Finite GK-dimensional pre-Nichols algebras of quantum linear spaces and of Cartan type
aa r X i v : . [ m a t h . QA ] F e b FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS OFQUANTUM LINEAR SPACES AND OF CARTAN TYPE
NICOLÁS ANDRUSKIEWITSCH AND GUILLERMO SANMARCO
Abstract.
We study pre-Nichols algebras of quantum linear spacesand of Cartan type with finite GK-dimension. We prove that out of ashort list of exceptions involving only roots of order 2, 3, 4, 6, any suchpre-Nichols algebra is a quotient of the distinguished pre-Nichols algebraintroduced by Angiono generalizing the De Concini-Procesi quantumgroups. There are two new examples, one of which can be thought of as G at a third root of one. Contents
1. Introduction 12. Preliminaries 43. Quantum linear spaces 104. Cartan type 155. On the open cases 32References 331.
Introduction
Overview.
Let k be a field. Let GK-dim be an abbreviation of Gelfand-Kirillov dimension, see [KL]. In this paper we contribute to the ongoingprogram of classifying Hopf algebras with finite
GK-dim . See [B+, G, L]and references therein.Let H be a Hopf algebra and let HH YD be the category of Yetter-Drinfeldmodules over H . Assume that H is pointed (similar arguments apply moregenerally if its coradical is a Hopf subalgebra). Basic invariants of H are(i) the group of grouplikes Γ = G ( H ) ,(ii) the diagram R = ⊕ n ∈ N R n , a graded connected Hopf algebra in k Γ k Γ YD ,(iii) the infinitesimal braiding V := R , an object in k Γ k Γ YD .See [AS3]. Assume that Γ has finite growth. In order to classify those H with finite GK-dim , one first needs to understand all such R with finite This material is based upon work supported by the National Science Foundation underGrant No. DMS-1440140 while N. A. was in residence at the Mathematical SciencesResearch Institute in Berkeley, California, in the Spring 2020 semester. The work of N.A. and G. S. was partially supported by CONICET and Secyt (UNC).
GK-dim . Since R is coradically graded and connected it is strictly gradedas in [Sw]. Strictly graded Hopf algebras R in k Γ k Γ YD with R ≃ V are calledpost-Nichols algebras of V ; also, graded Hopf algebras R in k Γ k Γ YD generatedby R ≃ V are called pre-Nichols algebras of V . See §2.5.The Nichols algebra B ( V ) is isomorphic to the subalgebra of R generatedby V . When char k = 0 and dim H < ∞ , it was conjectured in [AS2]that R = B ( V ) , which reduces our problem to classifying finite-dimensionalNichols algebras in k Γ k Γ YD . The conjecture was proved to be valid in [An1]assuming that Γ is abelian. But beyond those hypotheses this fails to be true.Thus, it does not seem to be avoidable to consider the following questions:(A) classify all V ∈ k Γ k Γ YD such that B ( V ) has finite GK-dim ,(B) for such V classify all post-Nichols algebras with finite GK-dim .Lemma 2.2 below from [AAH3] reduces Question (B) for V as in (A) to(C) classify all pre-Nichols algebras of V ∗ with finite GK-dim .As usual it is more flexible to deal with these questions considering classesof braided vector spaces rather than classes of groups Γ and correspondinglypre-Nichols algebras as braided Hopf algebras. For Question (C) we point outthat all pre-Nichols algebras of V form a poset Pre ( V ) with T ( V ) minimaland B ( V ) maximal and those with finite GK-dim form a saturated subposet
Pre fGK ( V ) . In the extreme case when char k = 0 and the braiding is theusual flip, the Nichols algebra is just the symmetric algebra and the pre-Nichols algebras with finite GK-dim are the universal enveloping algebras ofthe finite-dimensional N -graded Lie algebras generated in degree one. Thus Pre fGK ( V ) is hardly computable when dim V ≥ . Similar considerationsare valid when the braiding is the super flip of a super vector space, see§2.9.2. But if dim V = 1 , then Pre fGK ( V ) = Pre ( V ) has obviously a minimalelement. We say that a pre-Nichols algebra is eminent if it is a minimumin Pre fGK ( V ) . See Definition 2.3. We shall show that many other Nicholsalgebras with finite GK-dim have eminent pre-Nichols algebras.From now on we assume that k is algebraically closed and char k = 0 . Inthis paper we are concerned with Question (C) for braided vector spaces V of diagonal type, i.e. with braiding determined by a matrix q = ( q ij ) i,j ∈ I with entries in k × where θ ∈ N and I = { , . . . , θ } . See §2.8 for precisions.First we need to discuss Question (A) for this class. Finite-dimensionalNichols algebras of diagonal type, i.e. those with GK-dim = 0 , were classifiedin [H1] through the notion of (generalized) root system. More generally thelist of all Nichols algebras of diagonal type with finite root system is given in loc. cit.
It was conjectured in [AAH1] that Nichols algebras of diagonal typewith finite
GK-dim are those with finite root system. This conjecture wasverified in various cases [R, AA1, AAH2]. We shall assume in a few proofsthat the conjecture is true.
INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 3
Let B ( V ) be a finite-dimensional Nichols algebra of diagonal type withconnected Dynkin diagram. The distinguished pre-Nichols algebra e B ( V ) wasintroduced in [An1] as a tool for determining the defining relations of B ( V ) .Several aspects of these algebras were established in [An3]. Particularly itwas asked in [An3] (with the terminology just introduced) whether or notthe distinguished are eminent.1.2. The main results.
In the present paper we focus on braided vectorspaces of diagonal type of two kinds. Fix V of diagonal type, with braidinggiven by the matrix q = ( q ij ) i,j ∈ I .1.2.1. Quantum linear spaces.
Here we assume that q satisfies q ij q ji = 1 forall i = j ∈ I . The distinguished pre-Nichols algebra e B q is presented bygenerators ( x i ) i ∈ I and relations x i x j − q ij x j x i , for all i = j ∈ I . We needsome notation to state our first Theorem. Set I ∞ = { i ∈ I : q ii / ∈ G ∞ } , I N = { i ∈ I : ord q ii = N } , N ≥ , I t = [ N> I N , I ± = { i ∈ I : q ii = ± } = I ⊔ I . (1.1)Thus I = I ± ⊔ I ⊔ I t ⊔ I ∞ . For ⋆ ∈ N ∪ {± , t, ∞} , let V ⋆ be the subspace of V spanned by ( x i ) i ∈ I ⋆ and q ⋆ the restriction of q to V ⋆ . Then V = V ± ⊕ V ⊕ V t ⊕ V ∞ . The pre-Nichols algebras of V ± with finite GK-dim are described in §2.9.2.
Theorem 1.1. (a)
The distinguished pre-Nichols algebra e B ( V ⋆ ) is emi-nent, for ⋆ ∈ { , t, ∞} . (b) Let B be a finite GK-dimensional pre-Nichols algebra of V ; let B ± , ,respectively B t , B ∞ be the subalgebra of B generated by V ± ⊕ V ,respectively V t , V ∞ . Then there is a decomposition B ≃ B ± , ⊗ B t ⊗ B ∞ . (1.2)(c) Assume that V has a basis { x , x } with x ∈ V , x ∈ V . Then ˘ B ( V ) = T ( V ) / h (ad c x ) ( x ) , (ad c x ) ( x ) i is an eminent pre-Nichols algebra of V and has GK-dim = 6 . Parts (a) and (b) follow from Proposition 3.2. Part (c) is Proposition 3.3.Although ˘ B ( V ) of part (c) is not the distinguished pre-Nichols algebra ofthe quantum plane V , it can be thought of as the distinguished one of thebraided vector space of Cartan type G , but degenerated in the sense thatthe parameter is a primitive third root of unity. Via suitable bosonizations, ˘ B ( V ) provides new examples of pointed Hopf algebras with finite GK-dim .By (1.2) it remains to understand B ± , for B ∈ Pre fGK ( V ) . By Propo-sition 3.2, B , ≃ B ⊗ B . Towards B , we just know Part (c), see also§3.3. The next step would be the following: NICOLÁS ANDRUSKIEWITSCH AND GUILLERMO SANMARCO
Question 1.2.
Assume that V = V ⊕ V , dim V = 1 and dim V = 2 . Isthe distinguished pre-Nichols algebra e B ( V ) eminent?1.2.2. Connected Cartan type.
Here q is of finite Cartan type, i.e. q ij q ji = q a ij ii , i = j ∈ I , where a = ( a ij ) i,j ∈ I is a Cartan matrix of finite type with connected Dynkindiagram. In §4 we recall the possibilities for such q . They depend on an rootof unity q , whose order is denoted by N . Theorem 1.3. (a)
The distinguished pre-Nichols algebra e B q is eminent ex-cept in the following cases: A with N = 3 , A θ , θ ≥ , N = 2; D θ , θ ≥ , N = 2; G , N = 4 , . (1.3)(b) Suppose q is of type A with N = 3 . Then b B = k h x , x | x , x , x , x i is an eminent pre-Nichols algebra of q , and GK-dim b B = 5 . This answers (partially) a question in [An3]. The graded duals of thedistinguished pre-Nichols algebras have been presented by generators andrelations in [AAR].The proof of (a) is given in Lemmas 4.12, 4.13, 4.15, 4.16, 4.17, 4.18. Forthe cases listed in (1.3) the determination of the poset
Pre fGK ( q ) remainsan open problem. See Section 5 for partial results; answers to Questions 5.2,5.5, 5.7, 5.9 and 5.11 would shed light on the issue.The proof of (b) is given in Proposition 4.11. The eminent pre-Nichols al-gebra b B is introduced and studied in §4.2.2. There we show that b B properlycovers the distinguished pre-Nichols algebra e B q , which has GK-dim e B q = 3 .In De Concini-Procesi quantum groups at roots of unity and, more gener-ally, in Angiono’s distinguished pre-Nichols algebras, powers of root vectorsgenerate a skew-central subalgebra [An3, §4.1]. Our b B has a slightly biggerskew central subalgebra Z that fits in an extension k → Z ֒ → b B ։ B q → k of braided Hopf algebras. We think that the same arguments may apply toproduce new examples in the case D θ at − . This will be treated in a sequel.2. Preliminaries
Conventions.
For n ≤ m ∈ N , put I n,m = { k ∈ N : n ≤ k ≤ m } and I m = I ,m . Given a positive integer N , we denote by G N the group of N -throots of unity in k × , and by G ′ N ⊂ G N the subset of those of order N . Thegroup of all roots of unity is denoted by G ∞ and G ′∞ := G ∞ − { } .The subalgebra generated by a subset X of an associative algebra is de-noted by k h X i .All Hopf algebras are assumed to have bijective antipode. If H is a Hopfalgebra, the group of group-like elements is denoted by G ( H ) , while P ( H ) is INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 5 the subspace of primitive elements. By gr H we mean the graded coalgebraassociated to the coradical filtration.If A and B are algebras in HH YD , we denote by A ⊗ B = ( A ⊗ B, µ A ⊗ B ) the algebra with multiplication µ A ⊗ B = ( µ A ⊗ µ B )(id A ⊗ c B,A ⊗ id B ) , where µ A and µ B are the multiplications of A and B , respectively.2.2. Gelfand-Kirillov dimension.
