# Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type

Iván Angiono<EMAIL_ADDRESS>and  Emiliano Campagnolo
<EMAIL_ADDRESS>
###### Abstract.

We classify graded pre-Nichols algebras of diagonal type with finite Gelfand-
Kirillov dimension. The characterization is made through an isomorphism of
posets with the family of appropriate subsets of the set of positive roots
coming from central extensions of Nichols algebras of diagonal type,
generalizing the corresponding extensions for small quantum groups in de
Concini-Kac-Procesi forms of quantum groups.

On the way to achieving this result, we also classify graded quotients of
algebras of functions of unipotent algebraic groups attached to semisimple Lie
algebras.

###### Key words and phrases:

Hopf algebras, Nichols algebras, Gelfand-Kirillov dimension.
MSC2020: 16T05, 16T20, 17B37, 17B62.

The work of both authors was partially supported by CONICET and Secyt (UNC)

##  1\. Introduction

Let $\Bbbk$ be an algebraically closed field of characteristic zero. The so
called quantized enveloping algebras $U_{q}(\mathfrak{g})$, where $q$ is a
parameter and $\mathfrak{g}$ is a semisimple Lie algebra, emerged after the
works of Drinfeld and Jimbo as examples of non-commutative and non-
cococommutative Hopf algebras over the field $\Bbbk(q)$: They were obtained by
_deforming_ the structure of the corresponding enveloping algebra over
$\Bbbk(q)$. When we consider the evaluation of the parameter $q$ in elements
of $\Bbbk$, we get a Hopf algebra over $\Bbbk$, which behaves as
$U(\mathfrak{g})$ in terms of representations if $q$ is not a root of unity.
In the nineties, de Concini, Kac and Procesi [DKP, DP] studied the case in
which $q$ is a root of unity of order $N$, under some mild conditions on $N$,
and found a completely different story. To begin with, the centre of
$U_{q}(\mathfrak{g})$ is larger than in the case where $q$ is not a root of
unity, and it contains a Hopf subalgebra $\mathcal{Z}_{q}$ which gives rise to
an extension of Hopf algebras

(1.1) $\displaystyle\mathcal{Z}_{q}\hookrightarrow
U_{q}(\mathfrak{g})\twoheadrightarrow\mathfrak{u}_{q}(\mathfrak{g}).$

Here $\mathfrak{u}_{q}(\mathfrak{g})$ is the Frobenius-Lusztig kernel, a
finite-dimensional pointed Hopf algebra. The name comes from the evaluation of
a different form of $U_{q}(\mathfrak{g})$ studied by Lusztig [L2, L1] in
connection with the representation theory of algebraic groups in positive
characteristic. He took an integral form generated by _divided powers_ of the
generators and the algebra $\mathcal{U}_{q}(\mathfrak{g})$ obtained after
evaluation in $q$ fits into an extension of Hopf algebras

(1.2)
$\displaystyle\mathfrak{u}_{q}(\mathfrak{g})\hookrightarrow\mathcal{U}_{q}(\mathfrak{g})\twoheadrightarrow
U(\mathfrak{g}).$

