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arXiv:2110.11217v1 [math.QA] 21 Oct 2021

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS OF

(SUPER)MODULAR AND UNIDENTIFIED TYPE

IV ´

AN ANGIONO, EMILIANO CAMPAGNOLO, AND GUILLERMO SANMARCO

Abstract. We show that every ﬁnite GK-dimensional pre-Nichols algebra for braidings

of diagonal type with connected diagram of modular, supermodular or unidentiﬁed type

is a quotient of the distinguished pre-Nichols algebra introduced by the ﬁrst-named au-

thor, up to two exceptions. For both of these exceptional cases, we provide a pre-Nichols

algebra that substitutes the distinguished one in the sense that it projects onto all ﬁnite

GK-dimensional pre-Nichols algebras. We build these two substitutes as non-trivial cen-

tral extensions with ﬁnite GK-dimension of the corresponding distinguished pre-Nichols

algebra. We describe these algebras by generators and relations, and provide a basis.

This work essentially completes the study of eminent pre-Nichols algebras of diagonal

type with connected diagram and ﬁnite-dimensional Nichols algebra.

1. Introduction

This paper is the third part of a series started in [AS] and continued in [ACS], where

we contribute to the classiﬁcation of Hopf algebras with ﬁnite Gelfand-Kirillov dimension

(abbreviated GKdim) over an algebraically closed ﬁeld kof characteristic zero.

Due to the vastness of that problem, further restrictions are usually put in place. For

instance, aﬃne Noetherian Hopf algebras with ﬁnite GKdim have been studied for more

than two decades. Also, substantial progress has been made towards the classiﬁcation of

Hopf algebras with small GKdim. See [BG, BZ, L, GZ, WZZ, G] and references therein.

We focus on pointed Hopf algebras with ﬁnite GKdim. Our point of view is inspired by

Andruskiewitsch-Schneider program [ASc2], which was originally set up for studying ﬁnite

dimensional pointed Hopf algebras but has proven itself fruitful also on the ﬁnite GKdim

setting, see [ASc1, AAH1, AAH2, AAM, A2] for example.

Recall that a Hopf algebra His pointed if the coradical (the sum of all simple subcoal-

gebras) is just the group algebra of the group-like elements. In this case, the coradical

ﬁltration is a Hopf algebra ﬁltration so the associated graded object gr His a graded Hopf

algebra. If Γ denotes the group of group-like elements of H, the Radford-Majid biproduct

(or bosonization) [Ra] yields a decomposition gr H≃R#kΓ, where R=⊕n>0Rnis a corad-

ically graded Hopf algebra in the braided tensor category kΓ

kΓYD of Yetter-Drinfeld modules

over kΓ. At this point we have encountered the three invariants that guide the ongoing

problem of classiﬁcation of pointed Hopf algebras: the group Γ of group-like-elements;

the Yetter-Drinfeld module R1, called the inﬁnitesimal braiding; and the diagram R, a

Hopf algebra in the category kΓ

kΓYD. This last invariant serves as the entryway for Nichols

algebras into the general problem of classiﬁcation of pointed Hopf algebras.

Key words and phrases. Hopf algebras, Nichols algebras, Gelfand-Kirillov dimension.

MSC2020: 16T05, 16T20, 17B37, 17B62.

The work of the three authors was partially supported by CONICET and Secyt (UNC).

1

2 ANGIONO, CAMPAGNOLO, AND SANMARCO

Given a Yetter-Drinfeld module Vover a group Γ, we can construct the Nichols algebra

B(V) of V, a coradically graded Hopf algebra in kΓ

kΓYD; via Radford-Majid’s biproduct

we get a pointed Hopf algebra B(V)#kΓ. This is a distinguished element of the class

of all pointed Hopf algebras with coradical kΓ and inﬁnitesimal braiding Vfor various

reasons. Most importantly for us, if His in this class, then the subalgebra of the diagram

Rgenerated by V≃R1is isomorphic to B(V). In other words, diagrams of a pointed

Hopf algebra with inﬁnitesimal braiding Vare post-Nichols algebras of V, see [AAR].

Since we are interested in pointed Hopf algebras with ﬁnite GKdim, a celebrated re-

sult of Gromov states that the potential groups of group-like elements, assuming ﬁnite

generation, necessarily contain a nilpotent subgroup of ﬁnite index. A careful analysis of

Nichols algebras over ﬁnitely generated nilpotent groups was carried out recently in [A2];

in particular, it was explained that much can be aﬀorded with abelian groups.

In this paper, however, the group Γ will only play an implicit role, since most of the

theory of Nichols algebras can be expressed in the language of braided Hopf algebras.

That being the case, it is customary to study families of braided vector spaces rather than

Yetter-Drinfeld modules over families of groups. We will focus on braided vector spaces

of diagonal type, which correspond to semisimple Yetter-Drinfeld modules over ﬁnitely

generated abelian groups. Such a braided vector space Vdepends on a square matrix qof

non-zero elements in the ground ﬁeld, so we denote Bqrather than B(V).

During the last two decades, remarkable progress has been achieved in the study of

Nichols algebras of diagonal type. These Nichols algebras are known to have a restricted

PBW basis, which allows to introduce the notion of root systems and certain reﬂections

that give rise to Weyl groupoids. Using these tools, Heckenberger classiﬁed all braided

vector spaces of diagonal type with ﬁnite root system. The list contains ﬁve families:

Cartan type, super type, standard type, (super)modular type and unidentiﬁed type [AA].

Later on, Andruskiewitsch, Angiono and Heckenberger stated the following:

Conjecture 1.1. [AAH1, 1.5] The root system of a Nichols algebra of diagonal type with

ﬁnite Gelfand-Kirillov dimension is ﬁnite.

While the converse implication holds due to a routinary argument involving PBW ba-

sis, the conjecture remains open. However there is strong evidence on its behalf, see

[AAH2, AGI]. Assuming the validity of this statement, as we will do throughout this

paper, we deduce that diagrams of pointed Hopf algebras with ﬁnite GKdim and inﬁnites-

imal braiding of diagonal type correspond precisely to ﬁnite GK-dimensional post-Nichols

algebras of braided vector spaces on Heckenberger’s classiﬁcation [H3].

As discussed in [AS, §2.6], the problem of classifying these post-Nichols algebras can be

rephrased in the terms of pre-Nichols algebras. The later language better ﬁts our purposes

for two reasons. The ﬁrst one is rather technical: pre-Nichols algebras of Vare certain

Hopf quotients of the tensor algebra T(V); this allow us to deﬁne pre-Nichols algebras

by generators and relations. Thus the family of pre-Nichols algebras becomes partially

ordered by projection, determining a poset with minimal element T(V) and maximal one

the Nichols algebra. As explained above, we are interested in describing the subposet of

pre-Nichols algebras with ﬁnite GKdim; as a initial step we look for minimum elements,

called eminent pre-Nichols algebras.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 3

There is a more conceptual reason to study pre-Nichols algebras. For each ﬁnite dimen-

sional Nichols algebra of diagonal type, the ﬁrst-named author introduced in [An2] the

distinguished pre-Nichols algebra, which has ﬁnite GKdim among other crucial features.

A natural question arises: is the distinguished pre-Nichols algebra eminent? For braided

vector spaces of Cartan, super or standard type, that is indeed the case up to very few

exceptions, as shown [AS, ACS]. Here we complete the work, studying (super)modular

and unidentiﬁed types. Summarizing, our main result, after the three papers, is:

Theorem 1.2. Let qbe a braiding matrix such that dim Bq<∞and the Dynkin diagram

of qis connected. Assume that Conjecture 1.1 holds.

(i) If qis not of type

•Cartan Aθor Dθwith q=−1,

•Cartan A2with q∈G′

3,

•A3(q|{2})or A3(q|{1,2,3}), with q∈G∞,

•g(2,3) with any of the following Dynkin diagram

d1:−1

◦ξ2ξ

◦ξ−1

◦, d2:−1

◦ξ−1

◦ξ−1

◦,

then the distinguished pre-Nichols algebra e

Bqis eminent.

(ii) If qis of type A3(q|{2})or A3(q|{1,2,3}), there is an eminent pre-Nichols algebra

b

Bqwhich is a braided central extension of e

Bqby a polynomial algebra in one variable.

(iii) If qis of type Cartan A2with q∈G′

3or g(2,3) with diagram d1or d2, then b

Bqis a

braided central extension of e

Bqby a q-polynomial algebra in two variables.

For braidings of Cartan type Aθor Dθwith q=−1, there are candidates for eminent

pre-Nichols algebras. The GKdim of these candidates is not known yet, see [AS, §5].

Proof. For (i), the Cartan case follows by [AS, Theorem 1.3 (a)]; super and standard types,

by [ACS, Theorem 1.2 (1)]; the remaining cases are covered by Theorem 3.1.