We refer to [KL] for general informa-tion on this topic. The following useful statement is immediate from thedefinition of
GK-dim . Let R be a ring and let M = ⊕ n ∈ N M n be a graded R -module such that each M n is free of finite rank (we say M is locally finite).The Poincaré series of M is P M = P n ∈ N rank M n X n ∈ Z [[ X ]] . Lemma 2.1.
Let L and F be fields and let T = ⊕ n ∈ N T n and U = ⊕ n ∈ N U n be two locally finite graded algebras generated in degree one over L and F respectively. If P T = P U , then GK-dim T = GK-dim U . (cid:3) Actually [KL, 12.6.2] shows that the Poincaré series of a graded finitelygenerated algebra provides its
GK-dim .2.3.
Braided Hopf algebras.
A pair ( V, c ) where V is a vector space and c ∈ GL ( V ⊗ ) satisfies the braid equation ( c ⊗ id)(id ⊗ c )( c ⊗ id) = (id ⊗ c )( c ⊗ id)(id ⊗ c ) is called a braided vector space. A braided vector space with compatiblealgebra and coalgebra structures as in [T] is called a braided Hopf algebra.For instance the tensor algebra T ( V ) has a canonical structure of (gradedconnected) braided Hopf algebra such that the elements of degree 1 areprimitive. Also the tensor coalgebra T c ( V ) becomes a braided Hopf algebraby the twisted shuffle product. There is a homogeneous morphism of braidedHopf algebras Ω : T ( V ) → T c ( V ) determined by Ω( v ) = v , v ∈ V ; its imageis the Nichols algebra of V , denoted B ( V ) .Another description: let J ( V ) be the largest element of the set S of gradedHopf ideals of T ( V ) trivially intersecting k ⊕ V . Then B ( V ) ≃ T ( V ) / J ( V ) .2.4. Principal realizations.
Theorems 1.1 and 1.3 are relevant for the clas-sification of Hopf algebras with finite
GK-dim . Indeed a braided vector spacearises (up to a mild condition) as a Yetter-Drinfeld module over a Hopf al-gebra; this is called a realization. Realizations are not unique and we singleout a class of them for braidings of diagonal type. Let H be a Hopf algebra.A YD-pair is a couple ( g, χ ) ⊂ G ( H ) × Hom
Alg ( H, k ) satisfying χ ( h ) g = χ ( h (2) ) h (1) g S ( h (3) ) , h ∈ H. Compare with [AS1, p. 671]. This compatibility guarantees that k χg (i. e. H acting and coacting on k by χ and g , respectively) is a Yetter-Drinfeld moduleover H . Let ( V, c q ) be a braided vector space of diagonal type. Following[AS1, p. 673], a principal realization of ( V, c q ) over H is a family ( g i , χ i ) i ∈ I ofYD-pairs such that q ij = χ j ( g i ) for all i, j . In this case V = L i k χ i g i ∈ k Γ k Γ YD . NICOLÁS ANDRUSKIEWITSCH AND GUILLERMO SANMARCO
Pre-Nichols and post-Nichols algebras.
We present in detail theobjects of interest in this paper. • Let B = L n ∈ N B n be a graded connected braided Hopf algebra with B ≃ V . Then B is a pre-Nichols algebra of V if it is generated by B .In this case there are epimorphisms of (graded) braided Hopf algebras T ( V ) ։ B ։ B ( V ) . Hence the set
Pre ( V ) of isomorphism classes of pre-Nichols algebras of V ispartially ordered with T ( V ) minimal and B ( V ) maximal: T ( V ) v v v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ (cid:15) (cid:15) (cid:15) (cid:15) ( ( ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ B . . . . . . B ′ . . . . . . B ′′ B ( V ) ( ( ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ (cid:15) (cid:15) (cid:15) (cid:15) v v v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ • Dually, a graded connected braided Hopf algebra E = L n ∈ N E n with E ≃ V is a post-Nichols algebra of V if it is coradically graded. Thus wehave monomorphisms of (graded) braided Hopf algebras B ( V ) ֒ → E ֒ → T c ( V ) . Hence the set
Post ( V ) of isomorphism classes of post-Nichols algebras of V is partially ordered with T c ( V ) maximal and B ( V ) minimal: B ( V ) I i v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ (cid:127) _ (cid:15) (cid:15) (cid:21) u ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ B . . . . . . B ′ . . . . . . B ′′ T ( V ) ( ( (cid:21) u ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ (cid:15) (cid:15) (cid:127) _ v v I i ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠
The only pre-Nichols which is also a post-Nichols algebra of V is B ( V ) itself.2.6. Eminent pre- and post-Nichols algebras.
For the purposes of clas-sifying Hopf algebras with finite
GK-dim , it is important to describe the(partially ordered) subset
Post fGK ( V ) of Post ( V ) consisting of post-Nicholsalgebras with finite GK-dim . In this paper we are mainly interested in the(partially ordered) subset
Pre fGK ( V ) of Pre ( V ) consisting of pre-Nichols al-gebras with finite GK-dim . The reason to start with this is given by thefollowing result:
Lemma 2.2. [AAH3]
Let B be a pre-Nichols algebra of V and let E = B d be the graded dual of B . Then GK-dim
E ≤
GK-dim B . If E is finitelygenerated, then the equality holds. (cid:3) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 7
A first approximation to the determination of
Post fGK ( V ) and Pre fGK ( V ) is through the following notion. Definition 2.3. (a) A pre-Nichols algebra b B is eminent if it is the min-imum of Pre fGK ( V ) ; i. e. there is an epimorphism of braided Hopfalgebras b B ։ B that is the identity on V for any B ∈ Pre fGK ( V ) .(b) A post-Nichols algebra b E is eminent if it is the maximum of Post fGK ( V ) ;that is for any E ∈
Post fGK ( V ) , there is a monomorphism of braidedHopf algebras E ֒ → b E that is the identity on V .Beware that there are braided vector spaces without eminent pre-Nicholsalgebras; e. g., if dim V > and the braiding is the usual flip, then Pre fGK ( V ) has infinite chains. An intermediate situation could be described as follows. Definition 2.4.
A family ( b B i ) i ∈ I ⊂ Pre fGK ( V ) is eminent if(a) for any B ∈ Pre fGK ( V ) , there exists i ∈ I and an epimorphism ofbraided Hopf algebras b B i ։ B that is the identity on V , and(b) ( b B i ) i ∈ I is minimal among the families in Pre fGK ( V ) satisfying (a).Eminent families of post-Nichols algebras are defined similarly.All the notions above about braided Hopf algebras related to braided vec-tor spaces have a counterpart for Yetter-Drinfeld modules. Namely, supposethat ( V, c ) is realized in HH YD for some Hopf algebra H . Then Pre H ( V ) is thesubset of Pre ( V ) of pre-Nichols algebras that belong to HH YD ; similarly wehave Pre H fGK ( V ) , Post H ( V ) , Post H fGK ( V ) , and also H -eminent pre-Nicholsor post-Nichols algebras.2.7. The adjoint representation and q -brackets. Any Hopf algebra R in HH YD comes equipped with the (left) adjoint representation ad c : R → End R ,given by (ad c x ) y = µ ( µ ⊗ S )(id ⊗ c )(∆ ⊗ id)( x ⊗ y ) , x, y ∈ R, where µ , ∆ and S denote the multiplication, comultiplication and antipodeof R , respectively. The adjoint action of a primitive element x ∈ R is (ad c x ) y = xy − ( x ( − · y ) x (0) , y ∈ R. Given x i , x i . . . , x i k ∈ R , put x i i ...i k = (ad c x i ) . . . (ad c x i k − ) x i k . (2.1)We also set x ( k h ) = x k ( k +1) ( k +2) ...h for k < h .On the other hand, the braided commutator is defined by [ x, y ] c = xy − ( x ( − · y ) x (0) , x, y ∈ R. We refer to [AA2, Introduction] for a more detailed treatment.
NICOLÁS ANDRUSKIEWITSCH AND GUILLERMO SANMARCO
Nichols algebras of diagonal type.
Fix a natural number θ and let I = I θ . Any matrix q = ( q ij ) i,j ∈ I with coefficients in k × determines a braidedvector space of diagonal type ( V, c q ) , where V has a basis ( x i ) i ∈ I , c q ( x i ⊗ x j ) = q ij x j ⊗ x i , i, j ∈ I . (2.2)The Dynkin diagram associated to q is a non-oriented graph with θ ver-tices. The vertex i is labelled by q ii , and there is an edge between i and j if and only if e q ij := q ij q ji = 1 ; in this case, the edge is labeled by e q ij . Thuswe may speak of the connected components of this diagram and by abuse ofnotation of q . The following useful result says that a connected componentwith at least 2 vertices one of them labelled by 1 gives rise to an infiniteGK-dimensional Nichols algebra. Lemma 2.5. [AAH1, Lemma 2.8]
Let U be a braided vector space of diagonaltype with Dynkin diagram q ◦ r ◦ , r = 1 . Then
GK-dim B ( U ) = ∞ . (cid:3) Let α , . . . , α θ be the canonical basis of Z θ . From the braiding matrix q we obtain a k × -valued bilinear form on Z θ , still denoted q and determinedby q ( α i , α j ) = q ij , i, j ∈ I . Put also e q ( α, β ) := q ( α, β ) q ( β, α ) , α, β ∈ Z θ . (2.3)For sake of brevity, we use q αβ = q ( α, β ) and e q αβ = e q ( α, β ) as well.The braided vector space ( V, c q ) as in 2.2 is realized in Z θ Z θ YD by declaring deg( x i ) = α i , α i · x j = q ij x j , i, j ∈ I . (2.4)The algebra T ( V ) becomes Z θ -graded. Thus any quotient algebra R of T ( V ) by a graded ideal inherits the grading: R = L α ∈ Z θ R α . We keep the notation deg for this degree. Furthermore, if R is an algebra obtained as a quotientof T ( V ) by a graded ideal I (thus a subobject in Z θ Z θ YD ), then the braidingon the homogeneous subspaces is given by c ( u ⊗ v ) = q α,β v ⊗ u, u ∈ R α , v ∈ R β . (2.5)The braided commutators satisfy [ u, vw ] c = [ u, v ] c w + q αβ v [ u, w ] c , (2.6) [ uv, w ] c = q βγ [ u, w ] c v + u [ v, w ] c , (2.7) (cid:2) [ u, v ] c , w (cid:3) c = (cid:2) u, [ v, w ] c (cid:3) c − q αβ v [ u, w ] c + q βγ [ u, w ] c v, (2.8)for homogeneous elements u ∈ R α , v ∈ R β , w ∈ R γ . INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 9
In the diagonal setting (2.2) we set as usual J q = J ( V ) , B q = B ( V ) ,etc. Nichols algebras of diagonal type (i. e. those arising from braided vectorspaces of diagonal type) have been intensively studied. The classification ofall matrices q such that B q has finite root system was provided in [H1]; thedefining relations of these Nichols algebras are given in [An1, An2]. Clearly,finite dimensional Nichols algebras of diagonal type have finite root system.It was conjectured that those of finite GK-dim share the same property.