Coming back to (1.1), $U_{q}(\mathfrak{g})$ is $\mathbb{Z}^{\theta}$-graded,
where $\theta$ is the rank of $\mathfrak{g}$, and has a triangular
decomposition $U_{q}(\mathfrak{g})\simeq U_{q}^{+}(\mathfrak{g})\otimes
U_{q}^{0}(\mathfrak{g})\otimes U_{q}^{-}(\mathfrak{g})$, where
$U_{q}^{0}(\mathfrak{g})$ is the group algebra of a free abelian group in
generators $K_{i}$, $1\leq i\leq\theta$, and $U_{q}^{\pm}(\mathfrak{g})$ has a
PBW basis made of PBW generators $E_{\alpha}$, respectively $F_{\alpha}$, of
degree $\pm\alpha\in\varDelta_{+}$, where $\varDelta$ is the set of roots of
$\mathfrak{g}$ viewed as a subset of $\mathbb{Z}^{\theta}$. In addition,
$\mathcal{Z}_{q}$ is the subalgebra generated by $E_{\alpha}^{N}$,
$F_{\alpha}^{N}$, $\alpha\in\varDelta_{+}$, and $K_{i}^{N}$, $1\leq
i\leq\theta$, while $\mathfrak{u}_{q}(\mathfrak{g})$ has a _restricted_ PBW
basis with the same set of generators, but we restrict the powers up to $N$.
Also, the restriction of (1.1) to the corresponding positive parts gives an
extension $\mathcal{Z}_{q}^{+}\hookrightarrow
U_{q}^{+}(\mathfrak{g})\twoheadrightarrow\mathfrak{u}_{q}^{+}(\mathfrak{g})$
of Hopf algebras, but in the braided tensor category
${}^{\Bbbk\mathbb{Z}^{\theta}}_{\Bbbk\mathbb{Z}^{\theta}}\mathcal{YD}$ of
Yetter-Drinfeld modules over $\mathbb{Z}^{\theta}$.

A few years later, Andruskiewitsch and Schneider [AS1] introduced the so
called Lifting Method to classify finite-dimensional pointed Hopf algebras. In
a nutshell, this method is based on the decomposition of the associated
coradically graded Hopf algebra into the bosonization between a group algebra
$\Bbbk\Gamma$ and a coradically graded Hopf algebra $R$ in the category
${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$ by solving the following steps:
first, to classify all finite dimensional Nichols algebras (see §2.1 and
references therein for the definition and examples); then classify the
possible finite dimensional Hopf algebras $R$ extending the Nichols algebras
as before, called _post-Nichols algebras_ ; and finally to obtain all the
liftings of the corresponding bosonizations. This method was widely applied,
with the first main result [AS4] being the classification in the case of
abelian groups with moderate restriction on the order, where the examples are
certain deformations of the bosonizations of
$\mathfrak{u}_{q}^{+}(\mathfrak{g})$ with appropriate abelian groups. The
general answer for abelian groups involves the classification of finite-
dimensional Nichols algebras $\mathscr{B}_{\mathfrak{q}}$ of diagonal type
(depending on a braiding matrix
$\mathfrak{q}\in(\Bbbk^{\times})^{\theta\times\theta}$), which in turn is
contained in the classification of those with finite root system (i.e., a
finite number of PBW generators) given by Heckenberger [H2]. To do so, we
attach a (kind of) Dynkin diagram with labels depending on $\mathfrak{q}$ and
consider the connected components of this diagrams: the root system is, as
expected, the disjoint union of the root systems of the connected components,
so the list in [H2] contains just those matrices $\mathfrak{q}$ with connected
Dynkin diagram. In [AA] this list was split into the following families:

  *   * •

Cartan type

  * •

Super type

  * •

Standard type

  * •

Modular type

  * •

Supermodular type

  * •

UFO.

Here, Cartan type is essentially the case of quantized enveloping algebras,
while super type is related with quantized enveloping Lie superalgebras.

Andruskiewitsch and Schneider also started the classification of pointed Hopf
algebras with finite Gelfand-Kirillov dimension (i.e. infinite-dimensional
ones with some kind of moderate growth) in [AS3] by classifying those which
are domains and satisfy a technical condition. To do so, they extend the
Lifting Method to this context and obtained that all possible Nichols algebras
are close to $U_{q}^{+}(\mathfrak{g})$ for $q$ not a root of unity, and the
unique possible Hopf algebras $R$ extending the Nichols algeras are just the
Nichols algebras themselves.