Now (ii) follows from [ACS, Theorem 1.2 (2), (3)]. Finally Propositions 4.2 and 4.3

together with [AS, Theorem 1.3 (b)] prove (iii).

The structure of the paper is the following. In Section 2 we ﬁx the notation and recall

general aspects of the theory of Nichols algebras for later use. Section 3 is devoted to

complete the proof of Theorem 1.2 (i), with a case-by-case analysis of the behaviour of

the generators for the deﬁning ideals of Nichols algebras of diagonal type, when considered

as elements of pre-Nichols algebras with ﬁnite GKdim. Finally, in Section 4 we introduce

and study the eminent pre-Nichols algebras for the two exceptional cases of type g(2,3)

described in Theorem 1.2 (iii).

2. Preliminaries

In this section we recall deﬁnitions and general aspects of the theory Nichols algebras,

with particular interest on the diagonal case. We refer to [T] for the deﬁnition of braided

vector spaces and braided Hopf algebras, and to [Ra, A1, AA, HS] for unexplained termi-

nology regarding Hopf and Nichols algebras. For preliminaries on GKdim see [KL].

4 ANGIONO, CAMPAGNOLO, AND SANMARCO

Notations. For each θ∈Nwe set Iθ={1,...,θ}; we shall write Iwhen no confusion is

possible. The canonical basis of the free abelian group ZIis denoted {αi:i∈I}; an element

Pi∈Ibiαiwill be denoted as 1b12b2···θbθ. Given β= 1b12b2···θbθand γ= 1c12c2···θcθ,

we say that β≤γif and only if bi≤cifor all i∈I; this deﬁnes a partial order on ZIthat

will be used without further mention.

We work over an algebraically closed ﬁeld kof characteristic 0. The subgroup of k×

consisting of N-th roots of unity is denoted by GN, and G′

Ndenotes the subset of those

of order N. The set of all roots of unity is G∞.

If Ais a Zgraded algebra, we denote by Anthe degree n-component. The subspace of

primitive elements of a (braided) Hopf algebra His P(H).

2.1. Braided action and braided commutator. Any braided Hopf algebra Badmits

aleft adjoint representation adc:B→End B,

(adcx)y=m(m⊗S)(id ⊗c)(∆ ⊗id)(x⊗y), x, y ∈B.

Also, the braided bracket [·,·]c:B⊗B→Bis the map given by

[x, y]c=m(id −c)(x⊗y), x, y ∈B.

2.2. Eminent pre-Nichols algebras. In a nutshell, a pre-Nichols algebra of a braided

vector space (V, c) is a N0-graded braided connected Hopf algebra that contains (V, c) as

the degree-1 component and is generated (as an algebra) by V. One can show that there

exists a unique pre-Nichols algebra that is moreover strictly graded, which means that the

subspace of primitive elements coincides with V; this is the Nichols algebra of (V, c).

There is a more concrete interpretation of these deﬁnitions in which one uses the braiding

of Vto induce in the tensor algebra T(V) = ⊕n>0V⊗na structure of braided Hopf algebra

such that the elements of Vare primitive. Then one can show that there exists a (unique)

maximal homogeneous Hopf ideal J(V) of T(V) generated by elements of degree ≥2, thus

the Hopf quotient B(V) := T(V)/J(V) is the Nichols algebra of V.

In this concrete interpretation, a pre-Nichols algebra of Vis a Hopf quotient Bof

T(V) by a braided homogeneous Hopf ideal contained in J(V). Hence, the identity of

Vextends to canonical epimorphisms T(V)։B։B(V). The family Pre(V) of pre-

Nichols algebras of Vis naturally equipped with a partial order. Namely, B1≤B2if

ker(T(V)։B1)⊆ker(T(V)։B2). Thus T(V) is minimal and B(V) is maximal.

Assume now that Vis such that GKdim B(V)<∞. Inside Pre(V) we get the (non-

empty) subposet PrefGK(V) containing all ﬁnite GK-dimensional pre-Nichols algebras. Un-

derstanding this subposet is a crucial problem in the classiﬁcation of pointed Hopf-algebras

with ﬁnite GK-dimension, see [AS] for details. As a ﬁrst step towards this direction, in

loc. cit. the authors introduced the notion of eminent pre-Nichols algebra of V, which is

a minimum of PrefGK(V). The existence of such minimal objects is not warrantied, see

[AS] for concrete examples related to Lie algebras. However, for the family of braidings

of diagonal type with connected diagram, this problem have been addressed in [AS, ACS].

We will get back to this in §3.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 5

2.3. Nichols algebras of diagonal type. A matrix q= (qij )i,j∈Iwith entries in k×is

called a braiding matrix, since it gives raise to a braided vector space (V, cq): ﬁxed a basis

(xi)i∈Iof V,cq∈GL(V⊗V) is determined by:

cq(xi⊗xj) = qijxj⊗xi, i, j ∈I.(2.1)

Such a braiding is called of diagonal type. The (Dynkin) diagram of qis a decorated graph.

The set of vertices is I, each vertex labelled with qii. There is an edge between vertices

i6=jif and only if eqij := qijqji 6= 1, in such case the edge is labelled with this scalar.

The braiding matrix qinduces a Z-bilinear form q:ZI×ZI→k×deﬁned on the canonical

basis by q(αj, αk) := qjk,j, k ∈I. For α, β ∈ZIand i∈I, we set

qαβ =q(α, β), qα=q(α, α), Nα= ord qα, Ni= ord qαi.(2.2)

In the diagonal setting, the Nichols algebra has a compatible N0-grading. Indeed, the

tensor algebra T(V) itself becomes a NI

0-graded algebra by the rule deg xi=αi,i∈I;

moreover, the braided structure induced from that of Vis homogeneous:

c(u⊗v) = qαβ v⊗u, u ∈T(V)α, v ∈T(V)β, α, β ∈NI

0.

Since the braided coalgebra structure only depends on the braiding, T(V) becomes a NI

0-

graded braided Hopf algebra. Furthermore, the deﬁning ideal Jq=J(V) turns out to be

homogeneuos, thus the Nichols algebra Bq=B(V) is a NI

0-graded braided Hopf algebra.

If Bis a NI

0-graded pre-Nichols algebra of q, the following equalities hold:

[u, vw]c= [u, v]cw+qαβ v[u, w]c,(2.3)

[uv, w]c=qβ γ [u, w]cv+u[v, w]c,(2.4) [u, v]c, wc=u, [v, w]cc−qαβ v[u, w]c+qβγ [u, w]cv,(2.5)

for all homogeneous elements u∈Bα,v∈Bβ,w∈Bγ.

Given at least two indexes i1,...,ik∈I, we denote

xi1···ik:= (adcxi1)xi2···ik=xi1xi2···ik−qi1i2···qi1ikxi2···ikxi1

(2.6)

as an element in the tensor algebra or any pre-Nichols algebra.

2.3.1. Classiﬁcation of arithmetic braidings. In this work we only consider arithmetic

braiding matrices q, which are those with connected diagram and ﬁnite generalized root

system ∆q; that is, those Nichols algebras admitting a (restricted) PBW basis with ﬁnite

set of generators, so ∆q

+is the set of degrees of a set of generators. This is precisely the

class that was classiﬁed in [H3], and includes all connected braiding matrices with ﬁnite di-

mensional Nichols algebra. Next we recall two results regarding the shape of the diagrams

for arithmetic braiding matrices q= (qij )i,j ∈Iθ.

Lemma 2.1. [H3, Lemma 9 (ii)] If θ= 3 and the root system is ﬁnite, then eq12eq13eq23 = 1

and (q11 +1)(q22 + 1)(q33 + 1) = 0. Moreover, if q22, q33 6=−1then q22 eq12 =q33eq13 = 1.

Lemma 2.2. [H3, Lemma 23] Assume that qhas ﬁnite root system. Then the diagram of

qdoes not contain cycles of length larger than 3.

6 ANGIONO, CAMPAGNOLO, AND SANMARCO

The classiﬁcation provided by [H3] consists on several tables containing the Dynkin dia-

gram of all arithmetic braidings. Later on, an organization from a Lie-theoretic perspective

was achieved in [AA], giving rise to ﬁve families: Cartan, super, standard, (super)modular,

and unidentiﬁed. The task of ﬁnding eminent pre-Nichols for the ﬁrst three families was

achieved in [AS, ACS], up to two exceptions. Here we focus on the remaining families.

2.3.2. Deﬁning relations. An explicit presentation of the Nichols algebras for arithmetic

braidings was achieved in [An1, Theorem 3.1], which serves as an implicit guidance through-

out this work and thus deserves a brief review. That result consists in a list of 29 homo-

geneous relations, each of them accompanied by a very speciﬁc set of conditions on the

entries of the braiding matrix qthat determine whether or not the relation needs to be

included in the presentation of Bq.

