (cid:1) ∈ J ( s ∗ p χ ) , for any p ∈ { , . . . , θ } . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 29 Proof. We denote x := [( E iij , E ij ] c . We begin with the case p = j . Note that m pi = 1(because q pp = − f q pi = 1), 3 α i + 2 α p / ∈ ∆ χ + , so s p (3 α i + 2 α p ) = 3 α i + α p / ∈ ∆ s ∗ p χ + . Using (71) and (ad c E i ) E p ∈ J ( s ∗ p χ ), we obtain that T p ( x ) = a a h(cid:2) E pi , E i (cid:3) c , E i i c ∈ J ( s ∗ p χ ) . Now let p = i . By Lemma 3.4 it is equivalent to prove that T p ( x ′ ) ∈ J ( s ∗ p χ ) , where x ′ := h E jp , (cid:2) E jp , E p (cid:3) c i c , because we have proved that T p apply the generating relations of degree less than x inelements of J ( s ∗ p χ ). By (36), T p ( x ′ ) = a a (cid:2) (ad c E p ) E j , E j (cid:3) c ∈ J ( s ∗ p χ ) , because it holds that q jj = − 1, or q jj q jp q pj = 1.Finally, let p = i, j ; the case m pi = m pj = 0 follows easily as in the previous Lemmata,so consider the case in which p , i , j determine a connected subdiagram of rank three. Wenote that q ii ∈ G .We take first m pi = 0, m pj = 0. The possible braidings verify that m pi = 1, so for theorder p < i < j , T p ( x ) = (cid:20)h E pi , (cid:2) E pi , E j (cid:3) c i c , (cid:2) E pi , E j (cid:3) c (cid:21) c = (cid:2) E p E i ( E p E i E j ) (cid:3) c , where we use that (ad c E p ) E j = 0 in U ( s ∗ p χ ). As q pi q ip q ii = 1 or q pi q ip = ± q ii , we have that q ii = − 1, or q ii q ip q pi = 1, or q ii q ip q pi = 1, or q ii = − q ip q pi ∈ G , so E p E i E p E i E j E p E i E j can be expressed as a linear combination of greater words modulo J ( s ∗ p χ ), using thequantum Serre relations or (45). We deduce that T p ( x ) ∈ J ( s ∗ p χ ), using the Lemma 3.4.Now let m pi = 0, m pj = 0. We note that m pj = 1 for any possible diagram, and in U ( s ∗ p χ ) we have that: T p ( x ) = (cid:2) (ad c E i ) (ad c E p ) E j , (ad c E i )(ad c E p ) E j (cid:3) c = q ip (cid:2) (ad c E p )(ad c E i ) E j , (ad c E p )(ad c E i ) E j (cid:3) c = q ip (ad c E p ) (cid:2) (ad c E i ) E j , (ad c E p )(ad c E i ) E j (cid:3) c = q ip (ad c E p ) (cid:2) (ad c E i ) (ad c E j ) E p , (ad c E i ) E j (cid:3) c = 0 , by applying (69) (because (ad c E p ) E i = 0), (9), and also that (46) is a generator of J ( s ∗ p χ ),because q ii = q ii ∈ G , m ji = m jp = 1, and (45) is another generator by Lemma 3.8.Finally, consider m pi , m pj = 0. There exists just one possible braiding: q pp = − q pj q jp , q ii = − f q ij = q − pi q − ip . The diagram of s ∗ p χ is ◦ − − ◦ − q ii ◦ − , and thesolution is analogous to the previous case, but now we use (48). (cid:3) Lemma 3.14. Let i, j, k ∈ { , . . . , θ } be such that q ii = ± f q ij ∈ G , f q ik = 1 , and q jj f q ij = q jj f q jk = 1 or q jj = − , f q ij f q jk = 1 . Then, for any p ∈ { , . . . , θ } , T p (cid:0) [ E iijk , E ij ] c (cid:1) ∈ J ( s ∗ p χ ) . Proof. We denote x = [ E iijk , E ij ] c . We begin with the case p = k . In all the cases wehave that m kj = 1, and s ∗ p χ satisfies the same conditions, so (46) is a generator of J ( s ∗ p χ ).Then, T p ( x ) ∼ = (cid:2) (ad c E i ) ( a E j ) , (ad c E i )(ad c E k ) E j (cid:3) c ∼ = a q ik (cid:2) E iij , E kij (cid:3) c ∼ = a q ik (cid:2) E iijk , E ij (cid:3) c ∼ = 0 (mod J ( s ∗ p χ )) , where we apply first (71), then (69), (ad cb E k ) E i = 0 for the second line, y finally (9) plusthe fact that (45) is a generator of J ( s ∗ p χ ). The cases p = i , p = j are proved in a similarway to the case p = i of previous Lemma.Finally take p = i, j, k , and assume that p is connected with at least one of the othervertices; in other case the proof is easy as above. We have two possible cases: m pi = 1, m pj = m pk = 0, or m pk = 1, m pj = m pi = 0. For the first one, T p ( x ) = (cid:20)h E pi , (cid:2) E pi , E jk (cid:3) c i c , [ E pi , E j ] c (cid:21) c , and we have two possibilities: • if q pp = − 1, then q ii f q ip = 1, and q ii = − 1, so (48) is a generator of J ( s ∗ p χ ) for thesubdiagram determined by p, i, j . Therefore T p ( x ) ∈ J ( s ∗ p χ ). • if q pp = − 1, then q pp = q − ip q − pi = q ii , so s ∗ p χ is twist equivalent to χ and (46)is a generator of J ( s ∗ p χ ). Then T p ( x ) ∈ J ( s ∗ p χ ), because it is obtained after toapply (ad cp E p ) to (46) and multiply by a non-zero scalar, where we use also thequantum Serre relations involving E p .For the second case, we use (ad c E p ) E i , (ad c E p ) E j ∈ J ( s ∗ p χ ) to obtain that T p ( x ) ∼ = (cid:2) (ad c E i ) (ad c E j )(ad c E p ) E k , (ad c E i ) E j (cid:3) c ∼ = q ip q jp (cid:2) (ad c E p )(ad c E i ) (ad c E j ) E k , (ad c E i ) E j (cid:3) c ∼ = q ip q jp (ad c E p ) (cid:16)(cid:2) (ad c E i ) (ad c E j ) E k , (ad c E i ) E j (cid:3) c (cid:17) ∼ = 0 (mod J ( s ∗ p χ )) , by (69) and the fact that (46) is a generator of J ( s ∗ p χ ). (cid:3) Lemma 3.15. Let i, j, k ∈ { , . . . , θ } be such that f q ik , f q ij , f q jk = 1 . Then, for any p , T p (cid:18) E ijk − − f q jk q kj (1 − f q ik ) [ E ik , E j ] c − q ij (1 − f q jk ) E j E ik (cid:19) ∈ J ( s ∗ p χ ) . Proof. Let x = E ijk − − f q jk q kj (1 − f q ik ) [ E ik , E j ] c − q ij (1 − f q jk ) E j E ik . By a direct computation weobtain the same relation, up to an scalar, if we permute the vertices i , j , k , where we usethat f q ik f q ij f q jk = 1, so it is enough to consider one of these permutations for each p .Consider then p = k , which is analogous to take p = i or p = j . Note that { m pi , m pj } = { , } , or { m pi , m pj } = { , } , so we fix m pj = 1, m pi ∈ { , } . By (71), T p ( E ijk ) = ( q − pp q − pj q − jp − q jp q pp (cid:2) (ad c E p ) m pi E i , E j (cid:3) c ,T p (cid:0) [ E ik , E j ] c (cid:1) = ( q − − m pi pp q − pi q − ip − q ip q pp (cid:2) (ad c E p ) m pi − E i , (ad c E p ) E j (cid:3) c ,T p ( E j E ik ) = ( q − − m pi pp q − pi q − ip − q ip q pp (ad c E p ) E j (ad c E p ) m pi − E i . If m pi = 2, or m pi = 1, q pp = − 1, then q ik q ki , q ij q ji , q jk q kj = 1 and we deduce that T p ( x ) ∈ J ( s ∗ p χ ) from the fact that (47) is a generator of J ( s ∗ p χ ), because we can writethen E m pi p E i E j as a linear combination of greater words (for the order on the letters N NICHOLS ALGEBRAS OF DIAGONAL TYPE 31 p < i < j ), modulo J ( s ∗ p χ ), y apply then Lemma 3.4. If q pp = − q ij q ji = 1, so(ad c E i ) E j ∈ J ( s ∗ p χ ). By a direct computation, there exists a ∈ k × such that T p ( x ) = a (ad c E p )(ad c E i ) E j ∈ J ( s ∗ p χ ) . Let p = i, j, k . We note that p is not connected with any of the other vertices (sothe proof follows easily as in the previous Lemmata), or p is connected just with one ofthese vertices. For the last case we can assume that m pi = 0, so the unique possibility is m pi = m ip = 1. Then f q ik = q ik q ki , f q ij = q ij q ji , f q jk = q kj q jk = 1, so (47) is a generator of J ( s ∗ p χ ). By Lemma 3.7 and the relations (ad c E p ) E j = (ad c E p ) E k = 0, we deduce that T p ( x ) is obtained, up to a non-zero scalar, after to apply (ad c E p ) to (47), modulo J ( s ∗ p χ ),so T p ( x ) ∈ J ( s ∗ p χ ). (cid:3) Lemma 3.16. Let i, j, k ∈ { , . . . , θ } be such that ( i ) q ii = q jj = − , f q ij = f q jk − , f q ik = 1 , or ( ii ) f q ij = q jj = − , q ii = − f q jk ∈ G , f q ik = 1 , or ( iii ) q kk = f q jk = q jj = − , q ii = − f q ij ∈ G , f q ik = 1 , or ( iv ) q jj = − , f q ij = q − ii , f q jk = − q − ii , f q ik = 1 , or ( v ) q ii = q jj = q kk = − , ± f q ij = f q jk ∈ G , f q ik = 1 ,Then, for any p , T p (cid:0)(cid:2) [ E ij , E ijk ] c , E j (cid:3) c (cid:1) ∈ J ( s ∗ p χ ) .Proof. Denote x = (cid:2) [ E ij , E ijk ] c , E j (cid:3) c ; we analyze each case.