If we want to get all Hopf algebras including those which are not domains, the
answer is fully different: indeed, Lusztig examples provide post Nichols
algebras $\mathcal{U}_{q}^{+}(\mathfrak{g})$ properly extending the Nichols
algebras $\mathfrak{u}_{q}^{+}(\mathfrak{g})$. Taking graded duals, the
extension between the positive parts in (1.2) becomes that of (1.1), where
$U_{q}^{+}(\mathfrak{g})$ is a pre-Nichols algebra (a graded intermediate
quotient between the tensor algebra and the Nichols algebra
$\mathfrak{u}_{q}^{+}(\mathfrak{g})$) of finite $\operatorname{GKdim}$, and
classifying post-Nichols algebras with finite $\operatorname{GKdim}$ is
related to classifying pre-Nichols algebras with finite
$\operatorname{GKdim}$. In general, finite-dimensional Nichols algebras
$\mathscr{B}_{\mathfrak{q}}$ of diagonal type fit into an exact sequence of
braided Hopf algebras
$\mathcal{Z}_{\mathfrak{q}}^{+}\hookrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\mathscr{B}_{\mathfrak{q}}$,
generalizing the one between positive parts in (1.1), where
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is the distinguished pre-Nichols
algebra [An3]: it has a PBW basis with the same set of generators as
$\mathscr{B}_{\mathfrak{q}}$, but where we allow the powers of some of them to
be arbitrary as in (1.1). Thus we may identify first those Nichols algebras of
diagonal type with finite $\operatorname{GKdim}$ and then obtain all possible
pre-Nichols algebras of finite $\operatorname{GKdim}$ covering each one of
these Nichols algebras. For the first question the answer was given in [AnG],
following the conjecture made in [AAH]: a Nichols algebra of diagonal type has
finite $\operatorname{GKdim}$ if and only if its root system is finite, i.e.
it appears in the lists in [H2]. Thus we have to classify all pre-Nichols
algebras $\mathscr{B}$ with finite $\operatorname{GKdim}$ covering the Nichols
algebras $\mathscr{B}_{\mathfrak{q}}$ with finite root system.

Fixing a braiding matrix $\mathfrak{q}$, the corresponding set of pre-Nichols
algebras form a poset whose maximal element is $\mathscr{B}_{\mathfrak{q}}$
and we may wonder if there exists a minimal element between those of finite
$\operatorname{GKdim}$, called the _eminent_ pre-Nichols algebra
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ in [ASa]. This is the case for all
$\mathfrak{q}$ with connected diagrams up to Cartan types $A_{\theta}$,
$D_{\theta}$ with label $q=-1$, thanks to [ASa, ACSa, ACSa2, C]: in most cases
eminent and distinguished pre-Nichols algebras coincide, i.e.
$\widehat{\mathscr{B}}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}$.
In any case, $\widehat{\mathscr{B}}_{\mathfrak{q}}$ fits into an exact
sequence of braided Hopf algebras
$\widehat{\mathcal{Z}}_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\mathscr{B}_{\mathfrak{q}}$,
where $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is an algebra of $q$-polynomials
whose variables are homogeneous: we collect the
$\mathbb{N}_{0}^{\theta}$-degrees of these variables in a set denoted by
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$, which is the set of
positive roots of a classical root system when
$\widehat{\mathscr{B}}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}$
by [AAR]. As we will explain later, we are interested in subsets
$B\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ closed by
sums: it means if $\alpha,\beta\in B$ are such that
$\alpha+\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$, then
$\alpha+\beta\in B$. Let $\mathcal{P}_{c}(\mathfrak{q})$ be the set of all
subsets of $\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ closed by
sums, which is a subposet of the poset of subsets of
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$.

Due to the results stated above, the determination of the poset of pre-Nichols
algebras with finite $\operatorname{GKdim}$ is equivalent to the
characterization of all intermediate quotients $\mathscr{B}$ between
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ and $\mathscr{B}_{\mathfrak{q}}$. The
main result of this paper deals with this question:

######  Theorem 1.1.

Let $\mathfrak{q}$ be a braiding matrix whose connected components are not of
Cartan types $A_{\theta}$, $D_{\theta}$ with label $q=-1$ neither one
dimensional with label $q=\pm 1$. For each
$\underline{\beta}\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$
let $z_{\beta}$ be a generator of $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ of
degree $\underline{\beta}$, and for each $B\in\mathcal{P}_{c}(\mathfrak{q})$
let