Conjecture 2.6. [AAH1, Conjecture 1.5]
The root system of a Nichols al-gebra of diagonal type with finite GK-dimension is finite.
The validity of Conjecture 2.6 would imply the classification of finite GK-dimensional Nichols algebras of diagonal type. There is strong evidencesupporting it. The conjecture holds when θ = 2 [AAH2, Thm. 4.1], whenthe braiding is of affine Cartan type [AAH2, Thm. 1.2], or when q is generic,that is q ii / ∈ G ∞ , and q ij q ji = 1 or q ij q ji / ∈ G ∞ , for all i = j ∈ I [R, AA1].We include for completeness proofs of the following well-known results. Lemma 2.7.
Let = v, w ∈ T ( V ) be homogeneous primitive elements with deg v = α and deg w = β . Then (ad c v ) w is primitive if and only if e q αβ = 1 .Proof. Using (2.5), compute ∆((ad c v ) w ) = ∆( vw − q αβ wv ) == ( v ⊗ ⊗ v )( w ⊗ ⊗ w ) − q αβ ( w ⊗ ⊗ w )( v ⊗ ⊗ v )= vw ⊗ v ⊗ w + q αβ w ⊗ v + 1 ⊗ vw − q αβ ( wv ⊗ w ⊗ v + q βα v ⊗ w + 1 ⊗ wv )= (ad c v ) w ⊗ ⊗ (ad c v ) w + (cid:0) − e q αβ (cid:1) v ⊗ w. (cid:3) Lemma 2.8.
Let R be a graded braided Hopf algebra. If W is any braidedsubspace of R contained in P ( R ) then GK-dim B ( W ) ≤ GK-dim R .Proof. We follow [AS4, Lemma 5.4]. Since the elements of W are prim-itive, the subalgebra k h W i is a braided Hopf subalgebra of R ; by defini-tion of the Nichols algebra it follows that gr k h W i projects onto B ( W ) , so GK-dim B ( W ) ≤ gr k h W i . But GK-dim gr k h W i ≤ GK-dim k h W i by [KL,Lemma 6.5], and this proves the desired inequality. (cid:3) Pre-Nichols algebras of diagonal type.
Let ( V, c q ) be a braidedvector space of diagonal type associated to the matrix q = ( q ij ) i,j ∈ I . Recallthat e q ij = q ij q ji , i = j . We write Pre Z θ fGK ( V ) for Pre kZ θ fGK ( V ) , cf. (2.4).2.9.1. Pre-Nichols algebras under twist-equivalence.
Let p = ( p ij ) i,j ∈ I be an-other braiding matrix such that q ii = p ii , e q ij = e p ij , i, j ∈ I . In this case, ( V, c q ) and the braided vector space ( W, c p ) with basis ( y i ) i ∈ I are said to be twist-equivalent . Lemma 2.9.
There is an isomorphism of posets
Pre Z θ fGK ( W ) ≃ Pre Z θ fGK ( V ) . Proof.
Let σ : Z θ × Z θ → k × be the bilinear form, hence a 2-cocycle, givenby σ ( α i , α j ) = ( p ij q − ij , i ≤ j, , i > j . Let T ( V ) σ be the corresponding cocycledeformation of T ( V ) , i. e. with multiplication u. σ v = σ ( α, β ) uv, u ∈ T ( V ) α , v ∈ T ( V ) β , α, β ∈ Z θ . (2.9)By the proof of [AS3, Prop. 3.9] the linear map ϕ : W → V , ϕ ( y i ) = x i , i ∈ I , induces an isomorphism ϕ : T ( W ) → T ( V ) σ of Hopf algebras in Z θ Z θ YD .Let I be a Hopf ideal of T ( V ) that belongs to Z θ Z θ YD ; then it is also a Hopfideal of T ( V ) σ and GK-dim T ( V ) /I = GK-dim T ( V ) σ /I by Lemma 2.1. (cid:3) Pre-Nichols algebras of super symmetric algebras.
Assume that e q ij =1 = q ii , for all j = i ∈ I . Then V = V ⊕ V is a super vector space where V j is spanned by those x i ’s such that q ii = ( − j , j = 0 , . Let p = ( p ij ) i,j ∈ I bethe matrix corresponding to the associated super symmetry. Then • The pre-Nichols algebras of ( V, c p ) are the enveloping superalgebras U ( n ) ,where n = ⊕ j ∈ N n j is a graded Lie superalgebra generated by n ≃ V . • Pre fGK ( V, c p ) consists of the enveloping superalgebras U ( n ) , where n = ⊕ j ∈ N n j is a graded Lie superalgebra generated by n ≃ V with dim n < ∞ . • Hence
Pre Z θ fGK ( V, c p ) consists of the enveloping superalgebras U ( n ) , where n = ⊕ β ∈ Z θ n β is a finite-dimensional Z θ -graded Lie superalgebra generatedby n = ⊕ i ∈ I n α i ≃ V . In particular Pre Z θ fGK ( V, c p ) ( Pre fGK ( V, c p ) . • By Lemma 2.9,
Pre Z θ fGK ( V, c q ) is isomorphic as a poset to the set of iso-morphism classes of finite-dimensional Z θ -graded Lie superalgebras as inthe previous point.2.9.3. Distinguished pre-Nichols algebras.
Assume that dim B q < ∞ . The distinguished pre-Nichols algebra of V introduced in [An3] is the quotient e B q := T ( V ) / I q , where I q is the ideal of T ( V ) generated by the definingrelations of J q given in [An1] but excluding the powers of root vectors andincluding the quantum Serre relations at Cartan vertices. A detailed presen-tation of J q and I q is available in [AA2, §4].3. Quantum linear spaces
In this section we investigate finite GK-dimensional pre-Nichols algebrasof quantum linear spaces. These are Nichols algebras of braided vector spacesof diagonal type with totally disconnected Dynkin diagram. More precisely,fix a matrix q = ( q ij ) i,j ∈ I and a vector space V with basis ( x i ) i ∈ I and braidinggiven by c q ( x i ⊗ x j ) = q ij x j ⊗ x i , i, j ∈ I . In this section we assume that q ij q ji = 1 , i = j ∈ I . (3.1)Then B q is presented by generators ( x i ) i ∈ I and relations x ij = 0 , if i < j, (3.2) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 11 x N i i = 0 , if q ii ∈ G ′∞ , where N i := ord q ii ∈ N ∪ ∞ ; (3.3)here we are using the notation (2.1). It has a PBW-basis: { x a x a · · · x a θ θ : 0 ≤ a i < N i if q ii ∈ G ′∞ ; 0 ≤ a i otherwise } . (3.4)The distinguished pre-Nichols algebra e B q of V is presented by generators ( x i ) i ∈ I and relations (3.2); it is a domain of GK-dim = θ . Recall the partition I = I ± ⊔ I ⊔ I t ⊔ I ∞ where as in (1.1) we set I ∞ = { i ∈ I : q ii / ∈ G ∞ } , I ± = { i ∈ I : q ii = ± } , I N = { i ∈ I : q ii ∈ G ′ N } , N ≥ , I t = [ N> I N . For ⋆ ∈ {± , , t, ∞} , let V ⋆ be the subspace of V spanned by ( x i ) i ∈ I ⋆ and q ⋆ the restriction of q to V ⋆ . Then V = V ± ⊕ V ⊕ V t ⊕ V ∞ . As we have seen in§2.9.2 the Z θ -graded pre-Nichols algebras of V ± are twistings of envelopingalgebras of nilpotent Lie superalgebras with suitable properties, particularlythere is no eminent pre-Nichols algebra of V ± .3.1. Reduction to order ≤ . Remark . Let i = j ∈ I . Recall that x ij := (ad c x i ) x j = x i x j − q ij x j x i .The braiding of the 3-dimensional subspace k x i + k x ij + k x j ⊂ T ( V ) is easilycomputed, and the corresponding Dynkin diagram is either q ii ◦ i q ii q ii q jj ◦ ij q jj q jj ◦ j , (3.5)or it is disconnected if the label of some edge is . Proposition 3.2.
Let i, j ∈ I such that < ord q ii + ord q jj . Then x ij = 0 holds in any finite GK-dimensional pre-Nichols algebra of q .Proof. Let B be a pre-Nichols algebra of q , so there is a braided Hopf algebramap T ( V ) → B . Let y , y , y denote the image of x i , x j , x ij , respectively,and consider W := k y + k y + k y . By Lemma 2.7 we have W ⊂ P ( B ) ,hence Lemma 2.8 warranties GK-dim B ( W ) ≤ GK-dim B .Assume y = 0 , so W is 3-dimensional by a degree argument and itsDynkin diagram D is (3.5). We show that GK-dim B ( W ) = ∞ .Consider the subspaces V = k y ⊕ k y , V = k y ⊕ k y ⊂ W ; denote theircorresponding Dynkin diagrams by D and D , respectively. From q ij q ji = 1 it follows x ij = − q ij x ji , so the image of x ji in B is not zero.We split the proof in several cases according to the possibilities for ord q ii and ord q jj . Case 1: q ii / ∈ G ∞ or q jj / ∈ G ∞ . This essentially goes back to [R]. Assumefirst q ii / ∈ G ∞ . If GK-dim B < ∞ , it follows from [AAH1, Lemmas 2.6 and2.7] that there exists a natural number k such that ( k ) ! q ii Q k − h =0 (1 − q hii ) = 0 ,which contradicts q ii / ∈ G ∞ . The case q jj / ∈ G ∞ is similar: since the imageof x ji is not zero, we may apply the same argument as with q ii . Case 2: q jj = 1 . By the previous case, we might assume q ii is a root ofunity, and by hypothesis its order must be N i > . The diagram D is q ii ◦ i q ii q ii ◦ ij , Cartan type (cid:18) − N i − N i (cid:19) . If GK-dim B ( V ) < ∞ then [AAH2] implies that the Cartan matrix is offinite type. Thus we conclude N i = 3 , a contradiction. Case 3: q jj = − . By Case 1, assume that q ii is a root of unity. Its orderis ≥ . By [AAH2], GK-dim B ( V ) = ∞ since the Dynkin diagram of V is q ii ◦ i q ii − q ii ◦ ij and this does not appear in [H1, Table 1]. Case 4: q ii , q jj / ∈ G ′ . Now W has a connected Dynkin diagram: D = q ii ◦ i q ii q ii q jj ◦ ij q jj q jj ◦ j . If the Nichols algebra of V is finite GK-dimensional, by exhaustion of [H1,Table 1] we conclude that q ii , q jj and D satisfy one of the following:(1) q ii ∈ G ′ , q ii q jj = − , q ii ◦ i q ii − ◦ ij (2) q ii ∈ G ′ , q jj = q ii , q ii ◦ i − − ◦ ij (3) q ii ∈ G ′ , q jj = ± q ii , q ii ◦ i q ii ± q ii ◦ ij (4) q ii ∈ G ′ , q jj = q ii , q ii ◦ i q ii ◦ ij (5) q ii ∈ G ′ , q ii q jj = − , q ii ◦ i q ii − ◦ ij (6) q ii ∈ G ′ , q jj = q ii , q ii ◦ i q ii q ii ◦ ij (7) q jj ∈ G ′ , q ii = q jj , q jj ◦ i q jj q jj ◦ ij (8) q ii ∈ G ′ , q ii q jj = − , q ii ◦ i q ii − ◦ ij In the rest of the proof, we discard one by one all these possibilities.(2) Now W is of Cartan type with Dynkin diagram and Cartan matrix: D = q ii ◦ i − − ◦ ij − q ii ◦ j , q ii ∈ G ′ ; − − − − . Since this matrix is of affine type,
GK-dim B ( W ) = ∞ by [AAH2].(3) Assume first q jj = − q ii . Then D = − q ii ◦ ij q ii − q ii ◦ j , q ii ∈ G ′ , INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 13 which is not arithmetic. By [AAH2] we see that
GK-dim B ( V ) = ∞ . Next,when q jj = q ii , W is of Cartan type with Dynkin diagram and Cartan matrix: D = q ii ◦ i q ii q ii ◦ ij q ii q ii ◦ j , q ii ∈ G ′ ; − − − − , which is affine, so GK-dim B ( W ) = ∞ by [AAH2]. (4) Since q ii = 1 , wehave GK-dim B ( V ) = ∞ by [AAH1, Lemma 2.8]. (6) In this case D = q ii ◦ ij q ii q ii ◦ j , q ii ∈ G ′ , is of indefinite Cartan type, so GK-dim B ( V ) = ∞ by [AAH2]. (7) Similarly, D = q jj ◦ ij q jj q jj ◦ j , q jj ∈ G ′ , is indefinite Cartan, so GK-dim B ( V ) = ∞ . In the remaining cases, D is(1) D = ω ◦ i ω − ◦ ij ω − ω ◦ j , ω ∈ G ′ . (5) D = − ω ◦ i ω − ◦ ij ω ω ◦ j , ω ∈ G ′ . (8) D = ζ ◦ i ζ − ◦ ij ζ − ζ ◦ j , ζ ∈ G ′ . Now (1) and (5) are equal up to permutation of the indexes. Only herewe need to assume the validity of Conjecture 2.6. Indeed, these diagrams donot appear in [H1, Table 2], so
GK-dim B ( W ) = ∞ in all cases. (cid:3) A pre-Nichols algebra of type G . We assume ( V, c q ) has the fol-lowing Dynkin diagram ω ◦ ◦ , ω ∈ G ′ . Proposition 3.3.