( i ) We begin with the case p = k ; by (71) and as E i , E j , (ad c E i ) E p are generators of J ( s ∗ p χ ) (note that s ∗ p χ is twist equivalent to χ ), we have that T p ( x ) ∼ = a h(cid:2) E ipj , E ij (cid:3) c , E pj i c ∼ = a q ip h(cid:2) E pij , E ij (cid:3) c , E pj i c ∼ = a h(cid:2) E pji , E ji (cid:3) c , E pj i c ∼ = a (cid:2) E p E j E i E j E i E p E j (cid:3) c (cid:0) mod J ( s ∗ p χ ) (cid:1) , for some a ∈ k × , where we use the order on the letters p < j < i . As (48) is alsoa generator of J ( s ∗ p χ ), we can write E j E i E j E i E p E j as a linear combination of otherwords, greater than this word or beginning with E p . Multiplying on the left by E p andusing the quantum Serre relations E p E j E i E j E i E p E j is expressed as a linear combinationof greater words modulo J ( s ∗ p χ ), so by Lemma 3.4, T p ( x ) ∈ J ( s ∗ p χ ).Let p = j ; note that m pi = m pk = 1. Also, q − ii = q ik q ki , so (ad c E i ) E k ∈ J ( s ∗ p χ ); use(71) and work as in the case p = i of Lemma 3.13 to obtain that T p ( x ) = a (cid:2) E i , E ipk (cid:3) c F p L − p − a q ip q kp F p L − p (cid:2) E i , E ipk (cid:3) c = b (ad c E i ) E k ∈ J ( s ∗ p χ ) , for some b ∈ k × .Let now p = i . As in the previous Lemmata, it is enough to prove the statement for x ′ := (cid:2) [ E kjp , E jp ] c , E j (cid:3) c . We apply (71) to obtain, for the order on the letters k < i < j , T p ( x ′ ) = (cid:20)h(cid:2) E k , a E j (cid:3) c , E j i c , (ad c E p ) E j (cid:21) c = a (cid:2) E k E j E p E j (cid:3) c . As q jj q ji q ij = q jj q jk q kj = 1, we deduce by (50) if q jj ∈ G , or by (ad c E j ) E p =(ad c E j ) E k = 0, if q jj / ∈ G , that E k E j E p E j is a linear combination of greater words, so T p ( x ′ ) ∈ J ( s ∗ p χ ), and then T p ( x ) ∈ J ( s ∗ p χ ). If p = i, j, k , then p is not connected with these three vertices, or p is connected justwith i , or p is connected with two vertices. For the second case we have that q pp f q pi = 1,or q pp = − f q ip f q ij = 1, so T p ( x ) ∼ = h(cid:2) E pij , E pijk (cid:3) c , E j i c ∼ = (ad c E p ) (cid:18)h(cid:2) E pij , E ijk (cid:3) c , E j i c (cid:19) ∼ =(ad c E p ) (cid:16)(cid:2) E p E i E j E i E j E k E j (cid:3) c (cid:17) mod J ( s ∗ p χ )by using first Lemma 3.7, then (ad c E p ) E j , (ad c E p ) E j , (ad c E p ) E j ∈ J ( s ∗ p χ ), and fixingthe order p < i < j < k . We conclude that T p ( x ) ∈ J ( s ∗ p χ ) by using (56) if q pp = − 1, orusing the quantum Serre relations corresponding to ad c E p to write E p E i E j E i E j E k E j asa linear combination of greater words and apply Lemma 3.4 to deduce that (cid:2) E p E i E j E i E j E k E j (cid:3) c ∈ J ( s ∗ p χ ) . For the last case, we have two possibilities: • p is connected with i and j , in which case the proof follows by the fact that (58)is a generator of the ideal, or • p is connected with j and k , in which case it follows because (57) is a generator of J ( s ∗ p χ ).( ii ) , ( iii ) , ( iv ) , ( v ) If p ∈ { i, j, k } the proof is completely analogous to the previous case.Let p = i, j, k , so p is not connected with any of these vertices, or it connected onlywith i , or only with k . The first case is easy. For the second case, m pi = 1, because q pp = 1or q − pp = f q ip = − 1, and the solution follows as in the previous case. For the last case, m pk = 1 and the proof is also analogous, considering the previous x ′ . (cid:3) Lemma 3.17. ( i ) Let i, j, k, l ∈ { , . . . , θ } be such that q kk = − , f q jk = f q lk − = q ll , q jj f q ij = q jj f q jk = f q ik = e q il = f q jl = 1 . Then, for all p , T p (cid:16)(cid:2)(cid:2) [ E ijkl , E k ] c , E j (cid:3) c , E k (cid:3) c (cid:17) ∈J ( s ∗ p χ ) . ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = q jj = − , f q ij = f q jk − = q kk = ± , f q ik = 1 .Then, for all p , T p (cid:16)(cid:2)(cid:2) E ij , [ E ij , E ijk ] c (cid:3) c , E j (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) .Proof. ( i ) The proof is analogous to ( i ) of the previous Lemma, because if p = i, j, k, l isconnected with some of them, then p is connected only with i with the same conditions.( ii ) If p ∈ { i, j, k } the proof is completely analogous to the previous Lemma. If p = i, j, k isconnected with some of them, then p is connected only with i and q pp = − f q pi = − f q ij ∈ G . Anyway, the proof is analogous to the previous Lemma. (cid:3) Lemma 3.18. ( i ) Let i, j, k, l ∈ { , . . . , θ } be such that q ll = f q lk − = q kk = f q jk − = q , f q ij = q − ii = q for some q ∈ k × , q jj = − , f q ik = e q il = f q jl = 1 . Then, for all p , T p (cid:16)(cid:2)(cid:2) [ E ijk , E j ] c , [ E ijkl , E j ] c (cid:3) c , E jk (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) . ( ii ) Let i, j, k, l ∈ { , . . . , θ } be such that f q jk = f q ij = q − jj ∈ G ′ ∪ G ′ , q ii = q kk = − , f q ik = e q il = f q jl = 1 , f q jk = f q lk . Then, for all p , T p (cid:16)(cid:2)(cid:2) E ijk , [ E ijkl , E k ] c (cid:3) c , E jk (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) .Proof. ( i ) Let x = (cid:2)(cid:2) E ijk , [ E ijkl , E k ] c (cid:3) c , E jk (cid:3) c . If p / ∈ { i, j, k, l } , then p is not connectedwith i, j, k, l , so we consider the case p ∈ { i, j, k, l } . If p = i , the diagram for s ∗ p χ is thesame, and T p ( x ) corresponds to a relation of degree 3 α p + 5 α j + 3 α k + α l , obtained after N NICHOLS ALGEBRAS OF DIAGONAL TYPE 33 to apply ad c E p to (58). Similar situations hold when p = k and p = l . If p = j , then T p ( x ) corresponds to the relation (48), and the proof follows.( ii ) The proof is very similar to previous item, because p ∈ { i, j, k, l } , or p is not connectedwith i, j, k, l . (cid:3) Lemma 3.19. Let i, j, k, l ∈ { , . . . , θ } be such that one of the following situations hold: ◦ q kk = − , q ii = f q ij − = q jj , f q kl = q − ll = q jj , f q jk = q − jj , f q ik = e q il = f q jl = 1 , ◦ q ii = f q ij − = − q − ll = − f q kl , q jj = f q jk = q kk = − , f q ik = e q il = f q jl = 1 , or ◦ q jj = f q jk − ∈ G , q ii = f q ij − = q ll = f q kl − = − q jj , q kk = − , f q ik = e q il = f q jl = 1 .Then, T p (cid:16)(cid:2) [ E ijkl , E j ] c , E k (cid:3) c − q jk ( f q ij − − q jj ) (cid:2) [ E ijkl , E k ] c , E j (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) , for all p .Proof. No one of these diagrams can be extended in order to have a connected diagramwith finite root system. In consequence, it is enough to consider (as in the previousLemmata) the cases p ∈ { i, j, k, l } . The proof for these cases is similar to the Lemma 3.15,up to consider the necessary relations under the conditions of these new situations. (cid:3) Lemma 3.20. ( i ) Let i, j, k ∈ { , . . . , θ } be such that q kk = q jj = f q ij − = f q jk − ∈ G , f q ik = 1 , q ii = q kk . Then, for all p , T p (cid:0)(cid:2) [ E iij , E iijk ] c , E ij (cid:3) c (cid:1) ∈ J ( s ∗ p χ ) . ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = f q ij − ∈ G , q jj = f q jk − = q ii , f q ik = 1 , q kk = q ii . Then, for all p , T p (cid:0) [[ E ijk , E j ] c , E k ] c − (1 + f q jk ) − q jk (cid:2) [ E ijk , E k ] c , E j (cid:3) c (cid:1) ∈ J ( s ∗ p χ ) .Proof. For both cases, there exist no extensions of these diagrams. In consequence, it isenough to consider the case p ∈ { i, j, k } .( i ) The cases p = k and p = j follows easily because the diagram for s ∗ p χ in these casescoincide with the one for χ .If p = i , then T p (cid:0)(cid:2) [ E ppj , E ppjk ] c , E pj (cid:3) c (cid:1) is, up to an scalar, [ E pjk , E j ] c , which belongsto J ( s ∗ p χ ) because (ad c E j ) E p , (ad c E j ) E k ∈ J ( s ∗ p χ ) (we have that q jj f q jk = q jj f q jp = 1).( ii ) The proof is similar to the one for Lemma 3.15. (cid:3) Lemma 3.21. ( i ) Let i, j, k ∈ { , . . . , θ } be such that q jj = f q ij = f q jk ∈ G , f q ik = 1 .Then, for all p , T p (cid:0)(cid:2) [ E ijk , E j ] c , E j (cid:3) c (cid:1) ∈ J ( s ∗ p χ ) . ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q jj = f q ij = f q jk ∈ G , f q ik = 1 . Then, for all p , T p (cid:16)(cid:2)(cid:2) [ E ijk , E j ] c , E j (cid:3) c , E j (cid:3) c (cid:17) ∈ J ( s ∗ p χ ) .Proof. ( i ) Let x = (cid:2) [ E ijk , E j ] c , E j (cid:3) c . For the case p = k , note that m pj = 1 in all thecases, so for the order i < p < j on the letters we have by (71): T p ( x ) = a h(cid:2) E ij , E pj (cid:3) c , E pj i c = (cid:2) E i E j E p E j E p E j (cid:3) c . As (ad c E p ) E j ∈ J ( s ∗ p χ ), or (48) is a generator of J ( s ∗ p χ ), and also E j , (ad c E k ) E i ∈J ( s ∗ p χ ), E i E j E k E j E k E j can be written as a linear combination of greater words modulo J ( s ∗ p χ ), so T p ( x ) ∈ J ( s ∗ p χ ).Consider now p = j . By (71) and the relations defining J ( s ∗ p χ ), T p ( x ) = (cid:20)h(cid:2) E i , (ad c E p ) E k (cid:3) c , F p L − p i c , F p L − p (cid:21) c = a (ad c E i ) E k , for some a ∈ k × , so T p ( x ) ∈ J ( s ∗ p χ ).Let p = i . It is equivalent to prove that T p ( x ′ ) ∈ J ( s ∗ p χ ), where x ′ = (cid:2) [ E kjp , E j ] c , E j (cid:3) c .We note that m ij = 1 for all the possible diagrams, so T p ( x ) = a h(cid:2) E kj , E pj (cid:3) c , E pj i c , anda proof similar to the case p = k tells us that T p ( x ′ ) ∈ J ( s ∗ p χ ), so T p ( x ) ∈ J ( s ∗ p χ ).Finally, if p = i, j, k , then p is not connected to any of these vertices, or p is connectedonly with i , or it is connected only with k . The proof of the first case is again trivial, andfor the other two cases T p ( x ) ∈ J ( s ∗ p χ ), using Lemma 3.7 and the fact that s ∗ p χ is twistequivalent to χ .( ii ) The proof is analogous to ( i ) . (cid:3) Lemma 3.22. ( i ) Let i, j, k ∈ { , . . . , θ } be such that q ii = f q ij = − , q jj = f q jk − = − , f q ik = 1 . Then, for all p , T p (cid:0) [ E ij , E ijk ] c (cid:1) ∈ J ( s ∗ p χ ) . ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q jj = q kk = f q jk = − , q ii = − f q ij ∈ G , f q ik = 1 .Then, for all p , T p (cid:0) [ E iijk , E ijk ] c (cid:1) ∈ J ( s ∗ p χ ) .Proof. ( i ) Let x = [ E ij , E ijk ] c . If p = k , m pj = 1 in all the possible diagrams and then: T p ( x ) = a [ E ikj , E ij ] c = a q ip [ E kij , E ij ] c . We consider the two possible values of q pp . When q pp = − 1, the diagram is of Cartan C type, associated to a root of order 4, and s ∗ p χ is twist equivalent to χ . Therefore T p ( x ) ∈ J ( s ∗ p χ ), using that (54) is again a generator of J ( s ∗ p χ ). If q pp = − 1, then q jj ∈ G ∪ G ∪ G , and q jj = f q ij = q ii = − 1, so (43) is a generator of J ( s ∗ p χ ) and T p ( x ) = a q ip [ E kij , E ij ] c = a q ip ad c E k (cid:0) E ij (cid:1) ∈ J ( s ∗ p χ ) . If p = j , we consider the different possible orders of q pp . If q pp ∈ G ∪ G , then p is aCartan vertex and s ∗ p χ is twist equivalent to s ∗ p χ , and m pi = 2 , T p ( x ) = a m pi (cid:2) (ad c E p ) m pi − E i , [(ad c E p ) m pi − E i , E pk ] c (cid:3) c = [ E m pi − p E i E m pi − p E i E p E j ] c . We use the quantum Serre relations and (54) to write E m pi − p E i E m pi − p E i E p E j as a linearcombination of greater words modulo J ( s ∗ p χ ). If q pp ∈ G , then ◦ q ii = − − ◦ q pp q pp ◦ q kk = − ! s p ◦ q pp q pp ❏❏❏❏❏❏❏❏❏ ◦ − q pp − q pp ✇✇✇✇✇✇✇✇ − q pp ◦ q kk = − . The result follows in a similar way, but using that (45) and (47) are generators of J ( s ∗ p χ )in this case.If p = i , by Lemma 3.5, it is equivalent to prove that T p ( x ′ ) ∈ J ( s ∗ p χ ) , x ′ := [ E kjp , E jp ] c . Note that (ad c E j ) E k ∈ J ( s ∗ p χ ), because for all the possible diagrams q − jj = q jk q kj = − T p ( x ′ ) = a (cid:2) (ad c E k ) E j , E j (cid:3) c ∈ J ( s ∗ p χ ) . Finally, if p = i, j, k , then p is not connected with any of these vertices, or p is connectedjust with i and m pi = 1, or p is connected just with k and m pk = 1. The proof is analogousto the corresponding case in previous Lemmata. N NICHOLS ALGEBRAS OF DIAGONAL TYPE 35 ( ii ) If p = i, j, k , then p is not connected with them and the result follows easily. Inconsequence, we consider the case p ∈ { i, j, k } . If p = k , s ∗ p χ is twist equivalent to χ and(45) is a generator of J ( s ∗ p χ ). Therefore (71) implies that T p (cid:0) [ E iijk , E ijk ] c (cid:1) = a (cid:2) E iij , E ij (cid:3) c ∈ J ( s ∗ p χ ) . If p = j , we have that ◦ q ii − q ii ◦ − − ◦ − ! s p ◦ − − ❊❊❊❊❊❊❊❊ ◦ q ii − q ii ④④④④④④④④ q ii ◦ − . By (71) and fixing the order p < i < k , we have that T p (cid:0) [ E iijk , E ijk ] c (cid:1) = a h(cid:2) E pi , E pik (cid:3) c , E pik i c = [ E p E i E p E i E k E p E i E k ] c We write E p E i E p E i E k E p E i E k as a linear combination of greater words modulo J ( s ∗ p χ )using that (47), (45), E p , E k and E iik are generators of J ( s ∗ p χ ), so Lemma 3.4 impliesthat T p (cid:0) [ E iijk , E ijk ] c (cid:1) ∈ J ( s ∗ p χ ).Finally let p = i . By Lemma 3.5, it is equivalent to prove that T p ( x ′ ) ∈ J ( s ∗ p χ ) , x ′ := [ E kjp , E kjpp ] c . Note that s ∗ p χ is twist equivalent to χ , so x jp is a generator of J ( s ∗ p χ ). Applying (71), T p ( x ′ ) = a a (cid:2) E kjp , E kj (cid:3) c = a a (cid:2) E kj , E p (cid:3) c ∈ J ( s ∗ p χ ) . Therefore T p (cid:0) [ E iijk , E ijk ] c (cid:1) ∈ J ( s ∗ p χ ). (cid:3) Lemma 3.23. ( i ) Let i, j, k ∈ { , . . . , θ } be such that q ii = f q ij = − f q ik ∈ G , f q jk = 1 , q jj = − , q kk ∈ {− , f q ik − } . Then, for all p , T p (cid:0)(cid:2) E i , [ E ij , E ik ] c (cid:3) c + q jk q ik q ji [ E iik , E ij ] c + q ij E ij E iik (cid:1) ∈ J ( s ∗ p χ ) . ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = q kk = − , f q ik = 1 , f q ij ∈ G , q jj = − f q jk = ± f q ij . Then, for all p , T p (cid:16) [ E i , E jjk ] c − q − kj (1 + q jj ) [ E ijk , E j ] c + (1 + q jj )(1 + q jj ) E j E ijk (cid:17) ∈ J ( s ∗ p χ ) . Proof. If p = i, j, k , then p is not connected with these three vertices and the proof followseasily for both relations. We consider then p ∈ { i, j, k } for each item.( i ) If p = k , E p E i E p E i E j E i is a linear combination of greater words by (45) and thequantum Serre relations, depending on the value of q kk . In this way, there exist a, b ∈ k such that x ′ := [ E p E i E p E i E j E i ] c + a [ E p E i E p E i E j ] c + b [ E p E i E j ] c [ E p E i ] c ∈ J ( s ∗ p χ ) . On the other hand, by (71) and for the order on the letters p < i < j , T p (cid:0)(cid:2) E i , [(ad c E i ) E j , (ad c E i ) E p ] c (cid:3) c (cid:1) = a h (ad c E p ) E i , (cid:2) (ad c E p )(ad c E i ) E j , E i (cid:3) c i c = a [ E p E i E p E i E j E i ] c ,T p (cid:0)(cid:2) (ad c E i ) E p , (ad c E i ) E j (cid:3) c (cid:1) = a h(cid:2) (ad c E p ) E i , E i (cid:3) c , (ad c E p )(ad c E i ) E j i c = a [ E p E i E p E i E j ] c ; T p (cid:0) (ad c E i ) E j (ad c E i ) E p (cid:1) = a (ad c E p )(ad c E i ) E j (cid:2) (ad c E p ) E i , E i (cid:3) c = a [ E p E i E j ] c [ E p E i ] c . Calculating explicitly the scalars a, b , we notice that T p ( x ) = a x ′ ∈ J ( s ∗ p χ ). The case p = j is analogous.Finally the case p = i follows as the corresponding case in Lemma 3.15.( ii ) The proof is analogous to the previous case. (cid:3) Lemma 3.24. ( i ) Let i, j ∈ { , . . . , θ } be such that m ij , m ji > . Then, for all p , T p (cid:0) (1 − f q ij ) q jj q ji (cid:2) E i , [ E ij , E j ] c (cid:3) c − (1 + q jj )(1 − q jj f q ij ) E ij (cid:1) ∈ J ( s ∗ p χ ) . ( ii ) Let i, j ∈ { , . . . , θ } be such that q jj = − , q ii f q ij / ∈ G or m jj = 2 , and q ii ∈ G , m ij = 4 , or m ij ∈ { , } . Then, for all p , T p (cid:2) E i , [ E iij , E ij ] c (cid:3) c − − q ii f q ij − q ii f q ij q jj (1 − q ii f q ij ) q ji E iij ! ∈ J ( s ∗ p χ ) . ( iii ) Let i, j ∈ { , . . . , θ } be such that q jj = − , α i + 4 α j ∈ ∆ χ + . Let υ = f q ij , a = (1 − υ )(1 − q ii υ ) − (1 − q ii υ )(1 + q ii ) q ii υb = (1 − υ )(1 − q ii υ ) − a q ii υ,d = b − (1 + q ii )(1 − q ii υ )(1 + υ + q ii υ ) q ii υ a q ii q ij q ji . Then, for all p , T p (cid:16) [ E α i + α j , E α i +3 α j ] c − d E α i +2 α j (cid:17) ∈ J ( s ∗ p χ ) .Proof. ( i ) Let x be the relation we are considering here. We note that if p = i, j then m pi = m pj = 0, so the proof follows easily. Moreover the conditions about i, j are thesame but one relation implies the other holds in U ( χ ) too by Lemma 3.4. Therefore it isenough to consider one of cases p = i or p = j ; consider p = j , in order to apply (71).Note that m pi = 2 , m pi = 3, then m ip = 2 and we have that T p (cid:0)(cid:2) E i , [ E ip , E p ] c (cid:3) c (cid:1) = (cid:2) E pppi , E pi (cid:3) c . By (9) we can write T p ( x ) as a linear combination of h E p , (cid:2) E ppi , E pi (cid:3) c i c , E ppi ;note that h E p , (cid:2) E ppi , E pi (cid:3) c i c = [ E p E i E p E i ] c if we consider p < i . Using the quantum Serrerelations or (63), depending on the case (there exist two possible diagrams), E p E i E p E i is N NICHOLS ALGEBRAS OF DIAGONAL TYPE 37 expressed as a linear combination of greater words modulo J ( s ∗ p χ ), so T p ( x ) ∈ J ( s ∗ p χ ) byLemma 3.4.If m pi = 2, there exist three diagrams such that m ip = 2, and two such that m ip = 3.In all the cases, T p (cid:0) E ip (cid:1) = a E pi ,T p (cid:0)(cid:2) E i , [ E ip , E p ] c (cid:3) c (cid:1) = a a (cid:2) E ppi , E i (cid:3) c = a a h E p , (cid:2) E pi , E i (cid:3) c i c + a a q pi ( q pp − q ii ) E pi , where we use (9) for the last equality. If q pp = − ζ , f q ip = ζ , q ii = ζ , for some primitiveroot ζ ∈ G , then s ∗ p χ is twist equivalent to χ and (62) is a generator of J ( s ∗ p χ ), so T p ( x ) ∈ J ( s ∗ p χ ) by this relation and Lemma 3.4. For the other braidings q ii = − 1, so (cid:2) E pi , E i (cid:3) c ∈ J ( s ∗ p χ ) and the coefficient of E pi in the expression of T p ( x ) is zero. Then T p ( x ) ∈ J ( s ∗ p χ ).