A conceptual analysis of these relations (and their genesis) yields a separation in three

categories: quantum Serre relations, generalizations of these in up to four generators xi,

and the so-called powers of root vectors. Throughout this work, we will refer to this

particular set of generators for the ideal Jqsimply as the presentation of Bq.

2.3.3. Finite GK-dimensional Nichols algebras. Notice that some arithmetic braidings de-

pend on parameters that can take any but a small number of non-zero values in the ground

ﬁeld. In all such cases, the corresponding Nichols algebra is ﬁnite dimensional precisely

when these parameters are roots of unity; otherwise, it is just ﬁnite GK-dimensional. This

work and the prequels [AS, ACS] focus on braiding matrices with ﬁnite dimensional Nichols

algebras. The task of ﬁnding eminent pre-Nichols algebras when these parameters are not

roots of unity will be treated in a sequel.

From this perspective, Conjecture 1.1 states that the classiﬁcation of Nichols algebras

of diagonal type with connected diagram and ﬁnite GKdim is precisely the one in [H3].

Several steps towards proving the conjecture have been achieved. Namely, it is known to

be true for braidings of rank θ= 2,3 and of Cartan type, [AAH1, AGI] Some proofs in

this work and the prequels [AS, ACS] assume that the Conjecture holds. However, the

majority of those proofs belong to the realm where the Conjecture is known to be true.

As we assume GKdim Bq<∞, by [Ro, Lemma 20] for each i6=j∈Ithere exists

n∈N0such that (adcxi)n+1xj= 0. Then we set

cq

ij := −min{n∈N0: (adcxi)n+1xj= 0}=−min{n∈N0: (n+ 1)qii (1 −qn

ii eqij) = 0}.

Set also cq

ii = 2. Then Cq:= (cq

ij ) is a generalized Cartan matrix and one of the key

ingredients in the deﬁnition of the Weyl groupoid of q, cf. [H1, HY].

We end this subsection with two results that will be use several times in §3.

Lemma 2.3. [AAH1, Proposition 4.16] If Wis a braided vector space of diagonal type

with diagram 1

◦pq

◦,p6= 1, then GKdim B(W) = ∞.

Lemma 2.4. [AS, Lemma 2.8] Let Bbe a graded braided Hopf algebra. If Wis a braided

vector subspace of P(R), then GKdim B(W)≤GKdim R.

2.4. Pre-Nichols algebras of diagonal type. We collect preliminaries and notation

regarding pre-Nichols algebras for later use.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 7

2.4.1. Distinguished pre-Nichols algebras. Given a connected qwith ﬁnite dimensional Bq,

the distinguished pre-Nichols algebra e

Bqof qis the quotient of the tensor algebra by

the ideal that results from Jqby removing certain powers of root vectors and including

quantum Serre relations that could formerly be deduced from the power of root vectors.

This is a key tool in our search for eminent pre-Nichols algebras, since it is designed to admit

a PBW basis with the same set of generators as that of Bq, and satisﬁes GKdim e

Bq<∞,

see [An2]. The powers of root vectors removed from the presentation correspond to Cartan

roots, the subset of ∆q

+of those PBW generators with inﬁnite height in e

Bq.

2.5. Extensions and Hilbert series of graded braided Hopf algebras. We brieﬂy

introduce a tool that will be crucial in §4. For more details, see [ACS, §2.6]. Following

[AN, §2.5], a sequence of morphisms of braided Hopf algebras k→Aι

→Cπ

→B→kis

an extension of braided Hopf algebras if ιis injective, πis surjective, ker π=Cι(A+) and

A=Cco π. In this case, we just write Aι

֒→Cπ

։B.

In our examples, Cwill be connected (i.e. the coradical of Cis k). By [A+, 3.6], to

get extensions of Cit is enough to consider a surjective braided Hopf algebra morphism

Cπ

։Band set A=Cco π. This construction will be enough for our purposes.

2.5.1. Hilbert series. The Hilbert series of a Nθ

0-graded object Uwith ﬁnite-dimensional

homogeneous components is

HU=X

α∈Nθ

0

dim Uαtα∈N0[[t1,...,tθ]],

where tα=ta1

1···taθ

θfor α= (a1,··· , aθ). If U′is Nθ

0-graded object, we say that HU≤ HU′

if dim Uα≤dim U′

αfor all α∈Nθ

0.

The main reason for introducing these concepts is the following result.

Lemma 2.5. [ACS, Lemma 2.4] Fix a Hopf algebra Hwith bijective antipode. If Aι

֒→

Cπ

։Bis a degree-preserving extension of Nθ

0-graded connected Hopf algebras in H

HYD

with ﬁnite-dimensional homogeneous components, then HC=HAHB.

3. Defining relations and finite GK-dimensional pre-Nichols algebras

Throughout this section we assume that q= (qij )i,j∈Iθis a braiding matrix with con-

nected Dynkin diagram such that dim Bq<∞. In particular the root system is ﬁnite (see

[H1, §3]) and each qii is a root of unity, say of order Ni(necessarily Ni≥2 by Lemma 2.3).

Let Vqbe the braided vector space with basis (xi)i∈Iθand braiding cq(xi⊗xj) = qij xj⊗xi.

Theorem 3.1. If qis not of type

•Cartan Aθor Dθwith q=−1,

•A2with q∈G′

3,

•A3(q|{2})or A3(q|{1,2,3}), with q∈G∞,

•g(2,3) with any of the following Dynkin diagram

d1:−1

◦ξ−1

◦ξ−1

◦, d2:−1

◦ξ2ξ

◦ξ−1

◦,

then the distinguished pre-Nichols algebra e

Bqis eminent.

8 ANGIONO, CAMPAGNOLO, AND SANMARCO

Proof. If qis either of Cartan, super or standard type, then the proof follows by [AS,

Theorem 1.3] together with [ACS, Theorem 1.2]. Hence we reduce to the cases in which q

is of types either modular, supermodular or unidentiﬁed.

The presentation of the Nichols algebra of qgiven in [An1, Theorem 3.1] consists on a

list of 29 relations, each of them accompanied by speciﬁc conditions on the entries of q

that determine whether or not the relations needs to be included. Following the procedure

in §2.4.1, we get a set of relations that give a presentation of the distinguished pre-Nichols

algebra e

Bq. In the prequels [AS, ACS] we determined suﬃcient conditions on qto assure

that some of these relations hold in any ﬁnite GKdim pre-Nichols algebra of q, under some

mild assumptions. For these relations the only remaining task is to ensure their validity

without any assumption, which is achieved in Lemma 3.1. Finally, in §3.2 we deal with

the relations that where not considered in the prequels.

3.1. Relations already considered. In this subsection we extend some results estab-

lished in [AS, ACS] dropping superﬂuous assumptions. The organization goes as follows.

Each relation is studied in a diﬀerent item of Lemma 3.3, where we ﬁrst ﬁx the ele-

ments of Iθthat support the relation and then write down the conditions on qthat [An1,

Theorem 3.1] requires for including this relation in the presentation of Bq. If further hy-

pothesis on qare needed, they are included in a diﬀerent sentence. All such relations are

NI

0−homogeneous, so we will denote them by xβ, where β∈NI

0is the degree.

One of the tasks is to check that a relation is primitive in all pre-Nichols algebras with

ﬁnite GKdim. The following result will be useful for such proposal.

Remark 3.2. Let I⊂Jbe NI

0-graded Hopf ideals of T(V), let Sbe a system of NI

0-

homogeneous generators of J, and put B:= T(V)/I. Consider an homogeneous element

x∈J, and set Yx={y∈S: deg(y)<x}. If Yx⊂I, then x∈ P(B).

Proof. Since Jis a coideal and the coproduct is NI

0-homogeneous, there exist ay,by,cy,

dy,ey,fy∈T(V) such that

∆(x) = 1 ⊗x+x⊗1 + X

y∈Yx

ayyby⊗cy+dy⊗eyyfy∈T(V)⊗T(V).(3.1)

Then we use that y= 0 in Bfor all y∈Yx.

In particular we can apply Remark 3.2 to J=e

Jq, the deﬁning ideal of the distinguished

pre-Nichols algebra. This will help us to prove that an element is primitive in a pre-Nichols

algebra by studying its image onto e

Bq.

Now we provide a reﬁned treatment of some relations already considered in [AS, ACS].

Lemma 3.3. Let Bbe a ﬁnite GKdim pre-Nichols algebra of Bq.

(a) Let i, j ∈Iθbe such that cij = 0.Assume that one of the following hold:

•ord qii + ord qj j >4,

•qiiqj j = 1 and there exists k∈Iθ− {i, j}such that eqikeqjk 6= 1.

Then xij = 0 in B.

(b) Let i, j ∈Iθbe such that cij <0,eq1−cij

ij 6= 1.Assume that one of the following hold:

•the Dynkin diagram of kxi⊕kxjis diﬀerent from q

◦q−1q

◦, q ∈G′

3,

•cij =−1,qii =qjj =eq−1

ij ∈G′

3, and there exists k∈Iθ− {i, j}such that eq2

ik eqjk 6= 1.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 9

Then (adcxi)1−cij xj= 0 in B.