$\displaystyle\mathscr{B}(\mathfrak{q},B)$
$\displaystyle:=\widehat{\mathscr{B}}_{\mathfrak{q}}/\langle
z_{\underline{\beta}}|\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}-B\rangle.$

Then each $\mathscr{B}(\mathfrak{q},B)$ is an $\mathbb{N}_{0}^{\theta}$-graded
pre-Nichols algebra such that
$\operatorname{GKdim}\mathscr{B}(\mathfrak{q},B)=|B|$. The assignment
$B\mapsto\mathscr{B}(\mathfrak{q},B)$ gives an anti-isomorphism of posets
between

  * •

the set $\mathcal{P}_{c}(\mathfrak{q})$ of all subsets of
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ closed by sums, and

  * •

the set of all $\mathbb{N}_{0}^{\theta}$-graded pre-Nichols algebras of
$\mathfrak{q}$ with finite $\operatorname{GKdim}$.

In addition, we prove that the poset in the non-connected case decomposes as
the product of the corresponding posets of the connected components and give a
closed formula for the Hilbert series of each pre-Nichols algebra
$\mathscr{B}(\mathfrak{q},B)$ as well as a PBW basis.

Going back through the steps of the Lifting Method, we classify by taking
graded duals all post-Nichols algebras of diagonal type and finite
$\operatorname{GKdim}$ (up to few exceptions on the connected components),
which in turn give all coradically graded pointed Hopf algebras with abelian
coradical and finite $\operatorname{GKdim}$ after bosonization with suitable
abelian groups.

We can observe that any $\mathscr{B}(\mathfrak{q},B)$ fits into an exact
sequence of the Nichols algebra $\mathfrak{q}$ by a $q$-central Hopf
subalgebra, so we may ask:

######  Question 1.2.

Are there examples of pre-Nichols algebras of finite $\operatorname{GKdim}$
which are not a “central” extension of braided Hopf algebras of the
corresponding Nichols algebra?

Although the restriction to the $\mathbb{N}_{0}^{\theta}$-graded case may seem
very strong, it has both a _realisation-independence_ reason and also a
reduction to a problem with a closed answer: the general case depends strongly
on the realisation of the braided vector space of diagonal type as Yetter-
Drinfeld module, and a general answer may be somewhat unmanageable, see Remark
2.6.

The organization of the paper is the following. First we recall several
notions about Nichols algebras, distinguished and eminent pre-Nichols algebras
when the braiding is of diagonal type; we also summarize known results about
eminent pre-Nichols algebras and solve some questions on quotients of pre-
Nichols algebras, extensions of Nichols algebras by central subalgebras and
twist equivalence of braidings of diagonal type in §2. Motivated by the
results in this section we consider quotients of the algebras of functions of
unipotent algebraic groups which are the positive parts of semisimple ones;
hence, in §3 we give the classification of these quotients in terms of subsets
closed by sums of the set of positive roots of the associated semisimple Lie
algebra. Finally we attack the determination of the poset of
$\mathbb{N}_{0}^{\theta}$-graded pre-Nichols algebras with finite
$\operatorname{GKdim}$ of a matrix $\mathfrak{q}$ such that
$\mathscr{B}_{\mathfrak{q}}$ has a finite root system (or equivalently, such
that $\operatorname{GKdim}\mathscr{B}_{\mathfrak{q}}<\infty$). Due to the
results in §2, we can relate these quotients to those of the skew central Hopf
subalgebra $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ (the subalgebra of
coinvariants of the projection
$\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\mathscr{B}_{\mathfrak{q}}$),
and also we can move to the case in which
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is central. We attack first the
connected case: we apply results in §3 to solve all the cases where
$\widehat{\mathscr{B}}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}$,
and compute explicitly the poset for the few exceptions where
$\widehat{\mathscr{B}}_{\mathfrak{q}}\neq\widetilde{\mathscr{B}}_{\mathfrak{q}}$.
Then we deal with the non-connected case using tools from [ASa] and the answer
for the connected case.