( ii ) Let x be the relation we are considering in this item. First we consider p = j ; if q pp = − 1, then m pi = 1, so s p (3 α i + α p ) = 3 α i + 2 α p , s p (3 α i + 2 α p ) = 3 α i + α p ∈ ∆ s ∗ p χ + , so m ip ≥ 3. Applying (71) we have that: T p (cid:0) E iip (cid:1) = a (cid:2) E pi , E i (cid:3) c ,T p (cid:0)(cid:2) E i , [ E iij , E ij ] c (cid:3) c (cid:1) = a (cid:20) E pi , h(cid:2) E pi , E i (cid:3) c , E i i c (cid:21) c . As m ip ≥ 3, (63) is a generator of J ( s ∗ p χ ), or m ip = 3, q ii / ∈ G , so E p E i E p E i can bewritten as a linear combination of greater words modulo J ( s ∗ p χ ), for the order p < i , usingthe corresponding quantum Serre relation and E p . In both cases we apply Lemma 3.4 todeduce that T p ( y ) ∈ J ( s ∗ p χ ).If m pi = 2, then m ip = 3; in this case, T p (cid:0) E iip (cid:1) = a (cid:2) E ppi , E pi (cid:3) c ,T p (cid:0)(cid:2) E i , [ E iip , E ip ] c (cid:3) c (cid:1) = a h E ppi , E α p +3 α i i c . We have two possibilities for s ∗ p χ : • ◦ ζ ζ ◦ − , ζ ∈ G , so (68) is a generator of J ( s ∗ p χ ), • ◦ ζ − ζ ◦ − , ζ ∈ G , so (66) and E p are generators of J ( s ∗ p χ ).Then E p E i E p E i E p E i E p E i is written as a linear combination of greater words modulo J ( s ∗ p χ ) in both cases, so by Lemma 3.4 we have that T p ( y ) ∈ J ( s ∗ p χ ).Let p = i ; by Lemma 3.5, it is equivalent to prove that T p ( y ′ ) ∈ J ( s ∗ p χ ), where y ′ := (cid:2)(cid:2) E jp , [ E jp , E p ] c (cid:3) c , E p (cid:3) c − a (cid:0) [ E jp , E p ] c (cid:1) , and a ∈ k × is fixed. Note that T p (cid:16)(cid:2)(cid:2) (ad c E j ) E p , [(ad c E j ) E p , E p ] c (cid:3) c , E p (cid:3) c (cid:17) = a m pi a m pi − (cid:16)(cid:2) (ad c E p ) m pi − E j , (ad c E p ) m pi − E j (cid:3) c F p L − p − q jp q pp F p L − p (cid:2) (ad c E p ) m pi − E j , (ad c E p ) m pi − E j (cid:3) c (cid:17) T p (cid:0) [ E jp , E p ] c (cid:1) = a m pi a m pi − (ad c E p ) m pi − E i In any case, T p ( y ′ ) ∈ ker π s ∗ p χ is a linear combination of[ E m ip − p E i E m ip − p E i ] c , [ E m ip − p E i ] c , so by Lemma 3.4, T p ( y ′ ) ∈ J ( s ∗ p χ ), because (62), (respectively, (63), (68)) is a generatorof J ( s ∗ p χ ) if m pi = 3, (respectively, m pi = 4, m pi = 5).Finally we take p = i, j , so p is not connected with i and j (and the proof followseasily by Lemma 3.7), or p is connected only with i , and q ii = f q ij = f q pi − ∈ G , q pp = − p < i < j , so T p ( E iij ) = (cid:2) E pi , E pij (cid:3) c ,T p (cid:0)(cid:2) E i , [ E iij , E ij ] c (cid:3) c (cid:1) = (cid:20) E pi , h(cid:2) E pi , E pij (cid:3) c , E pij i c (cid:21) c = [ E p E i E p E i E p E i E j E p E i E j ] c . By (53), E i E p E i E p E i E j E p E i can be written as a linear combination of other wordsmodulo J ( s ∗ p χ ), which are greater than it or they begin with E p ; multiplying on the leftby E p , on the right by E j , and using that E p ∈ J ( s ∗ p χ ), E p E i E p E i E p E i E j E p E i E j isa linear combination of greater words modulo J ( s ∗ p χ ), so T p ( y ) ∈ J ( s ∗ p χ ) by a similarargument to the previous steps.( iii ) The proof is analogous to the previous items, where we note that in the two possiblecases q jj = − 1, and if p = i, j , then p is not connected with them. (cid:3) Lemma 3.25. ( i ) Let i, j ∈ { , . . . , θ } be such that α i + 3 α j / ∈ ∆ χ + , q jj = − or m ji = 2 ,and also m ij ≥ , or m ij = 2 , q ii ∈ G . Then, T p (cid:0) [ E α i +2 α j , E ij ] c (cid:1) ∈ J ( s ∗ p χ ) , for all p . ( ii ) Let i, j ∈ { , . . . , θ } be such that α i + 3 α j ∈ ∆ χ + , α i + 4 α j / ∈ ∆ χ + . Then, for all p , T p (cid:0) [ E α i +3 α j , E ij ] c (cid:1) ∈ J ( s ∗ p χ ) . ( iii ) Let i, j ∈ { , . . . , θ } be such that α i +2 α j ∈ ∆ χ + , α i +3 α j / ∈ ∆ χ + , and q ii f q ij , q ii f q ij = 1 .Then, T p (cid:0) [ E iij , E α i +2 α j ] c (cid:1) ∈ J ( s ∗ p χ ) for all p . ( iv ) Let i, j ∈ { , . . . , θ } be such that α i + 2 α j ∈ ∆ χ + , α i + 3 α j / ∈ ∆ χ + . Then, for all p , T p ([[ E iiij , E iij ] , E iij ] c ) ∈ J ( s ∗ p χ ) .Proof. For these four sets of conditions, if p = i, j then p is not connected with i and j , sothe proof follows easily using Lemma 3.7, or we have a diagram as in Lemma 3.24, ( ii ) ,and the proof is analogous to this one. In consequence we will consider p = i and p = j for each one of these cases.( i ) Let x = [ E α i +2 α j , E ij ] c , and take p = j . If m pi = 1 (that is, q pp = − q pp f q ip = 1),we have that s p (3 α i + 2 α p ) = 3 α i + α p ∈ ∆ s ∗ p χ + , s p (4 α i + 3 α p ) = 4 α i + α p / ∈ ∆ s ∗ p χ + . Therefore m ip = 3, so E i (respectively, (ad c E i ) E p ) is a generator of J ( s ∗ p χ ), if q ii belongs(respectively, does not belong) to G . By (71) and the previous relations, depending on N NICHOLS ALGEBRAS OF DIAGONAL TYPE 39 the case, T p ( x ) = a (cid:20)h(cid:2) E pi , E i (cid:3) c , E i i c , E i (cid:21) c ∈ J ( s ∗ p χ ) . The remaining case is m pi = 2, for which there exist two possible diagrams: ◦ − ζ ζ ◦ ζ , ζ ∈ G ; ◦ − ζ − ζ ◦ ζ , ζ ∈ G . In both cases q pp ∈ G , and also s p (3 α i + 2 α p ) = 3 α i + 4 α p ∈ ∆ s ∗ p χ + , s p (4 α i + 3 α p ) = 4 α i + 5 α p / ∈ ∆ s ∗ p χ + . Then (66) is a generator of J ( s ∗ p χ ) if 3 α i + 5 α p ∈ ∆ s ∗ p χ + , or (67) is a generator of J ( s ∗ p χ )in other case, so for both braidings E p E i E p E i E p E i E p E i is a linear combination of greaterwords modulo J ( s ∗ p χ ), and (66) belongs to J ( s ∗ p χ ). Therefore T p ( x ) = a (cid:20)h(cid:2) E ppi , E pi (cid:3) c , E pi i c , E pi (cid:21) c ∈ J ( s ∗ p χ ) . Consider now p = i , so by Lemma 3.5 it is enough to prove that T p ( x ′ ) ∈ J ( s ∗ p χ ) , x ′ := (cid:2) E jp , (cid:2) E jp , [ E jp , E p ] c (cid:3) c (cid:3) c . If m pj = 2, then s p (3 α p + 2 α j ) = α p + 2 α j ∈ ∆ s ∗ p χ + , s p (4 α p + 3 α j ) = 2 α p + 3 α j / ∈ ∆ s ∗ p χ + ,so m jp = 2, and (45) is a generator of J ( s ∗ p χ ); then T p ( x ′ ) = a a h E pj , (cid:2) E pj , E j (cid:3) c i c ∈ J ( s ∗ p χ ) . If m pj = 3, then s p (3 α p + 2 α j ) = 3 α p + 2 α j ∈ ∆ s ∗ p χ + , s p (4 α p + 3 α j ) = 5 α p + 3 α j / ∈ ∆ s ∗ p χ + ,so (67) is a generator of J ( s ∗ p χ ). By (71), T p ( x ′ ) = a a h E ppj , (cid:2) E ppj , E pj (cid:3) c i c ∈ J ( s ∗ p χ ) . If m pj = 4, then s p (3 α p + 2 α j ) = 5 α p + 2 α j ∈ ∆ s ∗ p χ + ; moreover, we note that 7 α p + 3 α j / ∈ ∆ s ∗ p χ + in both cases, and (65) is a generator of J ( s ∗ p χ ). By this relation and the quantumSerre relations, E p E j E p E j E p E j is written as a linear combination of greater words modulo J ( s ∗ p χ ), so T p ( x ′ ) = a a h E pppj , (cid:2) E pppj , E ppj (cid:3) c i c ∈ J ( s ∗ p χ ) . ( ii ) The proof is similar to ( i ) , but more simple: for p = j we have just one possibility, m pi = 1.