(c) Let i, j ∈Iθsuch that q1−cij

ii = 1.Assume that eqij =qii and one of the following hold

•qjj 6=−1,

•cij ≤ −2,

•cij =−1,qii =−1and there exists k∈Iθ− {i, j}such that eqjk ,eq2

ik eqjk 6= 1.

Then (adcxi)1−cij xj= 0 in B.

(d) Let i∈Iθbe a non-Cartan vertex. Then xNi

i= 0 in B.

(e) Let i, j ∈Iθbe such that qii =eqij =qjj =−1and there exists k∈Iθ− {i, j}such that

either eq2

ik 6= 1 or eq2

jk 6= 1.Assume that eq2

ik eq2

jk 6= 1. Then x2

ij = 0 in B.

(f) Let i, j, k ∈Iθbe such that qjj =−1,eqik =eqij eqjk = 1 and eqij 6=±1.Assume that one

of the following conditions holds:

•either qii =−1or qkk =−1,

•qiiqkk = 1 and there exists ℓ∈Iθ− {i, j, k}such that eqiℓ 6= 1 = eqjℓ =eqk ℓ,

•qiiqkk = 1 and there exists ℓ∈Iθ− {i, j, k}such that eq2

jℓ 6= 1 = eqiℓ =eqkℓ,

•qiiqkk = 1 and there exists ℓ∈Iθ− {i, j, k}such that eqkℓ 6= 1 = eqj ℓ =eqiℓ .

Then [xijk , xj]c= 0 in B.

(g) Let i, j ∈Iθbe such that qj j =−1,qii eqij ∈G′

3∪G′

6and either qii ∈G′

3or cij ≤ −3.

Then [xiij , xij ]c= 0 in B.

(h) Let i, j, k ∈Iθbe such that qii =±eqij ∈G′

3,eqik = 1 and either −qjj =eqij eqjk = 1 or

q−1

jj =eqij =eqjk 6=−1. Then [xiijk , xij ]c= 0 in B.

(i) Let i, j, k ∈Iθbe such that eqij ,eqik ,eqj k 6= 1. Then

xijk −qij (1 −eqjk)xjxik +1−eqj k

qkj (1 −eqik )[xik , xj]c= 0 in B.

(j) Let i, j, k ∈Iθbe such that qii =qj j =−1,eq2

ij =eq−1

jk 6= 1 and eqik = 1.Assume that

either q2

kk 6= 1 or eq3

ij 6= 1. Then [[xij , xijk ]c, xj]c= 0 in B.

(k) Let i, j, k ∈Iθbe such that qii =qjj =−1,eq3

ij =eq−1

jk and eqik = 1. Then

[[xij ,[xij , xijk ]c]c, xj]c= 0 in B.

(l) Let i, j, k ∈Iθbe such that qjj =eq2

ij =eqjk ∈G′

3and eqik = 1.Assume that either

qii 6=−1or qkk 6=−1. Then [[xijk, xj]c, xj]c= 0 in B.

(m) Let i, j, k ∈Iθbe such that qjj =eq3

ij =eqjk ∈G′

4,eqik = 1. Then [[[xijk, xj]c, xj]c, xj]c=

0in B.

(n) Let i, j, k ∈Iθbe such that qii =eqij =−1,qjj =eq−1

jk 6=−1and eqik = 1. Then

[xij , xijk ]c= 0 in B.

(o) Let i, j, k, ℓ ∈Iθbe such that qkk =−1,qjj eqij =qj j eqjk = 1,eqik =eqil =eqj ℓ = 1 and

eq2

jk =eq−1

kℓ =qℓℓ. Then [[[xijkℓ, xk]c, xj]c, xk]c= 0 in B.

(p) Let i, j, k, ℓ ∈Iθand q∈kbe such that qℓℓ =eq−1

ℓk =qkk =eq−1

jk =q2,eqij =q−1

ii =q3,

qjj =−1and eqik =eqiℓ =eqj ℓ = 1. Then [[[xij k, xj]c,[xijkℓ, xj]c]c, xj k ]c= 0 in B.

(q) Let i, j, k, ℓ ∈Iθbe such that qkk =−1,qii =eq−1

ij =q2

jj ,eqkℓ =q−1

ℓℓ =q3

jj ,eqjk =q−1

jj ,

eqik =eqiℓ =eqjℓ = 1. Then [[xij kℓ , xj]c, xk]c−qjk(eq−1

ij −qjj )[[xij kℓ , xk]c, xj]c= 0 in B.

(r) Let i, j, k ∈Iθbe such that ord qii >3,eqik = 1,qjj =−1,eqij =q−2

ii ,eqjk =q−1

kk =−q3

ii.

Then [xi,[xijk , xj]c]c−qij qkj

1−q−1

ii

[xij, xijk ]c−(qii +q2

ii)qij qik xij k xij = 0 in B.

10 ANGIONO, CAMPAGNOLO, AND SANMARCO

Remark 3.4. The extra assumptions imposed on the braiding matrix qare in fact nec-

essary. If one of these conditions do not hold, then the corresponding relation is not

necessarily zero in all ﬁnite GKdim pre-Nichols algebra, as we see in the following cases:

•(a), for A3(q|{2}) with q∈G∞;

•(b), for A2with q∈G′

3;

•(c), (e) and (f), for Cartan Aθor Dθwith q=−1;

•(f), for A3(q|{1,2,3}), with q∈G∞;

•(j), for g(2,3) with diagram d1;

•(l), for g(2,3) with diagram d2.

Proof. By [ACS, Theorem 1.2] if qis of Cartan, super or standard type all items are

satisﬁed. The statements (a), (b), (c), (d), (e), (f), (i) and (l) were proved in [ACS, §3].

Items (k), (m) and (n) have corresponding lemmas in [ACS, §3], but with extra hypoth-

esis on q. However, we verify by exhaustion that these conditions are satisﬁed for each

diagram in [H3]. For (g), (h), (j), (k), (o), (p), (q) and (r), it remains to prove that xβis

primitive. The proof is recursive on the degree β∈NI

0. Indeed we apply Remark 3.2 with

J=e

Jqand Ithe ideal generated by those generators of e

Jqof degree < β.

3.2. Verifying more relations. We study the remaining deﬁning relations of e

Bqgiven in

[An1, Theorem 3.1]. Here Bstands for a pre-Nichols algebra of qsuch that GKdim B<∞.

Let us outline the general strategy. For each one of the Lemmas below we assume that

qsatisﬁes the conditions required on [An1, Theorem 3.1] for including certain relation xβ

of degree β∈NI

0in the presentation of e

Bq. Next we suppose that (the image of) xβdoes

not vanish in Band prove that (the image of) xβis primitive in B. Thus we get a braided

vector space of diagonal type Vq⊕kxβ⊂ P(B) which satisfy GKdim B(Vq⊕kxβ)<∞

by Lemma 2.4. Now we compute the Dynkin diagram of a suitable chosen subspace of

Vq⊕kxβ; this is a straightforward task involving (2.2) and depending only on qand β.

Since we are assuming the validity of Conjecture 1.1, this diagram should belong to the

classiﬁcation given in [H3] and this allow us to arrive at a contradiction. Sometimes we

get a Dynkin diagram that belongs to a class in which the conjecture is known to hold

true, so we do not need any further assumption.

As in Lemma 3.3, the proofs of the lemmas in this subsection use that the relation xβ

under consideration is primitive in any pre-Nichols algebra Bwith ﬁnite GKdim. This

is proved recursively on βapplying Remark 3.2 for J=e

Jqand Ithe ideal generated by

those generators of e

Jqof degree < β.

Lemma 3.5. Let i, j, k ∈Iθbe such that one of the following conditions hold:

(a) eqij =qjj =−1and qii =−eq2

jk ∈G′

3,eqik = 1;

(b) qkk =eqj k =qj j =−1and qii =−eqij ∈G′

3,eqik = 1;

(c) qjj =−1,eqij =q−2

ii 6= 1,qkk =eq−1

jk =−q3

ii,eqik = 1;

(d) qii =qjj =qkk =−1,−eqij =eqj k ∈G′

3,eqik = 1.

Then [[xij , xijk ]c, xj]c= 0 in B.

Proof. Suppose that xβ:= [[xij , xij k ]c, xj]c6= 0 in B.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 11

(a) We have eqiβ =−qii 6= 1, eqj β =eqj k 6= 1, so the Dynkin diagram of kxi⊕kxj⊕kxβis

a (connected) triangle. Since GKdim B<∞, Lemma 2.1 implies that eqij eqjβ eqiβ = 1, but

this means qii eqj k = 1 which contradicts qii =−eq2

jk ∈G′

3.

