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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