( iii ) Let x = [ E iij , E α i +2 α j ] c . Consider p = j ; we consider i a non-Cartan vertex, becausein other case E i E p E i or E i E p E i can be written as a linear combination of other wordsusing the corresponding quantum Serre relation, and finally E i E p E i E p E i is a linearcombination of greater words modulo J ( s ∗ p χ ), so the relation is redundant in this case. Inconsequence we consider m pi = 1, and for this case E p E i E p E i is a linear combination ofgreater words modulo J ( s ∗ p χ ), so T p ( x ) = (cid:20)(cid:2) E pi , E i (cid:3) c , h(cid:2) E pi , E i (cid:3) c , E i i c (cid:21) c ∈ J ( s ∗ p χ ) . Let p = i ; as above, it is enough to prove that T p ( x ′ ) ∈ J ( s ∗ p χ ) , x ′ := (cid:2)(cid:2) E jp , [ E jp , E p ] c (cid:3) c , [ E jp , E p ] c (cid:3) c . If m pj = 2, then s p (3 α p + 2 α j ) = α p + 2 α j ∈ ∆ s ∗ p χ + , s p (5 α p + 3 α j ) = α p + 3 α j / ∈ ∆ s ∗ p χ + ,so (ad c E j ) E p (or E i ) is a generator of J ( s ∗ p χ ); therefore T p ( x ′ ) = a a h(cid:2) E pj , E j (cid:3) c , E j i c ∈ J ( s ∗ p χ ) . If m pj = 3, then s p (3 α p + 2 α j ) = 3 α p + 2 α j ∈ ∆ s ∗ p χ + , s p (5 α p + 3 α j ) = 4 α p + 3 α j / ∈ ∆ s ∗ p χ + ,so (66) is a generator of J ( s ∗ p χ ). Therefore T p ( x ′ ) = a a h(cid:2) E ppj , E pj (cid:3) c , E pj i c ∈ J ( s ∗ p χ ) . If m pj = 4, then s p (3 α p + 2 α j ) = 5 α p + 2 α j ∈ ∆ s ∗ p χ + , s p (5 α p + 3 α j ) = 7 α p + 3 α j / ∈ ∆ s ∗ p χ + ,so (65) is a generator of J ( s ∗ p χ ). In consequence, T p ( x ′ ) = a a h(cid:2) E pppj , E ppj (cid:3) c , E ppj i c ∈ J ( s ∗ p χ ) . ( iv ) The proof is analogous to the previous one. (cid:3) Now we are ready to prove that the Lusztig isomorphisms descend to the family ofalgebras U ( χ ), so we will look at the root system of this family of algebras. As we considerfinite root systems, they are univocally determined as sets of real roots, and using thisresult we will obtain the desired Theorem of presentation by generators and relations ofNichols algebras. Proposition 3.26. The morphisms (38) induce algebra isomorphisms T p , T − p : U ( χ ) → U ( s ∗ p χ ) , such that T p T − p = T − p T p = id U ( χ ) .Proof. By the definition of the ideals J ( χ ) and the previous Lemmata, T p ( J ( χ )) ⊂J ( s ∗ p χ ), so there exists an algebra morphism T p : U ( χ ) → U ( s ∗ p χ ). By φ = id and thedefinition of the ideal, φ ( J ( χ )) = J ( χ ), and also ϕ λ ( J ( χ )) = J ( χ ) for any λ ∈ ( k × ) θ ,because the ideal is Z θ -homogeneous. By (39) we have that T − p ( J ( χ )) ⊂ J ( s ∗ p χ ), so thereexists also an algebra morphism T − p : U ( χ ) → U ( s ∗ p χ ), induced by the correspondingmorphism.These algebras are generated by E i , F i , L i , K i , and the identities T p T − p = T − p T p = idhold for each one of these elements, so these identities hold for all the elements of thesealgebras, and these morphisms are isomorphisms. (cid:3) This result lets us to prove the main result of this Section. The proof is similar to theone for [A1, Thm.5.25]. Proof of Theorem 3.1. Set ∆ χ + := ∆ + ( U + ( χ )) \ { N α α : α ∈ ∆ χ + } . By the triangular decomposition, Lemma2.8, Theorem 2.9 and Proposition 3.26, we have that(72) H U + ( χ ) = H U ++ p ( χ ) q h ( E p ) = s p ( H U ++ p ( s ∗ p χ ) ) q h ( E p ) , for all p ∈ { , . . . , θ } , because deg( T p ( X )) = s p (deg X ) for each homogeneous element X ∈ U ( χ ). Recall that h ( E p ) ∈ { ord q pp , ∞} , so(73) ∆ + (cid:0) U + ( χ ) (cid:1) = s p (cid:0) ∆ + (cid:0) U + ( s ∗ p χ ) (cid:1) \ { α p , N p α p } (cid:1) ∪ S p , where S p = { α p } , or S p = { α p , N p α p } , so ∆ χ + = s p (cid:16) ∆ s ∗ p χ + \ { α p } (cid:17) ∪ { α p } . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 41 In this way, if we consider the sets ∆ χ + , for each χ in a Weyl equivalence class of afixed braiding with finite root system, then R = { ∆ χ + } is a root system for our Weylgroupoid, according with the Definition 1.17. As we have a finite root system, it followsthat ∆ χ + = ∆ χ + , for all χ , because by Proposition 1.21 all the roots are real. In this way,∆ + ( U + ( χ )) is obtained from ∆ χ + adding N α α , for some α ∈ ∆ χ + . Fix an order on the letters x i and consider the corresponding PBW basis. We have a projection π χ : U ( χ ) ։ u ( χ )of graded braided Hopf algebras, so the corresponding x α of the PBW basis of u ( χ ) aregenerators of the PBW basis of U ( χ ), by the definition of Kharchenko’s PBW basis. Onthe other hand, each simple root of a non-Cartan vertex satisfies E N i i = 0 in U ( χ ), so N i α i / ∈ ∆ + ( U + ( χ )). Therefore (72) implies that N α α / ∈ ∆ + (cid:0) U + ( χ ) (cid:1) , for all α ∈ ∆ χ + \ O ( χ ) , because α is of the way α = w ( α i ) for some w ∈ W and i ∈ { , . . . , θ } , i a non-Cartanvertex in the corresponding χ ′ . Analogously, for each Cartan vertex i , N i α i ∈ ∆ + ( U + ( χ )),because E N i i = 0 in U ( χ ), so N α α ∈ ∆ + (cid:0) U + ( χ ) (cid:1) , for all α ∈ O ( χ ) . Therefore ∆ + ( U + ( χ )) = ∆ χ + ∪ { N α α : α ∈ O ( χ ) } .Suppose that the degree N α α in ∆ + ( U + ( χ )) corresponds to a Lyndon word of thisdegree: we can assume that it is of minimal length, and we denote it by u ; set ( v, w ) =Sh( u ). In this way, deg v = β , deg w = γ , for some β , γ ∈ ∆ χ + , and β + γ = N α α . Asall the roots are real, we deduce that if β < γ , then β < α < γ , by a similar argumentto the convexity properties in [A2]. We can consider then the case β = α i , becauseif β = s i · · · s i k ( α i k +1 ), where w = s i · · · s i k is the beginning of the expression of theelement of maximal length, we apply w − to obtain that α i k +1 + γ ′ = N α α ′ for some α ′ , γ ′ ∈ ∆ χ + . Note also that N α > 2, because if we suppose N α = 2, then α is applied ina simple root α i corresponding a Cartan vertex by some element of the Weyl groupoid,and as N α is invariant by the action of the Weyl groupoid, it should be q ii = − 1, but itcorresponds to an isolated vertex or a non-Cartan vertex, which is a contradiction. Set α = θ X j =1 n j α j , γ = θ X j =1 m j α j , for some n j , m j ∈ N . Note that m i = N α n i − ≥ 2, and for j = i , m j = N α n j ≥ 3, so supp γ = supp α . Bysimplicity assume that supp α = { , . . . , θ } ; note that the vertices of supp β correspondsto a connected subdiagram, for any positive root β .By these considerations we reduce the problem to an analysis case by case of thepositive root systems of connected diagrams, and we do it with the help of the program SARNA [GHV]. We look for the possible γ such that all the coordinates, except at mostone, are divisible by an integer ≥ 3, and the remaining coordinate is ≥ 2, so we just havea few 3-uples in rank two or three. Analyzing each of these 3-uples( α, γ, i ) ∈ ∆ χ + × ∆ χ + × { , . . . , θ } such that there exists N ∈ N : α i + γ = N α, we note that N = N α for all of them. Therefore, there are no Lyndon words of degree N α α , so the generators of degree N α α are x N α α , and then the elements x n β · · · x n k β k , β i ∈ ∆ χ + , (cid:26) ≤ n j < N j , if β j / ∈ O ( χ )0 ≤ n j < ∞ , si β j ∈ O ( χ )are a PBW basis of U ( χ ). As x N α α = 0 in u + ( χ ), π χ induces a surjective morphism π ′ χ : U ( χ ) / h x N α α : α ∈ O ( χ ) i −→ u + ( χ ) , which applies the set { x n β · · · x n k β k , β i ∈ ∆ χ + , ≤ n j < N j } , generating linearly the quotient, to the corresponding PBW basis of u + ( χ ). Therefore π ′ χ is an isomorphism. (cid:3) Generation in degree one Now we answer positively the Conjecture 1, formulated by Andruskiewitsch and Schnei-der, but restricting to the case in which G ( H ) is abelian. The technique of the proof isthe same that these authors use in [AS4], extended in some works to other families. Inparticular, the first Lemmata of this Section correspond to relations generating the idealfor standard braidings as in [AGI], but the proof is made in a general context.In what follows Γ denotes a finite abelian group, and S = L n ≥ S ( n ) is a gradedbraided Hopf algebra in GG YD such that S (0) = k 1, generated as an algebra by V := S (1).Fix a basis { x , . . . , x θ } of V , so V has a braiding of diagonal type: we can assume that x i ∈ S (1) χ i g i for some g i ∈ Γ and χ i ∈ b Γ. Set then q ij := χ j ( g i ) ∈ k × .We will prove that if S is finite dimensional, then S is the Nichols algebra B ( V )associated to V . We will obtain then the main Theorem of this Section, answering thisConjecture.We begin by extending [AS4, Lemma 5.4] for a general quantum Serre relation, provingthat they hold in S , or S is infinite-dimensional. Proposition 4.1. Let S be as above. If there exist i, j ∈ { , . . . , θ } such that q m ij +1 