(b) In this case eqiβ =−qii and eqkβ =−1, so the Dynkin diagram of kxi⊕kxj⊕kxk⊕kxβ

contains a 4-cycle, which contradicts Lemma 2.2.

(c) Since eqiβ =q−2

ii 6= 1 and eqkβ =q−1

kk 6= 1, the previous argument applies again.

(d) In this case qββ = 1 and eqiβ =−1, so the Nichols algebra of kxi⊕kxβis inﬁnite

GK-dimensional by Lemma 2.3.

Lemma 3.6. Let i, j, k ∈Iθbe such that qkk =qjj =eq−1

ij =eq−1

jk ∈G′

9,eqik = 1 and

qii =q6

kk. Then [[xiij , xiijk]c, xij ]c= 0 in B.

Proof. Assume that xβ:= [[xiij , xiijk ]c, xij ]c6= 0. Since qβ β =q7

kk and eqk β =q−1

kk = (q7

kk)−4

the braided subspace W:= kxk⊕kxβ⊂ P(B) is of aﬃne Cartan type A(2)

2. Thus

GKdim B(W) = ∞by [AAH2, Theorem 1.2 (a)], a contradiction.

Lemma 3.7. Let i, j, k ∈Iθbe such that qii =eq−1

ij ∈G′

9,qjj =eq−1

jk =q5

ii,eqik = 1 and

qkk =q6

ii. Then [[xij k , xj]c, xk]c= (1 + eqj k)−1qjk [[xij k, xk]c, xj]cin B.

Proof. If xβ= [[xij k , xj]c, xk]c−(1 + eqj k )−1qjk[[xij k , xk]c, xj]c6= 0, then W:= kxk⊕kxβ⊂

P(B) has Dynkin diagram q

◦

k

qq3

◦

βwhere q:= q5

ii ∈G′

9. As this diagram does not

appear in [H3, Table 1], [AAH2, Theorem 1.2 (b)] assures that GKdim B(W) = ∞, a

contradiction with GKdim B<∞.

Lemma 3.8. Let i, j, k ∈Iθbe such that qii =qkk =−1,eqik = 1,eqij ∈G′

3and qjj =

−eqjk =±eqij . Then [xi, xjjk]c= (1 + q2

jj )q−1

kj [xijk, xj]c+ (1 + q2

jj )(1 + qj j )qij xjxij k in B.

Proof. Assume that xβ:= [xi, xjjk]c−(1 +q2

jj )q−1

kj [xijk, xj]c−(1 + q2

jj )(1+ qj j )qij xjxij k 6= 0.

As eqβi =eqβk =eq2

ij, the Dynkin diagram of W:= kxi⊕kxj⊕kxk⊕kxβ⊂ P (B) contains

a 4-cycle, so GKdim B(W) = ∞by Lemma 2.2. This contradicts GKdim B<∞.

Lemma 3.9. Let i, j, k, ℓ ∈Iθbe such that eqjk =eqij =q−1

jj ∈G′

4∪G′

6,qii =qkk =−1,

eqik =eqiℓ =eqjℓ = 1 and eq3

jk =eqℓk. Then [[xijk,[xij kℓ , xk]c]c, xj k]c= 0 in B.

Proof. If xβ:= [[xij k,[xijkℓ, xk]c]c, xj k ]c6= 0, then W=kxi⊕kxj⊕kxβhas diagram

q3qll

◦

β

−1

◦

i

q−1

q−1②

②

②

②

②

②

②

②q

◦

j,

q−5

❊

❊

❊

❊

❊

❊

❊

❊

q:= qjj ∈G′

4∪G′

6,

which does not have ﬁnite root system by Lemma 2.1. This contradicts GKdim B<∞.

Lemma 3.10. Let i, j, k, ℓ ∈Iθbe such that one of the following hold:

(i) qkk =−1,qii =eq−1

ij =q2

jj ,eqkℓ =q−1

ℓℓ =q3

jj ,eqjk =q−1

jj and eqik =eqiℓ =eqj ℓ = 1;

(ii) qii =eq−1

ij =−q−1

ℓℓ =−eqkl ,qjj =eqj k =qkk =−1and eqik =eqiℓ =eqj ℓ = 1;

12 ANGIONO, CAMPAGNOLO, AND SANMARCO

(iii) qjj =eq−1

jk ∈G′

3,qii =eq−1

ij =qℓℓ =eq−1

kl =−qjj,qkk =−1and eqik =eqiℓ =eqjℓ = 1.

Then [[xijkℓ , xj]c, xk]c=qjk(eq−1

ij −qjj )[[xij kℓ , xk]c, xj]cin B.

Proof. Assume that xβ:= [[xij kℓ , xj]c, xk]c−qjk(eq−1

ij −qjj )[[xij kℓ , xk]c, xj]c6= 0.

(i) This is [ACS, Lemma 3.24], we included the statement here for completeness.

(ii) Here, W:= kxi⊕kxj⊕kxk⊕kxℓ⊕kxβ⊂ P(B) has Dynkin diagram

−1

◦

β

q−1

ii

④

④

④

④

④

④

④

④−qii

❈

❈

❈

❈

❈

❈

❈

❈

qii

◦

i

q−1

ii −1

◦

j

−1−1

◦

k

−qii −q−1

ii

◦

ℓ.

This diagram does not appear in [H3, Table 4], a contradiction with GKdim B<∞.

(iii) Set W=kxj⊕kxβ⊂ P(B), which satisﬁes GKdim B(kxj⊕kxβ)<∞and thus

has ﬁnite root system [AAH2, Theorem 1.2 (b)]. The Dynkin diagram is qjj

◦

j

−qjj qjj

◦

β,

which does not belong to [H3, Table 1], a contradiction.

Lemma 3.11. Let i, j, k ∈Iθbe such that eqj k = 1,qii =eqij =−eqik ∈G′

3. Then

[xi,[xij , xik]c]c=−qj k qikqji [xiik , xij ]c−qij xij xiik in B.

Proof. Since kxj⊕kxi⊕kxkhas ﬁnite root system, [H3, Table 2] implies that qjj =−1

and qkk ∈ {−1,−q−1

ii }.

Assume that xβ:= [xi,[xij , xik ]c]c+qjkqik qj i[xiik , xij ]c+qij xij xiik 6= 0. The diagram

of kxi⊕kxβis d:= qii

◦

i

−q2

ii −qkk

◦

β. If qkk =−1, then GKdim B(kxi⊕kxβ) = ∞by

Lemma 2.3. In the case qkk =−q−1

ii , the root system of dis inﬁnite by [H3, Table 1], hence

GKdim B(kxi⊕kxβ) = ∞by [AAH2, Theorem 1.2 (b)].

Lemma 3.12. Let i, j, k ∈Iθbe such that qj j =qkk =eqjk =−1,qii =−eqij ∈G′

3and

eqik = 1. Then [xiij k, xijk ]c= 0 in B.

Proof. The degree of [xiij k , xijk ]cis β:= 3αi+ 2αj+ 2αk. Since qββ = 1 and eqiβ =q2

ii 6= 1,

it follows from Lemma 2.3 that [xiij k , xijk]c= 0.

Lemma 3.13. Let i, j ∈Iθbe such that −qii,−qjj ,eqij , qii eqij, qjj eqij 6= 1. Then the relation

[xi,[xij , xj]c]c=(1+qij)(1−qj j

eqij )

(1−eqij )qii qj i x2

ij holds in B.

Proof. By [H2, Corollary 13] the diagram qii

◦

i

eqij qjj

◦

jcannot be extended to a connected

diagram of rank 3 with ﬁnite root system. Moreover [H2, Proposition 9 (i)] warranties

that qii eq2

ij qjj =−1 and either qii ∈G′

3or qjj ∈G′

3. By symmetry we can assume

qii ∈G′

3. If xβ:= [xi,[xij , xj]c]c−(1+qij )(1−qjj

eqij )

(1−eqij )qii qj i x2

ij 6= 0, then the Nichols algebra of

kxi⊕kxj⊕kxβ⊂ P(B) has ﬁnite GKdim by Lemma 2.4. Hence its diagram must be

disconnected by the previous argument. But eqiβ =−q−1

jj 6= 1, a contradiction.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 13

Remark 3.14. The unique braiding matrix qwith connected diagram of rank at least 3,

with ﬁnite root system and such that 3αi+ 2αj∈∆q

+is

−1

◦q−1q

◦q−3q3

◦,ord q > 3,(3.2)

which is of type G(3). Furthermore, this is also the unique qsuch that mij ≥3 for

some i, j ∈I3, see [H3, Table 2]. Also, for all qin [H3, Table 2], 4αi+ 3αj, 5αi+ 3αj,

5αi+ 4αj/∈∆q

+. We will use these facts frequently in what follows.