The algebra ˘ B q := T ( V ) / h x , x i is an eminentpre-Nichols algebra of ( V, c q ) and GK-dim ˘ B q = 6 .Proof. We first claim that the elements x and x are primitive in T ( V ) .This is verified by a direct computation, see [S].Second, we claim that the relations x = 0 and x = 0 hold in anyfinite GK-dimensional pre-Nichols algebra B of ( V, c q ) .Assume first x = 0 in B . Then also x = 0 . From Lemma 2.7 andthe previous claim, we have a braided subspace W = k x + k x + k x ⊂ P ( B ) , so Lemma 2.8 gives GK-dim B ( W ) ≤ GK-dim B . By a degree argument, W has dimension three; from direct computation its Dynkin diagram is ω ◦ ω ❁❁❁❁❁❁❁❁❁ ω ✞✞✞✞✞✞✞✞✞ ω ◦ ω ω ◦ , Cartan type − − − − − − . (3.6)Since the Cartan matrix is of affine type A , we have GK-dim B ( W ) = ∞ by [AAH2, Theorem 1.2(a)]. Thus GK-dim B = ∞ .Assume now x = 0 in B . Then x = 0 , and since q q = 1 , wehave x = − q x = 0 . Consider W ′ = k x + k x + k x . We may nowuse the same argument as above. Indeed, W ′ ⊂ P ( B ) has Dynkin diagram(3.6) replacing x by x , so GK-dim B ( W ′ ) = ∞ by the same reasonas GK-dim B ( W ) = ∞ . Hence GK-dim B = ∞ . Thus ˘ B q ։ B .The verification of GK-dim ˘ B q = 6 is postponed to Proposition 4.5. (cid:3) A further reduction.
Let B be a finite GK-dimensional pre-Nicholsalgebra of B q . We are naturally led to consider E := { ( i, j ) : i ∈ I , j ∈ I , x ij = 0 in B } . (3.7) Remark . If ( i, j ) ∈ E , the braided vector space k x i ⊕ k x ij ⊂ B is ofCartan type A by Remark 3.1. Lemma 3.5. If ( i, j ) , ( i, j ) ∈ E then j = j .Proof. Since x ij and x ij are Z θ -homogeneous, c ( x ij ⊗ x ij ) = q ( α i + α j , α i + α j ) x ij ⊗ x ij = q ii q ij q j i q j j x ij ⊗ x ij . Assume j = j . Then x i , x ij and x ij have pairwise different Z θ -degrees, sothey span a 3-dimensional braided subspace W = k x i + k x ij + k x ij ⊂ P ( B ) .Now the Dynkin diagram of W is q ii ◦ i q ii ✾✾✾✾✾✾✾✾ q ii ✝✝✝✝✝✝✝✝ q ii ◦ ij q ii q ii ◦ ij , q ii ∈ G ′ , Cartan type − − − − − − . Since the Cartan matrix is of affine type A (1)2 , we have GK-dim B ( W ) = ∞ by [AAH2, Theorem 1.2(a)]. Thus GK-dim B = ∞ , a contradiction. (cid:3) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 15 Cartan type
In this section we determine the finite GK-dimensional pre-Nichols alge-bras of braided vector spaces of finite Cartan type under some restrictions.We fix a matrix q = ( q ij ) i,j ∈ I of non-zero scalars such that q ii = 1 for all i ∈ I and a braided vector space ( V, c q ) with braiding given by c q ( x i ⊗ x j ) = q ij x j ⊗ x i , i, j ∈ I θ , in a basis { x , . . . , x θ } . Let N i = ord q ii ∈ N ∪ ∞ .Recall that q , or ( V, c q ) , is of Cartan type if there exists a Cartan matrix a = ( a ij ) i,j ∈ I such that q ij q ji = q a ij ii for all i, j . Let i ∈ I . If N i = ∞ , then a ij are uniquely determined. Otherwise, we impose N i < a ij ≤ , j ∈ I . (4.1)In this way we say that ( V, c q ) , is of Cartan type a .We follow the terminology of [K]. Cartan matrices are arranged in threefamilies, namely: finite, affine and indefinite. We say that q , or ( V, c q ) ,belongs to one of these families if the corresponding a does.In this section we assume that q is of connected finite Cartan type . Hereare the possibilities for the Dynkin diagram of q : A θ : q ◦ q − q ◦ q ◦ q − q ◦ , B θ : q ◦ q − q ◦ q ◦ q − q ◦ C θ : q ◦ q − q ◦ q − q ◦ q ◦ q − q ◦ ,D θ : q ◦ q ◦ q − q ◦ q ◦ q − q ◦ q − q − q ◦ ,E θ : q ◦ q ◦ q − q ◦ q ◦ q − q − q ◦ q − q ◦ , θ ∈ I , ,F : q ◦ q − q ◦ q − q ◦ q − q ◦ , G : q ◦ q − q ◦ . Here q is a root of unity in k ; set N = ord q . We refer to the survey [AA2]for restrictions on N and precise features of B q on each case.The quantum Serre relations are the following elements of T ( V ) : (ad c x i ) − a ij x j , i, j ∈ I , i = j. (4.2)By [AS2, Lemma A.1] these are primitive in any pre-Nichols algebra. Let e B q = T ( V ) / I q be the distinguished pre-Nichols algebra of ( V, c q ) , see §2.9. Remark . From the detailed presentation in [AA2, §4] we see that thequantum Serre relations (4.2) generate I q in the following cases: • when a is of type A or B [AA2, pp. 397, 399, 400], • when a is of type G and N = 4 , [AA2, pp. 410, 411], • when a is simply-laced and N > [AA2, pp. 397, 404, 407], • when a is of type B, C, or F and N > [AA2, pp. 399, 402, 409].4.1. Quantum Serre relations.
Let a = ( a ij ) i,j ∈ I be a symmetrizableindecomposable generalized Cartan matrix and d ∈ GL θ ( Z ) diagonal suchthat da is symmetric; this is equivalent to an irreducible Cartan datum asin [Lu, 1.1.1] by setting · : I × I → Z , i · j = d i a ij , i, j ∈ I . Let g = g ( a ) be the associated Kac-Moody algebra which have a triangulardecomposition g ( a ) = g + ⊕ h ⊕ g − .Let q ∈ k × and consider the Dynkin diagram q di ◦ i q diaij q dj ◦ j (4.3)Let q be any matrix with Dynkin diagram (4.3) and ( V, c q ) be the corre-sponding braided vector space with basis ( x i ) i ∈ I . Notice that q is of Cartantype but it is not necessarily of type a as (4.1) may not hold.Let ˘ B q = T ( V ) modulo the ideal K q generated by the quantum Serrerelations (ad c x i ) − a ij ( x j ) , i = j ∈ I , which is a pre-Nichols algebra of V . Proposition 4.2.
GK-dim ˘ B q ≥ dim g + .Proof. If ξ ∈ k , ξ = q , then p = ( ξ d i a ij ) has Dynkin diagram (4.3). Let ( W, c q ) be the corresponding braided vector space with basis ( b x i ) i ∈ I . Claim 1.
GK-dim ˘ B q = GK-dim ˘ B p .Proof. By the proof of [AS3, Proposition 3.9] (or the proof of Lemma 2.9)there is a homogeneous linear isomorphism ψ : T ( V ) → T ( W ) determinedby ψ ( x i ) = b x i for all i ∈ I and satisfying [AS3, Remarks 3.10]. Hence ψ ( K q ) = K p and ψ induces a homogeneous linear isomorphism ψ : ˘ B q → ˘ B p .Then apply Lemma 2.1. (cid:3) Let now f be the Q ( v ) -algebra defined in [Lu, 1.2.5], where v is an indeter-minate and let A f be the A := Z [ v, v − ] -subalgebra spanned by the quantumdivided powers of the generators of f [Lu, 1.4.7]. By [Lu, 14.4.3], A f is a free A -module and P A f = P f . (4.4)Consider k as A -module via v ξ . Then we have the algebras k f = k ⊗ A A f and k e f defined in [Lu, 33.1.1] (which is nothing else than ˘ B p ). By [Lu, 1.4.3],the quantum Serre relations hold in k f , hence we have a surjective algebramap ˘ B p = k e f ։ k f . Thus GK-dim ˘ B p ≥ GK-dim f . (4.5) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 17
On the other hand, let k be k as A -module via v . Then k e f ≃ U ( g + ) by [Lu, 33.1.1] and dim Q ( v ) f ν = dim k ( k e f ν ) by [Lu, 33.1.3]; that is GK-dim f = GK-dim( k e f ) = dim g + , (4.6)where the first equality holds by Lemma 2.1. The Proposition follows. (cid:3) Example 4.3.