### Notation

We fix $\theta\in\mathbb{N}$ and set
$\mathbb{I}=\mathbb{I}_{\theta}:=\\{1,2,...,\theta\\}$. Let
$(\alpha_{j})_{j\in\mathbb{I}}$ be the canonical basis of
$\mathbb{Z}^{\theta}$ and $\alpha_{ij}$ denote $\alpha_{i}+\cdots+\alpha_{j}$,
$i\leq j$. Let
$\beta=\sum_{i\in\mathbb{I}}a_{i}\alpha_{i}\in\mathbb{Z}^{\theta}$, sometimes
also denoted $\beta=1^{a_{1}}\cdots\theta^{a_{\theta}}$ to shorten
expressions. The _support_ and the _height_ of $\beta$ are given by

$\displaystyle\operatorname{supp}\beta$
$\displaystyle=\\{i\in\mathbb{I}|a_{i}\neq 0\\},$
$\displaystyle\operatorname{ht}(\beta)=\sum_{i\in\mathbb{I}}a_{i}\in\mathbb{Z}.$

If $\gamma=\sum_{i\in\mathbb{I}}b_{i}\alpha_{i}\in\mathbb{Z}^{\theta}$ is such
that $a_{i}\leq b_{i}$ for all $i\in\mathbb{I}$, then we say that
$\beta\leq\gamma$.

If $N\in\mathbb{N}$ and $v\in\Bbbk^{\times}$, then
$(N)_{v}:=\sum_{j=0}^{N-1}v^{j}$. We denote by $\mathbb{G}_{N}$ to the group
of roots of unity of order $N$ in $\Bbbk$, and $\mathbb{G}_{N}^{\prime}$ the
subset of primitive roots of order $N$.

Let $A$ be an associative algebra (with unit). We denote by
$\operatorname{GKdim}A$ the Gelfand-Kirillov dimension of $A$. We refer to
[KL] for the definition and properties.

We will deal with $\mathbb{N}_{0}^{\theta}$-graded objects
$U=\oplus_{\alpha\in\mathbb{N}_{0}^{\theta}}U_{\alpha}$. The Hilbert series of
$U$ is

$\displaystyle{\mathcal{H}}_{U}$
$\displaystyle=\sum_{\alpha\in\mathbb{N}_{0}^{\theta}}\dim
U_{\alpha}\,t^{\alpha}\in\mathbb{N}_{0}[[t_{1},\dots,t_{\theta}]]$
$\displaystyle\text{where }t^{\alpha}=t_{1}^{a_{1}}\cdots
t_{\theta}^{a_{\theta}}\text{ for }\alpha=(a_{1},\cdots,a_{\theta}).$

Given
${\mathcal{H}}=\sum_{\alpha\in\mathbb{N}_{0}^{\theta}}a_{\alpha}\,t^{\alpha},{\mathcal{H}}^{\prime}=\sum_{\alpha\in\mathbb{N}_{0}^{\theta}}b_{\alpha}\,t^{\alpha}\in\mathbb{N}_{0}[[t_{1},\dots,t_{\theta}]]$,
we say that ${\mathcal{H}}\geq{\mathcal{H}}^{\prime}$ if $a_{\alpha}\geq
b_{\alpha}$ for all $\alpha\in\mathbb{N}_{0}^{\theta}$. Thus, if
$U^{\prime}\subseteq U$ as $\mathbb{N}_{0}^{\theta}$-graded objects, then
${\mathcal{H}}_{U}\geq{\mathcal{H}}_{U^{\prime}}$.

Let $C$ be a coalgebra. We will use Sweedler notation for $C$ and any (left)
comodule $V$; explicitly, $\Delta(c)=c_{(1)}\otimes c_{(2)}\in C\otimes C$ for
all $c\in C$, and if $\rho:V\to C\otimes V$ is the coaction, then
$\rho(v)=v_{(-1)}\otimes v_{(0)}$ for all $v\in V$.