ii = 1 ,and also ad c ( x i ) m ij ( x j ) = 0 , then S is infinite-dimensional.Proof. By definition of m ij , we have that q m ij ii f q ij = 1. We begin the proof as in [AS4,Lemma 5.4]. To simplify the notation, call m = m ij , q = q ii , y := x i , y := x j , y :=ad c ( x i ) m ( x j ). Set also h = g i , h = g j , h = g m +1 i g j ,η = χ i , η = χ j , η = χ m +1 i χ j , so y k ∈ S η k h k , 1 ≤ k ≤ 3. If W = k y + k y + k y , then W ⊂ P ( S ) (because y is alsoprimitive), so there exists a monomorphism B ( W ) ֒ → S . We compute the correspondingbraiding matrix ( Q kl = η l ( h k )) ≤ k,l ≤ , and consider the corresponding generalized Dynkindiagram:(74) ◦ q jj q − m ( m +1) q jj ❏❏❏❏❏❏❏❏ ◦ qq − m ⑤⑤⑤⑤⑤⑤⑤⑤ q m +2 ◦ q m +1 q jj . We will consider the different possible cases and prove that no one of them are in [H3],so B ( W ), and in consequence S , is infinite-dimensional. Suppose that the diagram (74) isin Heckenberger’s list: Case I: Q kl Q lk = 1 for all 1 ≤ k < l ≤ 3. By [H3, Lemma 9], 1 = Q k 1. Note that q = − m = 1 (and we assume q m +1 = 1). Also q jj = q m +1 q jj by hypothesis,so there is only one vertex labeled with − • If q jj = − 1, then 1 = ( q m +1 q jj )( q − m ( m +1) q jj ) = − q − m and m = 1 by the sameHeckenberger’s Lemma, which is a contradiction. N NICHOLS ALGEBRAS OF DIAGONAL TYPE 43 • If q m +1 q jj = − 1, then 1 = qq m +2 = q m +3 by the same result, and also1 = q jj ( q − m ( m +1) q jj ) = q jj q − m ( m +3)+2 m = q jj q m , so we deduce that − − = q jj q m +3 = ( q jj q m ) q m +3 = q m +3 , which is also a contradiction. Therefore (74) does not belong to Heckenberger listfor this case. Case II : Q Q = q − m = 1. In this case m = 0, so (74) becomes:(75) ◦ q q ◦ qq jj q jj ◦ q jj . If q jj = − ◦ q q ◦ − q , which has no verticeslabeled with − 1, and these labels are different. Such diagram is not of finite Cartan typeand moreover it does not correspond to any diagram without − B ( W ) is infinite-dimensional.If q jj = − q = − 1, we have an analogous situation, so q = − qq jj = − 1, then [H3, Lemma 9] implies that oneof the following situations holds: • the diagram is of finite Cartan type, so it contains a subdiagram of Cartan type A . Then 1 = qq = ( qq jj ) q , or 1 = q jj q jj = ( qq jj ) q jj , so q = 1 or q jj = 1; • q = 1, q jj , q jj q ∈ G ∪ G , and q jj q jj = 1 or q jj = 1, q, q jj q ∈ G ∪ G , qq = 1.No one of these situations hold, so qq jj = − 1. If this diagram is in [H3, Table 2], it followsthat Q ii Q i Q i = 1 for some i ∈ { , } in all the possible cases. We can assume then i = 1, q = 1. By [H3, Lemma 9], one of the following situations holds: • q jj = 1, but also q jj = − q − = − • q jj = 1, • q jj = − q .No one of these situations are possible, so we obtain a contradiction in this case too. Case III: Q Q = q m +2 = 1. We obtain the diagram: ◦ q q ◦ q jj q − q jj ◦ q − q jj . Such diagram is the corresponding to (75), but changing q jj by q jj q − , so it does notbelong to [H3, Table 2]. Then q m +2 = 1. Case IV: Q Q = 1. This means q jj = q m ( m +1) , so we have the diagram:(76) ◦ q jj q − m ◦ q q m +2 ◦ q m +1 q jj . This diagram is connected by the previous cases. As m = 0, q m +1 = 1, it follows that q = − 1. Consider the different possible values of the labels of the vertices: q jj = q m + q jj = − : that is, q m +1 = 1 and we have the diagram: ◦ − q ◦ q q ◦ − , which is not in Heckenberger’s list. q jj = − , q m + q jj = − : By [H3, Table 2], it should be 1 = Q Q Q = q m +3 , andwe should have the diagram ◦ − q − q − ◦ q q ◦ − . Moreover, 1 = q jj = q m ( m +1) = q m = q − . Note that q = 1 because q m = 1, so q ∈ G .But this diagram is not in Heckenberger’s list. q jj = − , q m + q jj = − : as above, 1 = Q Q Q = q − m . By definition it shouldbe m = 1, with the same diagram of the previous case and q ∈ G , so we obtain the samecontradiction. q jj , q m + q jj = − : By [H3, Lemma 9], one of the following situations holds: • the diagram is of Cartan type. Then, q = q jj and m = 1, or q = q m +1 q jj = q − m − .In both cases we obtain the same diagram, ◦ q q − ◦ q q ◦ q . We discard easily the cases A , C , because q, q = q . If it is of type B , q =( q ) = q − , which is a contradiction. • q jj ∈ G , q ∈ G ∪ G and 1 = q − m = q jj q m +3 . Then m = 1 and q = q − jj ,so q = 1, but we obtain then a contradiction with the fact that q ∈ G ∪ G isprimitive. • q m +1 q jj ∈ G , q ∈ G ∪ G y 1 = q jj q − m = q m +3 . Again q = 1, and we obtainthe same contradiction.In consequence, (74) is not in Heckenberger’s list, and S is infinite-dimensional. (cid:3) Now we continue with another Lemmata from [AGI], just adapted to this generalcontext. Lemma 4.2. Let j, k, l ∈ { , . . . , θ } be such that q kk = − , f q kj = f q kl − = 1 , f q jl = 1 . If [ x jkl , x k ] c = 0 is a primitive element of S , then S is infinite-dimensional.Proof. Set u := [ x jkl , x k ] c , g u := g j g k g l ∈ Γ, χ u := χ j χ k χ l ∈ b Γ, q := f q lk ; we work then asin the previous Lemma.We compute the braiding corresponding to the primitive elements y = x j , y = x k , y = x l and y = u , with the corresponding elements h i ∈ Γ, η i ∈ b Γ; we will prove thatsuch braiding has an associated Nichols algebra of infinite dimension, and so S has infinitedimension. The corresponding generalized Dynkin diagram to ( Q rs = η s ( h r )) ≤ r,s ≤ is:(77) ◦ q jj q − q jj q − ◦ − q ◦ q jj q ll q ll q ◦ q ll . Suppose that such diagram is in Heckenberger’s list. If q = − 1, then (77) contains (75)as a subdiagram, so it does not appear in the list. Therefore q = − 1. As each diagram in[H3, Table 3] does not contain a 4-cycle, it follows that q jj q − = 1, or q ll q = 1. As theconditions are symmetric, it is enough to consider the case q jj = ± q .If we also have q ll = ± q − , and as Q = q jj q ll = 1, the diagram contains the following ◦ q q − ◦ − q ◦ − q − , N NICHOLS ALGEBRAS OF DIAGONAL TYPE 45 which is a contradiction with [H3, Lemma 9]. In consequence we have: ◦ ± q q − ◦ − q ◦ q ll q ll q ◦ q jj q ll . Suppose that q jj = − q . As Q Q Q = 1, we deduce from [H3, Table 3] that m = 2;that is, 0 = (1 − Q )( Q Q Q − 1) = (1 + q )( q − , which gives conditions about q , but each diagram in [H3, Lemma 9] does not satisfy thiscondition.Therefore q jj = q . We look at [H3, Table 3] but a diagram in such list does not satisfy Q = − Q = Q Q − = q = ± 1, so (77) is not in the list. In consequence, S hasinfinite dimension. (cid:3) Lemma 4.3. ( i ) Let i, j ∈ { , . . . , θ } be such that q jj = − , q ii f q ij ∈ G , and also q ii ∈ G or m ij ≥ . If [ x iij , x ij ] c ∈ P ( S ) \ { } , then S is infinite-dimensional. ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = ± f q ij ∈ G , f q ik = 1 , and also − q jj = f q ij f q jk = 1 or q − jj = f q ij = f q jk = − . If [ x iijk , x ij ] c ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. ( i ) We follow the same scheme of proof. Set y = x i , y = x j , y = [ x iij , x ij ] c , and h i ∈ Γ, η i ∈ b Γ, i = 1 , , Q rs = η s ( h r )) ≤ r,s ≤ appears in Heckenberger’s list. The associated generalizedDynkin diagram is ◦ q ii qq ❈❈❈❈❈❈❈❈ ◦ − ◦ q ii q ④④④④④④④④ , q := f q ij . Then Q = q ii = 1, so m ij ≥ 3. Moreover the diagram is connected, so it is of type super G (3), the unique diagram of rank three such that some m rs is ≥ 3. Therefore 1 = Q Q = q , which is a contradiction, so the diagram associated to ( Q rs ) does not correspond to afinite-dimensional Nichols algebra. In consequence S is infinite-dimensional.