Let a = (cid:18) − − (cid:19) . Then (4.3) takes the form q ◦ q − q ◦ with q ∈ k × . If q ∈ k × and q := q − q − , then q = (cid:18) q q q q (cid:19) has theDynkin diagram above. Here ˘ B q = k h x , x i modulo the relations x x − q ( q ) x x x + qq x x ,x x − q x x x + 3 q x x x − q x x . In this setting Proposition 4.2 gives
GK-dim ˘ B q = ∞ . Example 4.4.
Let a = (cid:18) − − (cid:19) . Then (4.3) takes the form q ◦ q − q ◦ with q ∈ k × . If q ∈ k × and q := q − q − , then q = (cid:18) q q q q (cid:19) has theDynkin diagram above. Here ˘ B q = k h x , x i modulo the relations x x − q (2) q x x x + q q x x ,x x − q (4) q x x x + q q (cid:18) (cid:19) q x x x − q (4) q x x x + q q x x . In this situation Proposition 4.2 establishes
GK-dim ˘ B q ≥ .This last example gains more relevance when the parameter q ∈ k × spe-cializes to a root of unity with small order. Proposition 4.5.
Let a and q ∈ k × as in Example 4.4. (a) If q ∈ G ′ then GK-dim ˘ B q = 6 . (b) If q ∈ G ′ then x = 0 in ˘ B q .Proof. Put x = [ x , x ] . By direct computation, in ˘ B q the followingrelations hold: x x = q q x x ,x x = q q x x + q q ( q − q x ,x x = q q x x + q q ( q − q − x + q q ( q − q x x ,x x = q q x x + q q ( q − (2) q x ,x x = q q x x + x x − q q x x , (2) q x x = q q (2) q x x + q q ( q − q x . (a) In this case the last two relations above become x x = q x x , x x = q x x . These imply more commutations: x x = q x x , x x = q x x ,x x = q x x , x x = q x x . Now we claim that ˘ B q is linearly spanned by B = { x n x n x n x n x n x n : 0 ≤ n , . . . , n } . Denote by I the linear span of B . Since ∈ I , it is enough to show that I isleft ideal of ˘ B q . If we multiply x n x n x n x n x n x n by x on the left, wecan use the previously deduced commutations between the (powers of the) x α ’s to successively rearrange the terms until we get a linear combination ofelements in B . The claim follows.Consider the N -filtration F on ˘ B q induced by B , and denote by gr F ( ˘ B q ) the associated graded algebra. There is a natural projection from a poly-nomial algebra k [ y , . . . , y ] ։ gr F ( ˘ B q ) , hence GK-dim gr F ( ˘ B q ) ≤ . By[KL, Proposition 6.6] and Example 4.4 we also have GK-dim gr F ( ˘ B q ) =GK-dim ˘ B q ≥ , so the equality holds.(b) This follows by specialization at q = − in the relation (2) q x x = q q (2) q x x + q q ( q − q x . (cid:3) Remark . Let us point out the relevance of (b). By Kharchenko’s theory[Kh], ˘ B q has a PBW-basis. By Proposition 4.2 we know GK-dim ˘ B q ≥ but, when q ∈ G ′ , the root α + α will not contribute to GK-dim ˘ B q by(b) above. So even if a is of type G , one should not expect that the PBWgenerators are just those related to the six positive roots of G , as was thecase in the proof of (a).4.2. Type A . In this and the next subsections we seek for eminent (familiesof) pre-Nichols algebras in order to determine finite
GK-dim pre-Nicholsalgebras of braidings of finite Cartan type. The distinguished pre-Nicholsalgebra will serve as the principal guide in our exploration.4.2.1.
Type A with N > . Lemma 4.7.
Assume a is of Cartan type A with N > . If B is a finiteGK-dimensional pre-Nichols algebra of q , then x = 0 and x = 0 in B ,i. e. the distinguished pre-Nichols algebra e B q is eminent, cf. Definition 2.3.Proof. Assume x iij = 0 for some i = j ∈ I ; the 3-dimensional braidedsubspace W := k x j ⊕ k x i ⊕ k x iij ⊂ P ( B ) has GK-dim B ( W ) < ∞ . INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 19
Consider the braided subspace W = k x i ⊕ k x iij ⊂ W . By direct com-putation, the braiding on W is of Cartan type with the following Dynkindiagram and Cartan matrix: q ii ◦ i q ii q ii ◦ iij , A = (cid:18) − N − M (cid:19) , M = ( N/ , if | N,N, otherwise . If either N = 5 or N > , it is evident that the Cartan matrix A is notfinite, so GK-dim B ( W ) = ∞ by [AAH2, Theorem 1.2 (b)]. This contradicts GK-dim B < ∞ .For the remaining cases (i. e. N = 4 and N = 6 ), we consider the whole W .Since e q ( α j , α i + α j ) = ( q ji q ij ) q jj = 1 , the braiding on W is of Cartan typewith the following Dynkin diagram and Cartan matrix q ii ◦ j q − ii q ii ◦ i q ii q ii ◦ iij , A = − − − N − M . Now it is straightforward to verify that if N = 4 or , then A is of affine type,which contradicts GK-dim B ( W ) < ∞ by [AAH2, Theorem 1.2 (b)]. (cid:3) Type A with N = 3 . Here is the first restriction.
Lemma 4.8.
Assume a is of Cartan type A with N = 3 . Let B ∈ Pre fGK .Then x iiij = 0 and x jiij = 0 in B for all i = j ∈ I .Proof. Since x iij is primitive, using that e q ( α i , α i + α j ) = q q − = 1 and e q ( α j , α i + α j ) = q q − = 1 , we get x iiij , x jiij ∈ P ( B ) by Lemma 2.7.Assume first x iiij = 0 in B . The braided subspace k x i ⊕ k x j ⊕ k x iiij ⊂ P ( B ) has finite GK-dim
Nichols algebra. The Dynkin diagram is q ◦ iiij q − ✽✽✽✽✽✽✽✽ q − ✞✞✞✞✞✞✞✞ q ◦ i q − q ◦ j , Cartan type − − − − − − . The Cartan matrix is of affine type A (1)2 , and by [AAH2, Theorem 1.2(a)]this contradicts GK-dim B < ∞ .If x jiij = 0 , the same argument leads to a contradiction. Indeed, by directcomputations, the Dynkin diagram of U = k x i ⊕ k x j ⊕ k x jiij is q ◦ jiij q − ✽✽✽✽✽✽✽✽ q − ✞✞✞✞✞✞✞✞ q ◦ i q − q ◦ j . so GK-dim B ( U ) = ∞ by [AAH2, Theorem 1.2(a)]. (cid:3) Remark . Denote b B = T ( V ) / h x , x , x , x i . The definingideal of b B is a Hopf ideal by the proof of Lemma 4.8. Let π : b B → B ( V ) denote the natural projection. Let Z be the subalgebra of b B generated by z := x , z := x , z := x , z := x , z := x . The next results are devoted to prove that b B is eminent. Lemma 4.10. (a)
Given i = j ∈ I , the following relations hold in b B : [ x ij , x iij ] = 0 = [ x ij , x jji ]; [ x iij , x jji ] = 0 . (b) Z is a normal Hopf subalgebra of b B . (c) The z i ’s q -commute; B = { z n z n z n z n z n : n i ∈ N } is a basis of Z . (d) Z = co π b B .Proof. (a) Just compute using (2.8): (cid:2) x ij , x iij (cid:3) = (cid:2) x i , [ x j , x iij ] (cid:3) − q ij x j (cid:2) x i , x iij (cid:3) + q jj q ji (cid:2) x i , x iij (cid:3) x j = 0; (cid:2) x ij , x jji (cid:3) = (cid:2) x i , [ x j , x jji ] (cid:3) − q ij x j (cid:2) x i , x jji (cid:3) + q ii q ij (cid:2) x i , x jji (cid:3) x j = 0; (cid:2) x iij , x jji (cid:3) = (cid:2) [ x i , x ij ] , x jji (cid:3) = (cid:2) x i , [ x ij , x jji ] (cid:3) − q ii q ij x ij (cid:2) x i , x jji (cid:3) + q ii q ij (cid:2) x i , x jji (cid:3) x ij = (cid:2) x i , [ x ij , x jji ] (cid:3) = 0 . (b) We claim that the generators of Z are annihilated by the braidedadjoint action of b B . Fix i ∈ I . By definition (ad c x i ) z = 0 = (ad c x i ) z .In T ( V ) we have (ad c x i ) x i = x i − q ii x i = 0 , and if j = i then x jjji = (ad c x j ) x i = X k =0 ( − k q kji q k ( k − / jj (cid:18) k (cid:19) q jj x − kj x i x kj = x j x i − q ji x i x j = − q ji (ad c x i ) x j . (4.7)Thus ad c x i annihilates z and z . Finally, we proceed with z . From (a) weget the commutation x x = q q x x in b B . Then using (2.6) (ad c x ) z = x x + q q x x x + q q x x = q ( q + q + q ) x x = 0 . (4.8)For (ad c x ) z , notice that on the one hand (cid:2) x , − (cid:3) x = X k =0 ( − k ( q q ) k q k ( k − / (cid:18) k (cid:19) q x − k x x k = x x i − q x i x = − q (ad c x ) x . (4.9)On the other hand, using [ x , x ] = q q x and (a) we get h x , (cid:2) x , [ x , x ] (cid:3)i = q q h x , [ x , x ] i = h x , i = 0 , (4.10) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 21 so (ad c x ) z = 0 . This shows that Z is a normal subalgebra.Next we verify that ∆( z i ) ∈ Z ⊗ Z for i ∈ I . This is clear for i ∈ I ,because those elements are primitive in T ( V ) ; for i = 5 we compute in T ( V ) : ∆( x ) = x ⊗ ⊗ x + ( q − − q − ) x ⊗ x + (1 − q − ) q x ⊗ x + (1 − q ) q x ⊗ x − (1 − q − ) q − x ⊗ x + ( q − x ⊗ [ x , x ] − (1 − q − ) q [ x , x ] ⊗ x . (4.11)Using (a) and the defining relations of b B we see that Z is a Hopf subalgebra.(c) We show that any pair of generators of Z q -commute. By definitionof b B , both x and x q -commute with z and z , so z and z q -commutewith z and z . Secondly, (4.7) implies that z and z q -commute. Thirdly,(a) shows that z q -commutes with z and z , and also that z and z q -commute. Lastly, z q -commutes with z by (4.8), and with z by (4.9) and(4.10). Hence B linearly generates Z .The linear independence is proven by steps. Step 1.