Lemma 3.15. Assume that the diagram of qis one of the following:

brj(2,3) : −ζ2

◦

i

ζ−1

◦

j, ζ ∈G′

9;ufo(9) : ζ

◦

i

ζ−5−1

◦

j, ζ ∈G′

24;

brj(2,5) : −ζ−2

◦

i

ζ−2−1

◦

j, ζ ∈G′

5;Standard G2:ζ2

◦

i

−ζ−1−1

◦

j, ζ ∈G′

8;

ufo(10) : −ζ−2

◦

i

±ζ3−1

◦

j, ζ ∈G′

20;ufo(11) : ζ3

◦

i

−ζ4−ζ−4

◦

j, ζ ∈G′

15;

ufo(11) : ζ3

◦

i

−ζ2−1

◦

j, ζ ∈G′

15;ufo(12) : −ζ−2

◦

i

−ζ3−1

◦

j, ζ ∈G′

7.

Then [xi, x3αi+2αj]c=1−qii

eqij −q2

ii

eq2

ij qjj

(1−qii

eqij )qii x2

iij in B.

Proof. Notice ﬁrst that either mij ∈ {4,5}or else mij = 3, qjj =−1, qii ∈G′

4. For qof

standard type G2, the claim was proved in [ACS, Lemma 6.8]. For the remaining cases,

suppose that xβ:= [xi, x3αi+2αj]c−1−qii

eqij −q2

ii

eq2

ij qjj

(1−qii

eqij )qii x2

iij 6= 0. The diagram of kxi⊕kxj⊕

kxβ⊂ P(B) is connected since either eqjβ 6= 1 when qjj =−1, or else eqiβ 6= 1 when

qjj 6=−1. This is a contradiction with Remark 3.14.

Lemma 3.16. Assume that the diagram of qis one of the following:

ufo(7) : −ζ2

◦ζζ

◦, ζ ∈G′

12,ufo(8) : −ζ2

◦ζ3−1

◦, ζ ∈G′

12,

ufo(9) : ζ6

◦ζζ

◦, ζ ∈G′

24,ufo(11) : −ζ

◦−ζ3ζ5

◦, ζ ∈G′

15,

brj(2,3) : −ζ

◦ζ2ζ3

◦, ζ ∈G′

9,G2:ζ

◦−1−1

◦, ζ ∈G′

6.

Then [x3αi+2αj, xij ]c= 0 in B.

Proof. Notice ﬁrst that in all cases 3αi+ 2αj∈∆q

+but 4αi+ 3αj/∈∆q

+. For qof type G2,

the claim was proved in [ACS, Lemma 4.2]. For the ﬁve remaining cases, suppose that

xβ6= 0. Consider kxi⊕kxj⊕kxβ⊂ P (B). Since ˜qiβ =q8

ii eq3

ij 6= 1 one can verify, case-by-

case, that the diagram of that subspace is connected. But this contradicts Remark 3.14,

since none of these ﬁve rank-two diagrams is a subdiagram of 3.2. It must be xβ= 0.

14 ANGIONO, CAMPAGNOLO, AND SANMARCO

Lemma 3.17. Assume that the diagram of qis one of the following:

ufo(11) : −ζ3

◦−ζ4ζ4

◦, ζ ∈G′

15,brj(2,3) : ζ3

◦ζ−1

◦, ζ ∈G′

9.

Then [xiij , x3αi+2αj]c= 0 in B.

Proof. In these cases 3αi+ 2αj∈∆q

+but 5αi+ 3αj/∈∆q

+. Supose that xβ6= 0. If qis of

type brj(2,3) then eqj β =ζ46= 1, and for type ufo(11), eqiβ =−ζ46= 1 Hence the diagram

of kxi⊕kxj⊕kxβ⊂ P(B) is connected, a contradiction with Remark 3.14.

Lemma 3.18. Assume that the diagram of qis one of the following:

ufo(10) : ζ

◦±ζ3−1

◦, ζ ∈G′

20,ufo(11) : ζ3

◦−ζ4−ζ4

◦, ζ ∈G′

15,

ufo(11) : ζ5

◦−ζ2−1

◦, ζ ∈G′

15,brj(2,5) : ζ

◦ζ2−1

◦, ζ ∈G′

5.

Then [x4αi+3αj, xij ]c= 0 in B.

Proof. Both cases have 4αi+ 3αj∈∆q

+and 5αi+ 4αj6∈ ∆q

+. Suppose that xβ6= 0. We

check case-by-case that eqiβ =q10

ii eq4

ij 6= 1, so the diagram of kxi⊕kxj⊕kxβ⊂ P(B) is

connected, contradicting Remark 3.14. Thus xβ= 0.

Lemma 3.19. Assume that the diagram of qis one of the following:

ufo(10) : −ζ2

◦±ζ3−1

◦, ζ ∈G′

20,ufo(11) : ζ3

◦−ζ4−ζ4

◦, ζ ∈G′

15.

Then [[xiiij , xiij ]c, xiij ]c= 0 in B.

Proof. If xβ6= 0, consider kxi⊕kxj⊕kxβ⊂ P(B), which has connected diagram since

eqβi =q14

ii eq3

ij 6= 1. But this diagram has inﬁnite root system, because it does not appear in

[H3, Table 2], a contradiction.

Lemma 3.20. Assume that the diagram of qis one of the following:

ufo(9) : −ζ4

◦ζ5−1

◦, ζ ∈G′

24,ufo(12) : −ζ

◦−ζ3−1

◦, ζ ∈G′

7.

Then [xiij , x4αi+3αj]c=cqx2

3αi+2α2in B, where cq∈kis given in [An1, (3.29)].

Proof. Supose that xβ6= 0. As eqβ i =q12

ii eq4

ij 6= 1, the diagram of kxi⊕kxj⊕kxβ⊂ P(B)

is connected. Also, 5αi+ 4αjbelongs to the set of roots of kxi⊕kxj⊕kxβ, so the Nichols

algebra of this space has GKdim = ∞by Remark 3.14. A contradiction, thus xβ= 0.

4. Exceptional cases of type g(2,3)

Theorem 3.1 says, in particular, that for a braiding matrix qof modular, supermodular or

unidentiﬁed type, the distinguished pre-Nichols algebra e

Bqis eminent up to two exceptions.

In this section we present, by generators and relations, eminent pre-Nichols algebras b

Bq

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 15

for these two exceptions. We show that in both cases b

Bqﬁts in an exact sequence of

braided Hopf algebras Z֒→b

Bq։e

Bq, where Zis a q-polynomial algebra in two variables.

Even though the exposition will not make it explicit, the construction of these eminent

pre-Nichols algebras was, in some sense, recursive. Namely, we start with a candidate

pre-Nichols algebra that covers all ﬁnite GKdim pre-Nichols. Then we try to show that

this candidate has ﬁnite GKdim by exhibiting a PBW basis; at this point we may realize

that some commutation relation is missing. In that case we redeﬁne our candidate, and

start again. Luckily, at most two iterations of this process were needed.

The two exceptional diagrams of type g(2,3) depend on third-root of unity ξ; they are:

−1

◦ξ−1

◦ξ−1

◦(4.1)

−1

◦ξ ξ

◦ξ−1

◦(4.2)

4.1. Type g(2,3), diagram (4.1).Fix a braiding matrix qwith diagram (4.1). The

distiguished pre-Nichols algebra has the following presentation:

e

Bq=T(V)/hx2

1, x2

2, x2

3, x13,[[x12 , x123]c, x2]c,[[x123, x23 ]c, x2]ci

Notice that Lemma 3.5 deals with the last two relations, but under extra assumptions

which are not satisﬁed for this particular q. We will show, in particular, that there is a

pre-Nichols algebra with ﬁnite GKdim where these elements do not vanish. Set

xu:= [[x12, x123 ]c, x2]c, xv:= [[x123, x23]c, x2]c.(4.3)

As x2

1,x2

2,x2

3,x13 are primitive in T(V), they span a Hopf ideal I:= hx2

1, x2

2, x2

3, x13iof

T(V). Now xu, xv∈B:= T(V)/I are primitive elements by Remark 3.2. From [AS,

Lemma 2.7] we get that [x1, xu]c, [xu, x3]c, [x1, xv]cand [xv, x3]care also primitive in B.

Lemma 4.1. Let Ba pre-Nichols algebra of Bqwith ﬁnite GKdim. Then

[x1, xu]c= [xv, x3]c= [xu, x3]c= [x1, xv]c= 0 in B.

Proof. Let xβ∈ {[x1, xu]c,[xv, x3]c,[xu, x3]c,[x1, xv]c}. Since B։Band xβis primitive

in B, it is also primitive in B. Assume xβ6= 0 in B. By direct computation, eq1β=eq3β= 1,

eq2β=ξand qββ =−1, so the Dynkin diagram of kx1⊕kx2⊕kx3⊕kxβ⊂ P (B) is

−1

◦

β

ξ

−1

◦

1

ξ−1

◦

2

ξ−1

◦

3,

which is not in [H3, Table 3], contradicting GKdim B<∞(we assume Conjecture 1.1).