( ii ) Set w := [ x iijk , x ij ] c , and denote as above y = x i , y = x j , y = x k , y = w , W the subspace generated by these elements, and h i ∈ Γ, η i ∈ b Γ, i = 1 , , , B ( W ) is a finite-dimensional Nichols algebra.Set ζ = q ii ∈ G . We analyze each possible case. • q jj = − f q ij f q jk = 1: the diagram of ( Q rs ) becomes ◦ ζ ± ζζ ❉❉❉❉❉❉❉❉ ◦ − ∓ ζ ζ ◦ q kk q kk ζ ①①①①①①①① ◦ q kk ζ . As Q Q , Q Q , Q Q = 1, and the product of these three scalars is not 1,such diagram is not in Heckenberger’s list, by [H3, Lemma 9]. • q − jj = f q ij = f q jk = − 1: now we have the diagram ◦ ζ ± ζζ ❈❈❈❈❈❈❈❈ ◦ ± ζ ± ζ ◦ q kk q kk ζ ①①①①①①①① ◦ q kk ζ . The lack of 4-cycles in Heckenberger’s list implies that 1 = Q Q = q kk ζ , so q kk ζ = − 1, because Q = q kk ζ = 1. But this diagram does not appear in [H3,Table 3].We obtain a contradiction in all the cases, so S is infinite-dimensional. (cid:3) Lemma 4.4. Let i, j, k ∈ { , . . . , θ } be such that f q ik , f q ij , f q jk = 1 . Let w := x ijk − − f q jk q kj (1 − f q ik ) [ x ik , x j ] c − q ij (1 − f q jk ) x j x ik . If w ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. Set y = x i , y = x j , y = x k , y = w , W the subspace generated by these elements, h i ∈ Γ, η i ∈ b Γ, i = 1 , , , Q rs = η s ( h r )) ≤ r,s ≤ :suppose as above that B ( W ) is finite dimensional. Note that Q Q = q ii f q ij f q ik = q ii f q jk − , because f q ij f q ik f q jk = 1, by [H3, Lemma 9]. By the same Lemma at least one vertex islabeled with − 1. Then, if q ii = − 1, we have that Q Q = − 1; the same holds for theother vertices, so exactly one vertex is labeled with − B ( W ) is infinite-dimensional, and S too. (cid:3) Lemma 4.5. ( i ) Let i, j, k ∈ { , . . . , θ } be such that one of the following conditions holds: • q ii = q jj = − , f q ij = f q jk − , f q ik = 1 , or • f q ij = q jj = − , q ii = − f q jk ∈ G , f q ik = 1 , or • q kk = f q jk = q jj = − , q ii = − f q ij ∈ G , f q ik = 1 , or • q jj = − , f q ij = q − ii , q kk = f q jk − = − q ii , f q ik = 1 , or • q ii = q jj = q kk = − , ± f q ij = f q jk ∈ G , f q ik = 1 ,If (cid:2) [ x ij , x ijk ] c , x j (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional. ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = q jj = − , ( f q ij ) = ( f q jk ) − , f q ik = 1 . If (cid:2)(cid:2) x ij , [ x ij , x ijk ] c (cid:3) c , x j (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. ( i ) Set y = x i , y = x j , y = x k , y = (cid:2) [ x ij , x ijk ] c , x j (cid:3) c , W the subspace generatedby these elements, h i ∈ Γ, η i ∈ b Γ, i = 1 , , , Q rs = η s ( h r )) ≤ r,s ≤ the braiding matrix. We will consider the associated generalized Dynkindiagram for each case.For the first case, we have the following diagram, where q := f q ij : ◦ − qq ❋❋❋❋❋❋❋❋ ◦ − q − ◦ q kk q kk q − ✇✇✇✇✇✇✇✇ ◦ − q kk . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 47 Suppose that B ( W ) is finite-dimensional. Then Q Q Q = 1, so q kk = q and then Q Q = q − = 1. In consequence such diagram is of type super F (4). Then 1 = Q Q = q , which is a contradiction.For the second case, Q Q = q ii ∈ G , and Q Q = − 1, so the diagram contains a4-cycle and then B ( W ) is infinite-dimensional. An analogous situation holds for the thirdcase, because Q Q = − q ii ∈ G , and Q Q = − g Q = g Q = g Q = f q kj = 1 and g Q = g Q = f q ij = 1, so B ( W ) isinfinite-dimensional.For the last case, Q = 1, and then B ( W ) is infinite-dimensional.Therefore S is infinite-dimensional in all the cases.( ii ) We use the same notation, but in this case y = (cid:2)(cid:2) x ij , [ x ij , x ijk ] c (cid:3) c , x j (cid:3) c . So we havethe following diagram for ( Q rs ): ◦ − qq ❋❋❋❋❋❋❋❋ ◦ − q − ◦ q kk q kk q − ✇✇✇✇✇✇✇✇ ◦ − q kk , where q = f q ij . Suppose that B ( W ) is finite-dimensional. By [H3, Table 3], this diagramcannot be connected. In consequence, 1 = Q Q = Q Q , so q kk = ± 1. But then Q = 1, or Q = 1, which is a contradiction to the fact that B ( W ) is finite-dimensional.So S is infinite-dimensional. (cid:3) Lemma 4.6. Let i, j, k, l ∈ { , . . . , θ } be such that q jj f q ij = q jj f q jk = 1 , f q jk = f q kl − = q ll , q kk = − , f q ik = e q il = f q jl . If (cid:2)(cid:2) [ x ijkl , x k ] c , x j (cid:3) c , x k (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. We use a similar notation and consider the corresponding subspace W generatedby the corresponding primitive elements. Suppose that B ( W ) is finite-dimensional. Itsassociated Dynkin diagram is ◦ q ii q − q ii q − ❊❊❊❊❊❊❊❊ ◦ q q − ◦ − q ◦ q − q ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ ◦ − q ii , q = q jj . Note that 1 = Q Q , because there are no 5-cycles and q = 1. Therefore q ii = q , butthis diagram is not in Heckenberger’s list, so B ( W ) is infinite-dimensional, and S too. (cid:3) Lemma 4.7. Let i, j, k ∈ { , . . . , θ } be such that q jj = f q ij − = f q jk . ( i ) If q jj ∈ G and (cid:2) [ x ijk , x j ] c x j (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional. ( ii ) If q jj ∈ G and (cid:2)(cid:2) [ x ijk , x j ] c , x j (cid:3) c , x j (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. ( i ) Using the same notation as in previous Lemmata, we have the diagram ◦ q ii ζ q ii ●●●●●●●●● ◦ ζ ζ ◦ q kk q kk ✈✈✈✈✈✈✈✈✈ ◦ q kk q ii , ζ = q jj ∈ G , for y = x i , y = x j , y = x k , y = (cid:2) [ x ijk , x j ] c x j (cid:3) c , with corresponding matrix ( Q rs ), and W is the subspace generated by these elements. Note that • if q ii = q kk = − 1, then Q = 1; • if q ii , q kk = − 1, then the diagram contains a 4-cycle; • if q ii = − q kk = − 1, or q ii = − q kk = − 1, the diagram contains ◦ q q ◦ − q as a subdiagram (where q = q ii or q = q kk ), and this connected subdiagram ofrank two is not in [H3, Table 1].In all the cases B ( W ) is infinite-dimensional, so S too.( ii ) The proof is analogous. (cid:3) Lemma 4.8. ( i ) Let i, j, k, l ∈ { , . . . , θ } be such that q ll = f q lk − = q kk = f q jk − = q , f q ij = q − ii = q for some q ∈ k × , q jj = − , f q ik = e q il = f q jl = 1 .If (cid:2)(cid:2) [ x ijk , x j ] c , [ x ijkl , x j ] c (cid:3) c , x jk (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional. ( ii ) Let i, j, k, l ∈ { , . . . , θ } be such that f q jk = f q ij = q − jj ∈ G ′ ∪ G ′ , q ii = q kk = − , f q ik = e q il = f q jl = 1 , f q jk = f q lk . If (cid:2)(cid:2) x ijk , [ x ijkl , x k ] c (cid:3) c , x jk (cid:3) c ∈ P ( S ) \ { } , then S isinfinite-dimensional.Proof. We use the same notation and strategy as in the previous Lemmata.( i ) We have that g Q = q = 1, and the diagram corresponding to ( q st ) does not admitextensions with finite root systems. Therefore S is infinite-dimensional.( ii ) Now, g Q = q = 1, and we conclude the same as in ( i ) . (cid:3) Lemma 4.9. Let i, j, k, l ∈ { , . . . , θ } be such that one of the following situations hold: ◦ q kk = − , q ii = f q ij − = q jj , f q kl = q − ll = q jj , f q jk = q − jj , f q ik = e q il = f q jl = 1 , ◦ q ii = f q ij − = − q − ll = − f q kl , q jj = f q jk = q kk = − , f q ik = e q il = f q jl = 1 , or ◦ q jj = f q jk − ∈ G , q ii = f q ij − = q ll = f q kl − = − q jj , q kk = − , f q ik = e q il = f q jl = 1 .If (cid:2) [ x ijkl , x j ] c , x k (cid:3) c − q jk ( f q ij − − q jj ) (cid:2) [ x ijkl , x k ] c , x j (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. Using the same notation and strategy as in the previous Lemmata, we computethe diagrams for ( Q rs ) in each case, and note that the diagrams of ( q rs ) does not admitextensions with finite root systems. • In the first case, g Q = q jj = 1. • In the second case, g Q = f q ij = 1. • For the last one, g Q = − q jj = 1.Therefore S is infinite-dimensional in any case. (cid:3) Lemma 4.10. ( i ) Let i, j, k ∈ { , . . . , θ } be such that q kk = q jj = f q ij − = f q jk − ∈ G , f q ik = 1 , q ii = q kk . If (cid:2) [ x iij , x iijk ] c , x ij (cid:3) c ∈ P ( S ) \ { } , then S is infinite-dimensional. ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = f q ij − ∈ G , q jj = f q jk − = q ii , f q ik = 1 , q kk = q ii . If [[ x ijk , x j ] c , x k ] c − (1 + f q jk ) − q jk (cid:2) [ x ijk , x k ] c , x j (cid:3) c ∈ P ( S ) \ { } , then S isinfinite-dimensional.Proof. We use the same notation and strategy as in the previous Lemmata, and note alsothat these diagrams does not admit extensions with finite root systems.( i ) In this case, g Q = f q jk = − 1, so S is infinite-dimensional, because the diagram of ( Q rs )is connected and contains the diagram of ( q ij ).