The set { z n z n z n z n : n i ∈ N } is linearly independent.Proof. Consider the Hopf algebra b B kZ ; let A denote the subalgebra gen-erated by z , . . . , z and Z . Since all the generators of A are either skew-primitives or group-likes, it follows that A itself is a pointed Hopf algebra.Notice that z , . . . , z ∈ P ( Z ) are linearly independent. Indeed, they arenon-zero because their Z -degree is , so they are linearly independent sincetheir Z -degrees are pairwise different (here we are using that the definingideal of b B is a Hopf ideal generated by Z -homogeneous elements of Z -degree ). Hence the infinitesimal braiding of A contains the braided vector space k z ⊕ · · · ⊕ k z , which is quantum linear space with all points labeled by .Thus (cid:8) z n z n z n z n g : n i ∈ N , g ∈ Z (cid:9) ⊂ A is linearly independent. (cid:3) Step 2.
The element z does not belong to the left ideal b B h z , z , z , z i .Proof. We verify this using [GAP]. (cid:3)
The ideal b B h z , z , z , z i is a Hopf ideal because the generators are prim-itive. Denote the quotient by R and consider the projection π R : b B ։ R . Step 3.
The set { π R ( z ) n : n ∈ N } is linearly independent.Proof. Consider the Hopf algebra R Z . The subalgebra generated by π R ( z ) and Z is a pointed Hopf algebra. Moreover, its infinitesimal braid-ing contains π R ( z ) , which is a non-zero point by Step 2 and is labeled by .Now proceed as in the proof of Step 1. (cid:3) Step 4.
We have (id ⊗ π R )∆( z n ) = P nk =0 (cid:0) nk (cid:1) z k ⊗ π R ( z ) n − k for all n ∈ N . Proof.
The case n = 0 is obvious, and n = 1 follows from (4.11). An standardinductive argument for braided comultiplication yields the desired result. (cid:3) Step 5.
The set B is linearly independent.Proof. Let P n ,...,n ∈ N λ n ,...,n z n z n z n z n z n = 0 . Assume there exists n such that λ n ,...,n = 0 for some n , . . . , n ∈ N ; take N as the maximalone. By Step 3 there is a linear map f : R → k such that f ( π R ( z ) n ) = δ n,N for all n ∈ N . Now using Step 4 we compute ⊗ f )(id ⊗ π R )∆ X n ,...,n ∈ N λ n ,...,n z n z n z n z n z n = X n ,...,n ∈ N λ n ,...,n (id ⊗ f )(id ⊗ π R ) Y i =1 n i X j =0 (cid:18) n i j (cid:19) z ji ⊗ z n i − ji ∆ z n = X n ,...,n ∈ N , n ≤ N λ n ,...,n (id ⊗ f ) n X j =0 z n z n z n z n z j ⊗ π R ( z n − j ) = X n ,...,n ∈ N λ n ,...,n ,N z n z n z n z n ⊗ . This contradicts Step 1. (cid:3) (d) Since ∆( z i ) ∈ Z ⊗ Z and Z is normal, the right ideal b B Z + is a Hopfideal. By [A+, Proposition 3.6 (c)] we get that Z = co π b B is equivalent to B q ≃ b B / b B Z + . This last equality holds because the diagram J q (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) T ( V ) (cid:15) (cid:15) (cid:15) (cid:15) ❍❍❍❍❍❍❍❍❍ b B Z + (cid:31) (cid:127) / / b B π / / / / B q , commutes. (cid:3) Proposition 4.11. (a)
There is an extension of braided Hopf algebras k → Z ֒ → b B ։ B q → k . (b) The pre-Nichols algebra b B is eminent and GK-dim b B = 5 .Proof. (a) Follows from Lemma 4.10 (d).(b) We know that b B covers all elements of Pre fGK by Lemma 4.8; itremains to show that b B itself belongs to Pre fGK . By [A+, Proposition 3.6(d)] there is a right Z -linear isomorphism B q ⊗ Z ≃ b B . Since B q is finitedimensional, this implies that b B is finitely generated as a Z -module. Now[KL, Proposition 5.5] provides GK-dim b B = GK-dim Z = 5 . (cid:3) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 23
Type B .Lemma 4.12. Assume that a is of Cartan type B . Then the distinguishedpre-Nichols algebra e B q is eminent.Proof. Here
N > . We may fix a braiding matrix q such that q = q ,so q = q and f q = q − . Let B be a finite GK-dimensional pre-Nicholsalgebra of V . It is enough to prove that x = 0 = x in B .Assume first x = 0 , and consider the 3-dimensional braided subspace W := k x ⊕ k x ⊕ k x ⊂ P ( B ) . Then GK-dim B ( W ) < ∞ from Lemma2.8. We split the proof according to the several possibilities for N . ♥ N = 3 . Now the braiding on W is of Cartan type q ◦ q − q ◦ q − q ◦ , A = − − − − . Since A is of affine type C (1)2 , this contradicts [AAH2, Theorem 1.2(a)]. ♥ N = 6 . In this case the braiding on W := k x ⊕ k x ⊂ W is q ◦ q − q ◦ , Cartan type (cid:18) − − (cid:19) . The Cartan matrix is of indefinite type, and by [AAH2, Theorem 1.2(b)] thiscontradicts
GK-dim B ( W ) < ∞ . ♥ N = 3 , . The Dynkin diagram of W := k x ⊕ k x ⊂ W is D = q ◦ q q ◦ . Since
GK-dim B ( W ) < ∞ , it follows from [AAH2, Theorem 1.2(b)] thatthe associated root system is finite. Now D is connected; by exhaustion on[H1, Table 1], we deduce that it must be N = 4 or N = 8 . We turn again to W , whose Dynkin diagram is easily computed in each case: ♥♥ N = 4 : q ◦ q − q ◦ , Cartan type (cid:18) − − (cid:19) , ♥♥ N = 8 : q ◦ q − q ◦ , Cartan type (cid:18) − − (cid:19) . In any case the Cartan matrix is of affine type A (1)1 , so GK-dim B ( W ) = ∞ by [AAH2, Theorem 1.2(b)].Assume x = 0 in B . The subspace U := k x ⊕ k x ⊕ k x ⊂ P ( B ) has dimension 3 and GK-dim B ( U ) < ∞ . Now U := k x ⊕ k x ⊂ U hasconnected Dynkin diagram q ◦ q q ◦ , and it is finite by [AAH2, Theorem 1.2(b)]. By exhaustion on [H1, Table 1]we deduce that N = 4 . Then the Dynkin diagram of U is of Cartan type q ◦ − − ◦ − q ◦ , A = − − − − . Since A is of affine type C (1)2 , this contradicts [AAH2, Theorem 1.2(a)]. (cid:3) Type G .Lemma 4.13. Assume that a is of Cartan type G . Then the quantum Serrerelations hold in any B ∈ Pre fGK . In particular, e B q is eminent if N = 4 , .Proof. Here
N > . Let B ∈ Pre fGK ( V ) ; we show first that the quantumSerre relations x = 0 = x hold in B .Start assuming x = 0 . Then the 3-dimensional subspace W := k x ⊕ k x ⊕ k x ⊂ P ( B ) satisfies GK-dim B ( W ) ≤ GK-dim B by Lemma 2.8.The Dynkin diagram of W := k x ⊕ k x ⊂ W is D = q ◦ q q ◦ . Since
GK-dim B ( W ) < ∞ , it follows from [AAH2, Theorem 1.2(b)] thatthe root system of D is finite. We split the proof according to the severalpossibilities for N . ♥ N = 5 . The diagram D is disconnected, but we might consider instead W := k x ⊕ k x ⊂ W , that satisfies GK-dim B ( W ) < ∞ as well. Bydirect computation W is of indefinite Cartan type: q ◦ q − q ◦ , (cid:18) − − (cid:19) , which is in contradiction with [AAH2, Theorem 1.2(a)]. ♥ N = 5 . Now D is connected and finite; by inspection on [H1, Table 1],it must be N = 4 or N = 6 . ♥♥ N = 4 . In this case W is of Cartan type q ◦ q − q ◦ , (cid:18) − − (cid:19) , which is of affine type A (1)1 , now contradicting [AAH2, Theorem 1.2(b)]. ♥♥ N = 6 . In this case the Dynkin diagram of W is of Cartan type q ◦ q − q ◦ q − q ◦ , A = − − − − . By [AAH2, Theorem 1.2(b)] this contradicts
GK-dim B ( W ) < ∞ , since A is of affine type G (1)2 . INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 25
Assume now x = 0 in B . The subspace U := k x ⊕ k x ⊕ k x ⊂ P ( B ) has dimension 3 and GK-dim B ( U ) < ∞ . Consider two possibilities for N . ♥ N = 4 . Now U := k x ⊕ k x ⊂ U has connected Dynkin diagram q ◦ q − q ◦ . By exhaustion on [H1, Table 1] we conclude that this diagram is never finite,which contradicts [AAH2, Theorem 1.2(b)], as
GK-dim B ( U ) < ∞ . ♥ N = 4 . In this case the braiding on U is of Cartan type q ◦ q − q ◦ q − q ◦ , A = − − − − . Since A is of affine type G (1)2 , this contradicts [AAH2, Theorem 1.2(a)].Thus the quantum Serre relations hold in B . By Remark 4.1 this provesthe assertion regarding N = 4 , . (cid:3) Type A .Lemma 4.14. If a is of Cartan type A with N > , then e B q is eminent.Proof. As N > , the ideal I q is generated by the quantum Serre relations x = 0 and x iij = 0 for | j − i | = 1 , cf. [AA2, p. 397]. Let B ∈ Pre fGK ( q ) .Then x = 0 holds in B since the braided vector space k x ⊕ k x satisfiesthe hypothesis in Proposition 3.2.Turn to x iij for some fix i, j ∈ I with | j − i | = 1 ; in this case k x i ⊕ k x j is ofCartan type A . If N > , then x iij = 0 in B by Lemma 4.7. It only remainsthe case N = 3 . Now we have q (2 α i + α j , α i + α j ) = q ii f q ij q jj = q q − = 1 .Using [AAH1, Lemma 2.8], in order to guarantee x iij = 0 in B it is enoughto find k ∈ I such that e q ( α k , α i + α j ) = 1 . It is straightforward to verifythat the unique k ∈ I different from i and j does the trick. (cid:3) Types B and C .Lemma 4.15. The distinguished pre-Nichols algebra e B q is eminent if either (i) a is of type B , or (ii) a is of type C .Proof. Let B ∈ Pre fGK ( q ) . Then x = 0 holds in B . Indeed, the braidedvector space k x ⊕ k x satisfies the hypothesis in Proposition 3.2. Similarly,since k x ⊕ k x is of type B , it follows from Lemma 4.12 that the quantumSerre relations involving x and x hold in B . Step 1. If a is of Cartan type B , then the quantum Serre relations hold inany finite GK-dim pre-Nichols algebra.
Proof.