Now we have a candidate for eminent pre-Nichols algebra:

b

Bq=T(V)/hx2

1, x2

2, x2

3, x13,[x1, xu]c,[x1, xv]c,[xu, x3]c,[xv, x3]ci.(4.4)

Notice that this is indeed a braided Hopf algebra, because it is a quotient of the auxiliary

Bby an ideal generated by primitive elements.

16 ANGIONO, CAMPAGNOLO, AND SANMARCO

Proposition 4.2. Let qof type g(2,3) with Dynkin diagram (4.1). Then

(a) The pre-Nichols algebra b

Bqdeﬁned in (4.4) is eminent, with GKdim b

Bq= 6.

(b) Consider

x12232= [x123, x23]c, x1223= [x123, x2]c, x122332= [x123 , x1223]c, x12223= [x12, x123 ]c.

Then a basis of b

Bqis given by

B={xn1

3xn2

23 xn3

vxn4

2xn5

12232xn6

1223xn7

122332xn8

123xn9

uxn10

12223xn11

12 xn12

1:

n1, n4, n5, n7, n10, n12 ∈ {0,1}, ni∈N0otherwise}.

(c) There is a N3

0-homogeneous extension of braided Hopf algebras Z֒→b

Bq։e

Bq, where

Zis the subalgebra of b

Bqgenerated by xuand xv. The braided adjoint action of b

Bq

on Zis trivial, and Zis a polynomial algebra in two variables.

Proof. Lemmas 3.3 and 4.1 imply that the projection from T(V) onto each ﬁnite GKdim-

dimensional pre-Nichols algebra Bof qfactors through b

Bq. To ﬁnish the proof of (a), we

still need to show that GKdim b

Bq= 6. This will be achieved after several steps, where we

will simultaneously prove (b) and (c).

Step 1. The elements xuand xvdo not vanish in b

Bq.

Proof. We consider the following representation of B,ρ:B−→ k4×4,

ρ(x1) = 0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0 , ρ(x2) = 0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0 , ρ(x3) = 0 1 0 0

0 0 0 0

0 0 0 q13

0 0 0 0 .

Then ρ(xu)6= 0, ρ(xv)6= 0 since the ﬁrst rows of ρ(xu), ρ(xv) are not zero; hence xu, xv6= 0

in B. As Bn=b

Bn

qif n≤6, we have that xu, xv6= 0 in b

Bq.

Step 2. The adjoint action of b

Bqon Zis trivial, and Zhas basis {xm

uxn

v:m, n ∈N0}.

Proof. By Step 1, xu, xv6= 0. Moreover xn

u, xn

v6= 0 for all n∈Nsince they are primitive

elements such that quu =qvv = 1, and Zis a q-polynomial algebra in the variables xu, xv.

By deﬁnition of b

Bq, (adcxi)xu= (adcxi)xv= 0 for i= 1,3, and [xu, x2]c= [xv, x2]c= 0

since x2

2= 0 in b

B. So (adcx)xu= (adcx)xv= 0 for every homogeneous element x∈b

Bq

of positive degree.

Step 3. The linear span of Bis b

Bq.

Proof. It is enough to check that the subspace Ispanned by Bis a left ideal of b

Bq. From

[x1, xu]c= [xv, x3]c= 0 and (2.5) we get the equalities:

x3

12x3=q3

13q3

23x3x3

12, x1x3

23 =q3

12q3

13x3

23x1.

From these equalities we obtain the following:

[x12, x12223]c= 0,[x12232, x23]c= 0.(4.5)

Using (2.5) again and x13 =x2

1=x2

3= 0 we also get

[x23, x3]c= [x123 , x3]c= 0,[x12223, x3]c=ζ2q13q23 x2

123,

[x1, x12]c= [x1, x123]c= 0,[x1, x12232]c=ζ2q12q13x2

123.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 17

Using the last equality, together with [xu, x3]c= 0, (2.3) and (2.5), we get

0 = [x1,[xu, x3]c]c= [x1,[[x12223, x2]c, x3]c]c

=x1,[x12223, x23]c+q2

12q13x2x2

123 −ζ2q13q2

23x2

123x2c

=q2

12q13x12223x123 +ζq2

12q2

13q23x123x12223+q2

12q13x12x2

123 −ζ2q2

12q3

13q2

23x2

123x12

= 2q2

12q13x12x2

123 −2ζ2q3

13q2

23q2

12x2

123x12 ;

thus x12x2

123 =ζ2q2

13q2

23x2

123x12. Analogously, x2

123x23 =ζ2q2

12q2

13x23x2

123. From these two

equalities we deduce the following:

[x12223, x123]c= 0,[x123, x12232]c= 0.

Using the ﬁrst equation and the ﬁrst one of (4.5).

x2

12223=x12223(x12x123 −ζq13q23x123 x12) = −(x12x123 −ζ q13 q23x123 x12 )x12223=−x2

12223.

This computation and an analogous one for x12232imply that x2

12223=x2

12232= 0.

Next we check the following equations:

[x1, x1223]c=q12q32x12223+ (ζ−1)q13q23x123x12 ,

[x1, x122332]c=−q12q13x123 [x1, x1223]c+ζ2q12q32 [x1, x1223]cx123 = 0.

In a similar way we get the equalities:

[x1223, x2]c= 0,[x122332, x2]c= (1 −ζ2)q12q32x2

1223,

[x1223, x3]c=x12223,[x122332, x3]c= 0.

Using the relations involving x122332we obtain that x2

122332= 0.

A rutinary recursive proof shows that xαxβ=qαβ xβxα+ ordered products of interme-

diate PBW generators for each pair of roots α < β, so the step is proved.

Step 4. There is a degree-preserving extension of braided Hopf algebras Z֒→b

Bq։e

Bq.

Furthermore Bis a basis of b

Bqand GKdim b

Bq= GKdim e

Bq+ GKdim Z= 6.

Let Z′=b

Bco πfor b

Bq։e

Bq. Since Z ⊆ Z′, from [ACS, Lemma 2.4] we get

Hb

Bq=HZ′He

Bq≥ HZHe

Bq≥

≥1

(1 −t2

1t3

2t3)(1 −t1t3

2t2

3)

(1 + t1)(1 + t2)(1 + t2

1t2

2t3)(1 + t2

1t3

2t2

3)(1 + t1t2

2t2

3)(1 + t3)

(1 −t1t2)(1 −t1t2

2t3)(1 −t2t3)(1 −t2t3).

On the other hand b

Bqis spanned by B, so

Hb

Bq≤(1 + t1)(1 + t2)(1 + t2

1t2

2t3)(1 + t2

1t3

2t2

3)(1 + t1t2

2t2

3)(1 + t3)

(1 −t1t2)(1 −t1t2

2t3)(1 −t2t3)(1 −t2t3)(1 −t2

1t3

2t3)(1 −t1t3

2t2

3)

These inequalities between the Hilbert series say that

Hb

Bq=(1 + t1)(1 + t2)(1 + t2

1t2

2t3)(1 + t2

1t3

2t2

3)(1 + t1t2

2t2

3)(1 + t3)

(1 −t1t2)(1 −t1t2

2t3)(1 −t2t3)(1 −t2t3)(1 −t2

1t3

2t3)(1 −t1t3

2t2

3)

so Z=Z′,Bis a basis of b

Bqand GKdim b

Bq= GKdim e

Bq+ GKdim Z= 4 + 2 = 6.

18 ANGIONO, CAMPAGNOLO, AND SANMARCO

4.2. Type g(2,3), diagram (4.2).Let qbe a braiding matrix with Dynkin diagram (4.2).

In this case the distiguished pre-Nichols algebra is

e

Bq=T(V)/hx2

1, x2

3, x13,[x223 , x23]c, x221, x2223 ,[[x123, x2]c, x2]ci.

Notice that Lemma 3.3 (l) deals with the relation xu:= [[x123, x2]c, x2]cbut under extra

assumptions which are not satisﬁed for this particular q. We will see that xuis not zero

in at least one pre-Nichols algebra with ﬁnite GKdim. Set:

x1223= [x123, x2]c, x12232= [x123, x23]c, x12332= [x12232, x2]c, xv= [x123, x1223]c.(4.6)

Consider the algebra:

b

Bq=T(V)/hx2

1, x2

3, x13,[x223 , x23]c, x221, x2223 ,[xv, x3]c,[x12332, x2]c,[x12332, x3]ci.(4.7)

Next we prove that b

Bqis an eminent pre-Nichols algebra.

Proposition 4.3. Let qis of type g(2,3) with Dynkin diagram (4.2). Then

(a) The algebra b

Bqdeﬁned in (4.7) is an eminent pre-Nichols of q, with GKdim b

Bq= 6.