( ii ) Now, g Q = q jj = − 1, and S is also infinite-dimensional. (cid:3) N NICHOLS ALGEBRAS OF DIAGONAL TYPE 49 Lemma 4.11. ( i ) Let i, j, k ∈ { , . . . , θ } be such that q ii = q kk = f q ij = − , q jj = f q jk − , f q ik = 1 . If [ x ij , x ijk ] c ∈ P ( S ) \ { } , then S is infinite-dimensional. ( ii ) Let i, j, k ∈ { , . . . , θ } be such that q ii = q kk = − , f q ij ∈ G , q jj = ± f q ij = − f q jk , f q ik = 1 . If [ x i , x jjk ] c − (1 + q jj ) q − kj [ x ijk , x j ] c − (1 + q jj )(1 + q jj ) q ij x j x ijk ∈ P ( S ) \ { } ,then S is infinite-dimensional. ( iii ) Let i, j, k ∈ { , . . . , θ } be such that f q jk = 1 , q ii = f q ij = − f q ik ∈ G . If (cid:2) x i , [ x ij , x ik ] c (cid:3) c + q jk q ik q ji [ x iik , x ij ] c + q ij x ij x iik ∈ P ( S ) \ { } , then S is infinite-dimensional. ( iv ) Let i, j, k ∈ { , . . . , θ } be such that q jj = q kk = f q jk = − , q ii = − f q ij ∈ G , f q ik = 1 . If [ x iijk , x ijk ] c ∈ P ( S ) \ { } , then S is infinite-dimensional.Proof. We consider the same notation as before. We consider the subspace W generatedby y = x i , y = x j , y = x k and y (the primitive element corresponding to the relation),where y i ∈ W η i h i , for some h i ∈ Γ, η i ∈ b Γ, and set ( Q rs = η s ( h r )). We will prove again that B ( W ) is infinite-dimensional.( i ) The corresponding diagram of ( Q rs ) is ◦ − − ◦ qq q − ◦ − q − ②②②②②②②② ◦ − q , q = q jj ∈ G ∪ G ∪ G . If q ∈ G then Q = 1, so B ( W ) is infinite-dimensional. If q ∈ G ∪ G , the diagram isnot in [H3, Table 3], so B ( W ) is also again infinite-dimensional.( ii ) We note that Q Q = Q Q = q jj ∈ G . Therefore, the diagram corresponding to( Q rs ) contains a 4-cycle, and then B ( W ) is infinite-dimensional.( iii ) In this case, Q Q = − q ii and Q Q = − q kk = 1, because q kk ∈ {− , f q ik − } . Thediagram corresponding to ( Q rs ) r,s =2 , , is a 3-cycle such that g Q g Q g Q = 1. [H3, Lemma9] implies that B ( W ) is infinite-dimensional.( iv ) B ( W ) is infinite-dimensional because Q = 1. (cid:3) Lemma 4.12. Let i, j ∈ { , . . . , θ } be such that the satisfy one of the following conditions: ( i ) − q ii , − q jj , q ii f q ij , q jj f q ij = 1 , (cid:2) x i , [ x ij , x j ] c (cid:3) c − (1 + q jj )(1 − q jj f q ij )(1 − f q ij ) q jj q ji x ij ∈ P ( S ) \ { } ;( ii ) q jj = − , q ii f q ij / ∈ G , and also m ij ∈ { , } , or m ij = 3 , q ii ∈ G , (cid:2) x i , x α i +2 α j (cid:3) c − − q ii f q ij − q ii f q ij q jj (1 − q ii f q ij ) q ji x iij ∈ P ( S ) \ { } ;( iii ) 4 α i + 3 α j / ∈ ∆ χ + , q jj = − or m ji = 2 , and also m ij ≥ or m ij = 2 , q ii ∈ G , [ x α i +2 α j , x ij ] c ∈ P ( S ) \ { } ;( iv ) 3 α i + 2 α j ∈ ∆ χ + , α i + 3 α j / ∈ ∆ χ + , and q ii f q ij , q ii f q ij = 1 , [ x iij , x α i +2 α j ] c ∈ P ( S ) \ { } ; ( v ) 4 α i + 3 α j ∈ ∆ χ + , α i + 4 α j / ∈ ∆ χ + , [ x α i +3 α j , x ij ] c ∈ P ( S ) \ { } ; ( vi ) 5 α i + 2 α j ∈ ∆ χ + , α i + 3 α j / ∈ ∆ χ + , [[ x iiij , x iij ] , x iij ] c ∈ P ( S ) \ { } ; ( vii ) q jj = − , α i + 4 α j ∈ ∆ χ + , [ x iij , x α i +3 α j ] c − ax α i +2 α j ∈ P ( S ) \ { } , for some a ∈ k × .Then S is infinite-dimensional. Proof. Firstly we note that there exists just one connected generalized Dynkin diagramof rank three such that 3 α i + 2 α j ∈ ∆ χ + , for some pair i, j , which is exactly the uniqueone such that m kl ≥ k, l . Moreover, 4 α i + 3 α j , 5 α i + 4 α j , are not positiveroots for any pair i, j and any connected Dynkin diagram of rank 3.We consider as above the subspace W generated by y = x i , y = x j and y , therelation which is a primitive element by hypothesis, and analyze its generalized Dynkindiagram.( i ) If Q Q = 1 or Q Q = 1, then B ( W ) is infinite-dimensional. In other case, Q Q = q ii q ij q ji = 1 , Q Q = q jj q ij q ji = 1 , so Q = q ii q ij q ji q jj = 1, and B ( W ) is also infinite-dimensional.( ii ) If q ii ∈ G , f q ij = q ii = q − jj (and then ( q rs ) is Cartan of type G ), then Q = q ii q ij q ji q jj = 1 , so B ( W ) is infinite-dimensional. In other case, Q Q = 1, or Q Q = 1, or Q Q = q ii q ij q ji = 1 , Q Q = q ij q ji q jj , so Q = 1 , and therefore B ( W ) is infinite-dimensional.( iii ) Now we calculate Q = q ii q ij q ji q jj , Q Q = q ii q ij q ji , Q Q = q ij q ji q jj . If ( q rs ) is Cartan of type G and q ii ∈ G , f q ij = q jj = − 1, then B ( W ) is infinite-dimensional, because we have a connected diagram of rank three such that M = 3, andit is not of type super G (3). In other case, we will prove that Q Q = 1 or Q Q = 1to conclude that B ( W ) is infinite-dimensional. If m ji ≥ 2, we have the following possiblecases: • q ii = − ζ , f q ij = ζ , q jj = ζ , ζ ∈ G ; in such case, Q Q = ζ . • q ii = − ζ , f q ij = − ζ , q jj = ζ , ζ ∈ G ; therefore, Q Q = ζ .Also, if q ii = ζ , f q ij = ζ , q jj = − ζ ∈ G , then Q Q = ζ . In all the remaining cases, q jj = − f q ij / ∈ G , so Q Q = 1.( iv ) This relation is not redundant just in the following two cases: ◦ ζ ζ ◦ − , ζ ∈ G , ◦ η − η ◦ − η − , η ∈ G . Note that they are not contained in any connected diagram of rank three in [H3, Table2], so it is enough to verify that Q Q = 1 or Q Q = 1 to conclude that B ( W ) isinfinite-dimensional. For the first diagram, Q Q = ζ = 1; and for the second one, Q Q = − η − = 1.( v ) The proof is analogous to ( iii ) . Note that Q = q ii q ij q ji q jj , Q Q = q ii q ij q ji , Q Q = q ij q ji q jj . We have that q jj = − q ii = ζ ∈ G , f q ij = ζ , it follows that Q = 1. In the remaining cases, f q ij / ∈ G , so Q Q = − 1, and then B ( W ) is infinite-dimensional.( vi ) In this case, Q = q ii q ij q ji q jj , Q Q = q ii q ij q ji , Q Q = q ij q ji q jj . We have that q jj = − f q ij / ∈ G , so Q Q = 1. Therefore B ( W ) is infinite-dimensional.( vii ) The proof is analogous to the one for ( i ) . N NICHOLS ALGEBRAS OF DIAGONAL TYPE 51 We conclude that S is infinite-dimensional in all the cases. (cid:3) Now we can prove the main Theorem of this Section. Theorem 4.13. Let S = ⊕ n ≥ S ( n ) be a finite-dimensional graded Hopf algebra in k Γ k Γ YD ,where Γ is a finite abelian group, such that S (0) = k . If S is generated as an algebra by S (0) ⊕ S (1) , then S ∼ = B ( V ) .Proof. Fix a basis x , . . . , x θ of V := S (1), such that x i ∈ S (1) χ i g i for some g i ∈ Γ and χ i ∈ b Γ, and set q ij := χ j ( g i ).As S is generated as an algebra by S (0) ⊕ S (1), the canonical projection T ( V ) ։ B ( V ) = T ( V ) /I ( V ) induces a surjective morphism π : S ։ B ( V ) of graded braided Hopfalgebras; we can consider S = T ( V ) /I , for some graded braided Hopf ideal I of T ( V ),generated by homogeneous elements of degree ≥ I ⊆ I ( V ).Suppose that I ( V ) % I . Then at least one of the generators of I ( V ) from Theorem3.1 does not belong to I . We can assume that x ∈ I ( V ) \ I is one of these generators, ofminimal degree k . Then x is primitive in S by Lemma 3.2.By Proposition 4.1 and Lemmata 4.2-4.12, we deduce that x = x N α α for some α ∈ O ,or a simple root α = α i such that i is not a Cartan vertex, or α = α i + α j , such that N α = 2, q ii = q jj = f q ij = − 1. If g α ∈ Γ, χ α ∈ b Γ are the associated elements, we have that q α = χ α ( g α ), which is a root of unity of order N α . Therefore g N α α ∈ Γ and χ N α α ∈ b Γ arethe associated elements to x , and c ( x ⊗ x ) = g N α α · x ⊗ x = χ N α α (cid:0) g N α α (cid:1) x ⊗ x = x ⊗ x , so x generates in S an infinite-dimensional braided Hopf subalgebra, and we obtain acontradiction. In consequence, S ∼ = B ( V ). (cid:3) Remark . Note that we just use the fact that the braiding is diagonal, so we cangeneralize this Theorem to a general braided Hopf algebra R in HH YD , where H is afinite-dimensional Hopf algebra which acts diagonally over R (1).The following Theorem answers positively Conjecture 1 in the case that the group ofgroup-like elements is abelian. It extends [AS4, Thm. 5.5]. Theorem 4.15. Let H be a finite-dimensional pointed Hopf algebra over an abelian group Γ . Then H is generated by its group-like and skew-primitive elements.Proof. Let gr H = R k Γ, V = R (1). Then H is generated by its group-like and skew-primitive elements if and only if gr H satisfies this condition, which is equivalent to thefact that R is the Nichols algebra B ( V ). Let S be the graded dual R ∗ in the category GG YD , which is generated as an algebra by S (1) = V ∗ . By [AS3, Lemma 2.3] it is enoughto prove that S is the Nichols algebra B ( V ∗ ), which follows by Theorem 4.13. (cid:3) References [An] N. 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