Here k x ⊕ k x has Dynkin diagram q ◦ q − q ◦ , type A . Hence,if ord q > , we know from Lemma 4.7 that the quantum Serre relationsbetween x and x hold in B . Let us show that in the cases ord q = 2 , the same happens. ♥ ord q = 2 . If x = 0 in B , we get a subspace k x ⊕ k x ⊕ k x ⊂ P ( B ) of dimension with the following Dynkin diagram − ◦ − q ◦ − − ◦ , Cartan type − − − − . This matrix is of affine type C (1)2 , hence GK-dim B = ∞ , a contradiction.Similarly, the assumption x = 0 yields a subspace of P ( B ) with braiding − ◦ − ◦ − − ◦ − − q ◦ , Cartan type − − − − − − . The Cartan matrix is of affine type B (1)3 , and again GK-dim( B ) = ∞ . ♥ ord q = 3 . Notice that q (2 α + α , α + α ) = q = 1 , e q (2 α + α , α ) = q − = 1; q ( α + 2 α , α + 2 α ) = q = 1 , e q ( α + 2 α , α ) = q − = 1 . Assuming x = 0 in B we get k x ⊕ k x ⊂ P ( B ) with Dynkin diagram q ◦ q − ◦ . Then by [AAH1, Lemma 2.8] it follows GK-dim B = ∞ , acontradiction. By the same argument, it can not be x = 0 in B . (cid:3) The assertion (i) for
N > follows since, in that case, e B q is presented bythe quantum Serre relations, cf. Remark 4.1. Step 2. If a is of Cartan type B with N = 3 , then e B q is eminent.Proof. By [AA2, pp. 399, 400], e B q is presented by the quantum Serre rela-tions and [ x , x ] c = 0 . Given B ∈ Pre fGK , let us show that [ x , x ] c ∈P ( B ) . Using x = 0 an straightforward computation gives ∆( x ) = x ⊗ ⊗ x + (1 − q ) x ⊗ x + (1 − q ) q x ⊗ x + (1 − q )(1 − q ) x ⊗ x . with this we compute ∆([ x , x ] c ) =[ x , x ] c ⊗ ⊗ [ x , x ] c − (1 − q ) (1 − q ) q x ⊗ x − (1 − q ) q q q q x ⊗ x INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 27 + (1 − q ) q q q x ⊗ ( x − [ x , x ] c ) − (1 − q ) q q x ⊗ x + (1 − q ) q q [ x , x ] c ⊗ x + (1 − q ) q x ⊗ ( q q q [ x , x ] c + [ x , x ] c ) The third and fourth terms vanishes in B by Step 1. For the fifth term, anstraightforward computation involving x = 0 shows that x = [ x , x ] c .The last three terms also vanish, but they require a more detailed analysis. ♥ x = 0 in B . Notice that ∆( x ) = x ⊗ ⊗ x + (1 − q ) x ⊗ x − (1 − q ) q x ⊗ x , so this element is primitive in B by Step 1. Assuming x = 0 we get asubspace k x ⊕ k x ⊂ P ( B ) where the braiding is given by q ◦ q − q ◦ , Cartan type (cid:18) − − (cid:19) , affine type A (1)1 . this contradicts GK-dim B < ∞ by [AAH2, Theorem 1.2]. ♥ [ x , x ] c = 0 in B . Now we have ∆([ x , x ] c ) =[ x , x ] c ⊗ ⊗ [ x , x ] c − (1 − q ) q x ⊗ [ x , x ] c − (1 − q ) q q x ⊗ x . The element [ x , x ] c is primitive in B , so it vanishes by the same reasonthat x does (cf. proof of Lemma 4.12). So [ x , x ] c ∈ P ( B ) by Step 1.If it is non-zero, consider k x ⊕ k [ x , x ] c ⊂ P ( B ) where the braiding is q ◦ x q − q ◦ [ x ,x ] , Cartan type (cid:18) − − (cid:19) , affine type A (1)1 , thus we get the same contradiction as with x . ♥ q q q [ x , x ] c + [ x , x ] c = 0 . Denote this element by r . Then ∆( r ) = r ⊗ ⊗ r + (1 − q ) q q q x ⊗ ( x − [ x , x ])+ (1 − q ) q q q q (cid:2) x , [ x , x ] (cid:3) ⊗ x − (1 − q ) q q q q x ⊗ (cid:2) [ x , x ] , x (cid:3) . Since [ x , x ] = 0 and x − [ x , x ] = 0 in B , it follows that r is primitive.If r = 0 we consider k x ⊕ k r ⊂ P ( B ) . The Dynkin diagram is computed: q ◦ q − q ◦ r , Cartan type (cid:18) − − (cid:19) , affine type A (1)1 , thus we get the same contradiction as before. Using this three ♥ we get [ x , x ] c ∈ P ( B ) . If this element is non-zero,consider U = k x ⊕ k [ x , x ] c ⊂ P ( B ) . We compute the braiding: q ( α + 2 α + 3 α , α + 2 α + 3 α ) = 1 , e q ( α + 2 α + 3 α , α ) = q − = 1 . From [AAH1, Lemma 2.8] it follows
GK-dim B ( U ) = ∞ , but this contradicts GK-dim B < ∞ . Then [ x , x ] c = 0 in B and Step 2 holds. (cid:3) Step 3. If a is of Cartan type B with N = 4 , then e B q is eminent.Proof. By [AA2, pp. 399, 400], e B q is presented by the quantum Serre re-lations and [ x , x ] c = 0 . We claim that this element is primitive in any B ∈ Pre fGK . Indeed, using that x = 0 in B , we get ∆([ x , x ] c ) =[ x , x ] c ⊗ ⊗ [ x , x ] c − (1 − f q ) q x ⊗ x + (1 − f q ) q [ x , x ] c ⊗ x . By straightforward computations, [ x , x ] c ∈ P ( B ) and it vanishes by thesame reason that x does (cf. proof of Lemma 4.7). Since x = 0 , theclaim follows.Assume [ x , x ] c = 0 . Inside P ( B ) we have the 2-dimensional subspace U = k x ⊕ k [ x , x ] c where the braiding is given by q ◦ x − − q ◦ [ x ,x ] , Cartan type (cid:18) − − (cid:19) . Since this matrix is of affine type A (1)1 , from [AAH2, Theorem 1.2(b)] itfollows GK-dim B ( U ) = ∞ , contradicting B ∈ Pre fGK . (cid:3) Step 4. If a is of Cartan type C , then the quantum Serre relations hold inany finite GK-dim pre-Nichols algebra.Proof.
Now k x ⊕ k x has Dynkin diagram q ◦ q − q ◦ , type A . If N > ,then the quantum Serre relations in x and x hold by Lemma 4.7. For thecase N = 3 , let i, j such that { i, j } = { , } and suppose x iij = 0 in B .Since q (2 α i + α j , α i + α j ) = q q − = 1 and e q (2 α i + α j , α ) = f q i f q j = 1 ,we get GK-dim B = ∞ by [AAH1, Lemma 2.8]. (cid:3) The assertion (ii) for
N > follows since, in that case, e B q is presented bythe quantum Serre relations, see Remark 4.1. Step 5. If a is of Cartan type C with N = 3 , then e B q is eminent.Proof. Following [AA2, pp. 401, 402]) we see that e B q is presented by thequantum Serre relations and (cid:2) [ x , x ] c , x (cid:3) c = 0 . Given B ∈ Pre fGK , letus show that this element is primitive in B . Using x = 0 it follows ∆([ x , x ] c ) =[ x , x ] c ⊗ ⊗ [ x , x ] c + (1 − f q ) x ⊗ x + (1 − q ) x ⊗ x + (1 − f q ) x ⊗ ( x x − q x x )+ (1 − f q ) q [ x , x ] c ⊗ x . INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 29
By straightforward computations, [ x , x ] c = q qx in T ( V ) , and so [ x , x ] c vanishes in B by Step 4. Then we obtain ∆ (cid:0)(cid:2) [ x , x ] c , x (cid:3) c (cid:1) = (cid:2) [ x , x ] c , x (cid:3) c ⊗ ⊗ (cid:2) [ x , x ] c , x (cid:3) c + (1 − q ) q q [ x , x ] c ⊗ x + (1 − f q ) q q x ⊗ x , and now the claim follows from Step 4.If (cid:2) [ x , x ] c , x (cid:3) c = 0 , consider U = k x ⊕ k (cid:2) [ x , x ] c , x (cid:3) c ⊂ P ( B ) .By [AAH1, Lemma 2.8], since q ( α + 3 α + α , α + 3 α + α ) = q q − = 1 and e q ( α + 3 α + α , α ) = q q − = 1 , we have GK-dim B ( U ) = ∞ . Thiscontradicts B ∈ Pre fGK . (cid:3) Step 6. If a is of Cartan type C with N = 4 , then e B q is eminent.Proof. By [An1, Theorem 3.1], e B q is presented by the quantum Serre rela-tions and [ x , x ] c = 0 . Let us show that this element is primitive in anyfinite dimensional pre-Nichols algebra B .First we claim that [ x , x ] c = 0 in B : using that x = 0 we compute ∆ (cid:0) [ x , x ] c (cid:1) =[ x , x ] c ⊗ ⊗ [ x , x ] c + (1 − f q q q ) x ⊗ x + (1 − f q ) x ⊗ [ x , x ] c . Since [ x , x ] c ∈ P ( B ) , it vanishes in B by the same reason that x does(cf. proof of Lemma 4.12). So [ x , x ] c ∈ P ( B ) . Hence, if it is non-zero weget a subspace U = k x ⊕ [ x , x ] c ⊂ P ( B ) where the braiding is given by q ◦ x q − q ◦ [ x ,x ] , indefinite Cartan type (cid:18) − − (cid:19) . But then
GK-dim B ( U ) = ∞ by [AAH2, Theorem 1.2(b)], a contradiction.Next we compute ∆ (cid:0) [ x , x ] c (cid:1) = [ x , x ] c ⊗ ⊗ [ x , x ] c + (1 − f q ) q q q x ⊗ [ x , x ] + (1 − f q ) q x ⊗ [ x , x ] c . Using the previous claim and the fact [ x , x ] c = q qx = 0 (by Step4), we get [ x , x ] c ∈ P ( B ) . If [ x , x ] c = 0 , consider the subspace W = k x ⊕ k x ⊕ k [ x , x ] c ⊂ P ( B ) , where the braiding is q ◦ q − q ◦ q − q ◦ [ x ,x ] , Cartan type − − − − . Since the Cartan matrix is of affine type G (1)2 , it follows GK-dim B ( W ) = ∞ by [AAH2, Theorem 1.2(a)]. This contradicts B ∈ Pre fGK . (cid:3) The result follows. (cid:3)
Some cases in rank > . Here we assume that θ ≥ . Lemma 4.16.