(b) A basis of b

Bqis given by

B={xn1

3xn2

23 xn3

223xn4

2xn5

12332xn6

12232xn7

uxn8

1223xn9

123xn10

vxn11

12 xn12

1:n1, n3, n5, n6, n10, n11 ∈ {0,1}}

(c) There is a N3

0-homogeneous extension of braided Hopf algebras Z֒→b

Bq։e

Bq, where

Zis the subalgebra of b

Bqgenerated by xuand xv. The braided adjoint action of b

Bq

on Zis trivial, and Zis a polynomial algebra in two variables.

Proof. We proceed in several steps. The ﬁrst two steps are devoted to verify that the

deﬁning ideal of b

Bqis a Hopf ideal, and also that b

Bqproject onto an arbitrary pre-Nichols

algebra Bwith ﬁnite GKdim. Consider the following auxiliary algebra:

B:= T(V)/hx2

1, x2

3, x13, x221 , x2223,[x223, x23 ]c,[xv, x3]ci.

Step 1. Bis a braided Hopf algebra and the canonical projection T(V)→Binduces a

surjective Hopf algebra map π:B→B. Also, xuand xvare primitive.

Proof. Let B′=T(V)/hx2

1, x2

3, x13, x221 , x2223,[x223, x23 ]ci. As x2

1,x2

3,x13,x221 ,x2223 are

primitive in T(V), J=hx2

1, x2

3, x13, x221 , x2223 iis a Hopf ideal of T(V). Also, [x223, x23 ]c∈

T(V)/J is primitive by Remark 3.2, so B′is a braided Hopf algebra. By Lemma 3.3, the

canonical projection T(V)։Binduces a surjective Hopf algebra map B′։B.

Next we prove that xvis primitive in B′. By (3.1) applied to B′։e

Bq,

∆(xv)∈1⊗xv+xv⊗1 + B′⊗ hxui+hxui ⊗ B′.

Also, xuis primitive in B′by Remark 3.2, and using GAP,

[xu, x1]c= [xu, x3]c= 0.(4.8)

Notice that xuand xvare the superletters associated to the Lyndon words x1x2x3x2

2and

x1x2x3x1x2x3x2, respectively, according to the deﬁnitions in [Kh]. By [Kh, Lemma 13]

and these relations, there exist a, b, c ∈ksuch that

∆(xv) = 1 ⊗xv+xv⊗1 + ax1⊗xux3+bx1xu⊗x3+cx1x3⊗xu.

FINITE GK-DIMENSIONAL PRE-NICHOLS ALGEBRAS 19

As (∆ ⊗id)∆(xv) = (id ⊗∆)∆(xv) we have that a=b=c= 0. Now, [xv, x3]cis primitive

in B′by [AS, Lemma 2.7] so Bis a braided Hopf algebra.

Finally, suppose that xβ:= [xv, x3]c6= 0 in B. By inspection, if the diagram of a matrix

q′is connected and contains (4.2), then q′is not in [H3, Table 3]; thus GKdim Bq′=∞,

assuming Conjecture 1.1. Now, the diagram of kx1⊕kx2⊕kx3⊕kxβis connected since

eq2β=ζ, and we get a contradiction.

Step 2. [x12332, x2]c, [x12332, x3]care primitive in B, thus b

Bqis a braided Hopf algebra.

The canonical projection T(V)→Binduces a surjective Hopf algebra map π:b

Bq→B.

Proof. From x221 =x2223 = 0 we get x1x3

2=q3

12x3

2x1and x3x3

2=q3

32x3

2x3. From the last

two equalities we deduce the following:

[xu, x2]c= [[[x123, x2]c, x2]c, x2]c=x123 x3

2−q3

12q3

32x3

2x123 = 0.(4.9)

As in the proof of Step 1, we use [Kh, Lemma 13] to show that [x12332, x2]cand [x12332, x3]c

are primitive, hence b

Bqis a braided Hopf algebra since b

Bq=B/h[x12332, x2]c,[x12332, x3]ci.

Assume that xβ= [x12332, xi]c6= 0, i∈I2,3. The diagram of kx1⊕kx2⊕kx3⊕kxβis

connected since eq2β6= 1; the same argument as in Step 1 leads to a contradiction since we

assume Conjecture 1.1.

By Step 2, it is enough to prove that GKdim b

Bq<∞. To do so, we will see that Bis

a basis of b

Bqin three steps.

Step 3. The adjoint action of b

Bqon Zis trivial, and Zhas basis {xm

uxn

v:m, n ∈N0}.

Proof. Using GAP we check that xu, xv6= 0 in the pre-Nichols algebra B′introduced in the

proof of Step 1; hence xu, xv6= 0 in b

Bqsince (B′)α=b

Bα

qfor all α≤2α1+ 3α2+ 2α3.

Therefore, xn

u, xn

v6= 0 for all n∈Nsince xuand xvare primitive and quu =qvv = 1;

moreover {xm

uxn

v:m, n ∈N0}is a basis of Z, and Zis a Hopf subalgebra. By (4.8) and

(4.9), (adcxi)xu= 0 for all i∈I3; hence (adcx)xu= 0 for all x∈b

Bqhomogeneous of

degree >0.

Step 4. b

Bqis spanned by B.

Proof. To prove the statement, we will see that the subspace Ispanned by Bis a left ideal

of b

Bq. As x2

1=x2

3= 0, we also have:

x1x12 =−q12x12x1, x1x123 =−q12q13x123x1,

x23x3=−q23x3x23, x123 x3=−q13q23 x3x123.

From (adcx2)3x3= [x223, x23]c= 0 we deduce the following equality:

x2

223 =x223(x2x23 −ξq23x23 x2) = −ξ−2(x2x23 −ξq23x23x2)x223 =−ξ−2x2

223.

Hence x2

223 = 0.

From x221 = 0 we also have [x12, x2]c= 0. Using this relation and x2

1= 0 we check that

x2

12 = 0; therefore,

x12x123 =x12 (x12x3−q13 q23x3x12) = −q13 q23(x12x3−q13q23x3x12)x12 =−q13q23 x123x12.

20 ANGIONO, CAMPAGNOLO, AND SANMARCO

Using the relations already proved, (2.6), (4.6) and (2.5), the following relations also hold:

[x12, x1223]c= 0,[x12232, x3]c= 0,[x12332, x23]c= 0,

[x12332, x223]c= 0,[x12232, x23]c= 0,[x12232, x223]c= 0.

Similarly,

[x223, x3]c=ξ2q23 x2

23,[x1223, x3]c=x12232,

[x1, x1223]c= (ξ2−1)q12q13x123x12 ,[x1, x12232]c= (ξ2−1)q12q13 x2

123,

[x12, x23 ]c= (ξ−1)q12x2x123 −ξq23x1223,[x1223, x23]c=−q23x12332−q12q32 x2x12232.

Next we use (2.5) and [xv, x3]c= 0 to deduce that [x123, x12232]c= 0; also,

x2

12232= (x123x23 −q123,23x23x123 )x12232=−x12232(x123x23 −q123,23x23x123 ) = −x2

12232.

Hence x2

12232= 0. From here we check that [x12232, x12332]c= 0; this relation and

[x12332, x2]c= 0 imply that x2

12332= 0.

Again, we prove recursively that xαxβ=qαβxβxα+ ordered products of intermediate

PBW generators for each pair of roots α < β, so the step is proved.

Step 5. There is a degree-preserving extension of braided Hopf algebras Z֒→b

Bq։e

Bq.

Furthermore Bis a basis of b

Bqand GKdim b

Bq= GKdim e

Bq+ GKdim Z= 6.

The proof is analogous to the corresponding step in Proposition 4.2. Indeed, Step 3

shows that Zis a central Hopf subalgebra of b

Bqwith basis {xm

uxn

v:m, n ∈N0}. If

Z′:= b

Bco π, then Z ⊆ Z′and [ACS, Lemma 2.4] implies that Hb

Bq=HZ′He

Bq≥ HZHe

Bq.

On the other hand b

Bqis spanned by B, so we have an equality between the Hilbert series:

Hb

Bq=HZHe

Bq=(1 + t1)(1 + t1t2)(1 + t1t2

2t2

3)(1 + t1t3

2t2

3)(1 + t2

2t3)(1 + t3)

(1 −t1t2t3)(1 −t1t2

2t3)(1 −t2

1t3

2t2

3)(1 −t1t3

2t3)(1 −t2)(1 −t2t3).

Thus Z=Z′,Bis a basis of b

Bqand GKdim b

Bq= GKdim e

Bq+ GKdim Z= 6.

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FaMAF-CIEM (CONICET), Universidad Nacional de C´

ordoba, Medina Allende s/n, Ciudad

Universitaria, 5000 C´

ordoba, Rep´

ublica Argentina

Email address:ivan.angiono@unc.edu.ar

Email address:emiliano.campagnolo@mi.unc.edu.ar

Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Email address:sanmarco@iastate.edu