In any of the following cases, e B q is eminent. (a) a is of Cartan type with simply laced Dynkin diagram and N > . (b) a is of type B θ , C θ ( θ ≥ ) or F , and N > .Proof. By Remark 4.1 and the restrictions on N , e B q is presented by thequantum Serre relations. Let B ∈ Pre fGK ( q ) . If a ij = 0 , then x ij = 0 holdsin B by Proposition 3.2. If a ij = 0 , then there is k ∈ I such that { i, j, k } span a subdiagram of type A , B or C . Then (ad x i ) − a ij ( x j ) = 0 byLemmas 4.14 or 4.15. Thus e B q ։ B . (cid:3) In the next subsections we treat some remaining cases with small N .4.8. Types B θ , C θ , F , θ > , N = 3 , .Lemma 4.17. If a is of types B θ , C θ , with θ > , or F , and N = 3 or ,then e B q is eminent.Proof. We split the proof according to the type. Let B ∈ Pre fGK . ♥ Type F . Here e B q is presented by the quantum Serre relations and [ x , x ] c = [ x , x ] c = 0 if N = 4; [ x , x ] c = 0 if N = 3 . (4.12)Since N > we get x = 0 in B from Proposition 3.2. The subdiagramspanned by { , , } is of type C thus the quantum Serre relations involvingthis indices hold in B by Lemma 4.15 (ii). Finally, { , , } span a diagramof type B so the quantum Serre relations involving this indices hold in B byLemma 4.15 (i). Moreover (4.12) are defining relations of the distinguishedpre-Nichols algebra of type B or C for the corresponding N , hence Lemma4.15 implies that these also vanish in B . ♥ Type B θ . Here e B q is presented by the quantum Serre relations and [ x ( i i +2) , x i +1 ] c , i < θ − , if N = 4; [ x θθθ − θ − , x θθ − ] c , if N = 3 . (4.13)The relations involving the indices { θ − , θ − , θ } hold in B by Lemma 4.15(i); also x iθ = 0 for any i < θ − by Proposition 3.2. We are left to treat therelations involving { , . . . , θ − } . If N = 3 we only have the quantum Serrerelations, which hold by Lemma 4.14. Turn to N = 4 . Now { , . . . , θ − } form a subdiagram of type A θ − at a root of order . If θ − ≥ we applyLemma 5.6 to get all the Serre relations except for x and x θ − θ − θ − . Thelast one holds by Lemma 4.15 (i) and the first one falls since the diagram − ◦ − ◦ − − ◦ − − ❏❏❏❏❏❏❏❏❏❏ − ◦ − ◦ θ − − q ◦ θ is of indefinite Cartan type. Now [ x ( i i +2) , x i +1 ] c = 0 for i < θ − hold byLemma 5.6 (e). We treat separately the last case standing. INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 31 ♥♥ Type B with N = 4 . The relations x and x θ − θ − θ − hold by thesame reason as above. Moreover, we also have x = 0 . This follows from[AAH1, Lemma 2.8] since q ( α + α , α + α ) = 1 and e q ( α + α , α ) = 1 .Finally, using (5.1) and the relations deduced so far, we get that [ x (13) , x ] c is primitive in B . Notiche that q ( α + 2 α + α , α + 2 α + α ) = 1 and e q ( α + 2 α + α , α ) = 1 , so [AAH1, Lemma 2.8] applies again. ♥ Type C θ . Here e B q is presented by the quantum Serre relations and [ x ( θ − θ ) , x θ − θ ] c , if N = 4; (cid:2) [ x ( θ − θ ) , x θ − ] c , x θ − (cid:3) , if N = 3 . (4.14)As before, Proposition 3.2 gives x iθ = 0 for any i < θ − ; all the relationsinvolving the indices { θ − , θ − , θ } hold in B by Lemma 4.15 (ii). It remainsto verify the relations involving { , . . . , θ − } . Here we only have the Serrerelations. But this indices span a subdiagram of type A θ − , θ − ≥ , at aroot of unity of order or , so they hold by Lemma 4.14. (cid:3) Types E , E and E with N = 2 . By [AA2, p. 407] the distinguishedpre-Nichols algebra is presented by the quantum Serre relations and [ x ijk , x j ] c = 0 if i, j, k are all different and f q ij , f q jk = 1 . Lemma 4.18. If a is of type E , E or E with N = 2 , then e B q is eminent.Proof. Let B ∈ Pre fGK ( q ) . First we deal with the quantum Serre relations,which are always primitive. Fix i = j ∈ I θ . Consider two possibilities. ♥ f q ij = 1 . In this case choose k ∈ I θ different from i and j such that e q i = 1 but f q ik = 1 . We get q ( α i + α j , α i + α j ) = 1 and e q ( α i + α j , α k ) = 1 .By [AAH1, Lemma 2.8], this warranties x ij = 0 in B . ♥ f q ij = 1 . In this case i and j are consecutive vertices in a subdiagramof type A with N = 2 . By Lemma 5.6 (b) below, it follows that x iij = 0 except in the following cases: ( i, j ) ∈ { (2 , , ( θ − , θ ) , ( θ − , θ − } . Fixsuch ( i, j ) , assume x iij = 0 and consider k x ⊕ · · · ⊕ k x θ ⊕ k x iij ⊂ P ( B ) .Then the Dynkin diagram of this braided vector space is of indefinite Cartantype. We illustrate the case ( i, j ) = ( θ − , θ ) , the other cases being similar. − ◦ θ − ◦ θ − θ − θ − ◦ − − ◦ − ◦ θ − − − − tttttttttttt − ◦ θ − − − ◦ θ − . Thus Conjecture 2.6 and Lemma 2.8 imply
GK-dim B = ∞ .Finally, fix i, j, k different such that f q ij , f q jk = 1 . These are consecutivevertices in a suitable chosen subdiagram of type A . The Serre relations holdin B , so by Lemma 5.6 (c) below we get that also [ x ijk , x j ] c = 0 in B . (cid:3) On the open cases
This section contain partial results towards those braidings of finite Cartantype which are still open. The detailed proofs can be found in [S].5.1.
Type A with N = 2 .Lemma 5.1. Assume a is of Cartan type A with N = 2 . Let B be a finiteGK-dimensional pre-Nichols algebra of q . The following hold: (a) if B ∈ Pre Z fGK , then either x = 0 or x = 0 in B ; (b) for different i, j ∈ I , (ad c x i ) x j = 0 in B . (cid:3) Question 5.2.
Let b B = k h x , x | x , x i . By Lemma 5.1 any B ∈ Pre Z fGK ( V ) is a quotient of either b B or b B := k h x , x | x , x i . Clearly b B ≃ b B as algebras. Is GK-dim b B < ∞ ?5.2. Type A with N = 2 .Lemma 5.3. Assume a is of Cartan type A with N = 2 . Let B be a finiteGK-dimensional pre-Nichols algebra of q . Then the following hold in B : (a) x = 0 = x , x ⋆ = 0 , (b) x = 0 = x , x = 0 = x , (c) if B ∈ Pre Z fGK , then at most one of x , x , x , x is non-zero. (cid:3) Remark . The relation ⋆ is relevant because in the tensor algebra ∆([ x (13) , x ] c ) = [ x (13) , x ] ⊗ ⊗ [ x (13) , x ] − q x ⊗ x − q q x ⊗ x − q q x ⊗ x + 4 q q x ⊗ x . (5.1) Question 5.5.
By Lemma 5.3 every B ∈ Pre Z fGK is covered by one of b B = k h x , x , x | x , x , x , x , x , x , x i , b B = b B / h x i , b B = b B / h x i , b B = b B / h x i , b B = b B / h x i . Are
GK-dim b B or GK-dim b B < ∞ ? ( b B ≃ b B and b B ≃ b B as algebras).5.3. Type A θ , θ ≥ with N = 2 . In this setting e B q is presented by x ij = 0 , | i − j | > x iij = 0 , | i − j | = 1; [ x ( ii +2) , x i +1 ] c = 0 , i ∈ I θ − . Lemma 5.6.
Assume a is of Cartan type A θ , θ ≥ , with N = 2 . Thefollowing hold in any finite GK-dimensional pre-Nichols algebra B of q : (a) x ij = 0 for any | i − j | > ; (b) x iij = 0 for | i − j | = 1 and ( i, j ) = (2 , , ( θ − , θ ) ; (c) x iiiij = 0 for ( i, j ) = (2 , , ( θ − , θ ) ; (d) if B ∈ Pre Z θ fGK , then either x = 0 or x θ − θ − θ = 0 ; (e) if ( i, j ) ∈ { (2 , , ( θ − , θ ) } and x iij = 0 , then [ x ( i − i +1) , x i ] c = 0 . (cid:3) INITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 33
Question 5.7.
Let b B denote the quotient of T ( V ) by the relations x ij = 0 , | i − j | >
1; (ad c x θ − ) x θ = 0; x iij = 0 , | i − j | = 1 , ( i, j ) = ( θ − , θ ); [ x (13) , x ] c = 0 . Similarly, define b B by the relations x ij = 0 , | i − j | >
1; (ad c x ) x = 0; x iij = 0 , | i − j | = 1 , ( i, j ) = (2 , x ( θ − θ ) , x θ − ] c = 0 . (Clearly b B ≃ b B as algebras). Is GK-dim b B < ∞ ?5.4. Type D θ with N = 2 . Here (cf. [AA2, p. 404]) the distinguished pre-Nichols algebra e B q is presented by the quantum Serre relations and a bunchof q -brackets coming from the several subdiagrams of type A , namely: [ x ( i i +2) , x i +1 ] c , i ≤ θ −
3; [ x θ − θ − θ , x θ − ] c ; [ x θ θ − θ − , x θ − ] c . (5.2) Lemma 5.8.
Assume a is of Cartan type D with N = 2 . The followingrelations hold in any B ∈ Pre fGK : (a) if i = j and e q ij = − , then x iij = 0 ; (b) if i = j and e q ij = 1 , then x kij = 0 for all k ∈ I ; (c) if r is one of the elements in (5.2) , then (ad c x k ) r = 0 for all k ∈ I . (cid:3) Question 5.9.
Let b B denote the quotient of T ( V ) by the relations (a), (b)and (c) . Is GK-dim b B < ∞ ? Lemma 5.10.
Assume a is of Cartan type D θ with θ > and N = 2 . Thefollowing relations hold in any B ∈ Pre fGK ( V ) : (a) all the defining relations of e B q except x θ θ − and [ x θ θ − θ − , x θ − ] c ; (b) the relations x k θ θ − and (ad c x k )[ x θ θ − θ − , x θ − ] c for all k ∈ I θ . (cid:3) Question 5.11.
Let b B denote the quotient of T ( V ) by the relations x ij = 0 , e q ij = 1 , ( i, j ) = ( θ, θ − x θ − θ − θ , x θ − ] c = 0; x iij = 0 , e q ij = −
1; [ x ( i i +2) , x i +1 ] c = 0 , i ≤ θ − x k θ θ − = 0 , k ∈ I θ ; [ x k , [ x θ θ − θ − , x θ − ]] = 0 , k ∈ I θ . Is GK-dim b B < ∞ ? We conjecture that GK-dim b B = GK-dim e B q + 2 . Thiswill be treated in a subsequent paper. Acknowledgements.
We thank Iván Angiono and James Zhang for usefulcomments.
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Facultad de Matemática, Astronomía y Física, Universidad Nacional deCórdoba. CIEM – CONICET. Medina Allende s/n (5000) Ciudad Universi-taria, Córdoba, Argentina
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