##  2\. On pre-Nichols algebras of diagonal type

We start by recalling notions and results related with Nichols and pre-Nichols
algebras, with special focus on the diagonal case. Let $H$ be a Hopf algebra.
As usual, we denote by ${}^{H}_{H}\mathcal{YD}$ the category of (left) Yetter-
Drinfeld modules over $H$. We refer to [A2, R] for unexplained notions and
notations on Yetter-Drinfeld modules and braided vector spaces, and to [AA]
for more information on Nichols algebras of diagonal type, and to [KL] for
definitions and basic results on Gelfand-Kirillov dimension.

###  2.1. Nichols algebras and pre-Nichols algebras

Recall that ${}^{H}_{H}\mathcal{YD}$ is a braided tensor category: for each
pair $V,W\in{}^{H}_{H}\mathcal{YD}$, the braiding is given by

$\displaystyle c_{V,W}$ $\displaystyle:V\otimes W\to W\otimes V,$
$\displaystyle c_{V,W}(x\otimes y)$ $\displaystyle=x_{(-1)}\cdot y\otimes
x_{(0)},$ $\displaystyle x\in V,y\in W.$

Therefore each pair $(V,c_{V,V})$, $V\in{}^{H}_{H}\mathcal{YD}$, is a braided
vector space.

The tensor algebra $T(V)=\bigoplus_{n\geq 0}V^{\otimes n}$ becomes a graded
Hopf algebra in ${}^{H}_{H}\mathcal{YD}$ by declaring that every element in
$V$ is primitive. The Nichols algebra $\mathscr{B}(V)$ of $V$ is the quotient
of $T(V)$ by the maximal Hopf ideal $\mathcal{J}(V)=\oplus_{n\geq
2}\mathcal{J}^{n}(V)$ generated by homogeneous elements of degree $\geq 2$.
Hence, $\mathscr{B}(V)$ is an $\mathbb{N}_{0}$-graded Hopf algebra over
${}^{H}_{H}\mathcal{YD}$, where the degree one part is $V$, coincides with the
set of primitive elements and generates $\mathscr{B}(V)$.

It is known that the structure of the Nichols algebra $\mathscr{B}(V)$ depends
on the braiding $c:=c_{V,V}\in\operatorname{GL}(V^{\otimes 2})$, not really on
the realisation as a Yetter-Drinfeld module. This is why we consider braided
vector spaces throughout this paper: i.e. pairs $(V,c)$, where $V$ is a vector
space and $c\in\operatorname{GL}(V^{\otimes 2})$ is a solution of the braid
equation.

Prominent examples are braided vector spaces of _diagonal type_. It means that
there exist a basis $\\{x_{i}\\}_{i\in\mathbb{I}}$ and a matrix
$\mathfrak{q}=(q_{ij})\in(\Bbbk^{\times})^{\mathbb{I}\times\mathbb{I}}$ such
that the braiding is

$\displaystyle c^{\mathfrak{q}}:$ $\displaystyle V\otimes V\to V\otimes V,$
$\displaystyle c^{\mathfrak{q}}(x_{i}\otimes x_{j})$
$\displaystyle=q_{ij}\,x_{j}\otimes x_{i},\,i,j\in\mathbb{I}.$

The matrix $\mathfrak{q}$ is called the braiding matrix. The information of
$\mathfrak{q}$ is encoded in the associated Dynkin diagram. This is a labelled
graph with $\theta$ vertices, each of them labelled with $q_{ii}$, and a edge
between vertices $i$ and $j$ if $\widetilde{q}_{ij}:=q_{ij}q_{ji}\neq 1$,
labelled with this scalar. Different braiding matrices can have the same
Dynkin diagram: the associated Nichols algebras are not isomorphic but
equivalent in some sense as we will see in §2.4.

Nichols algebras of diagonal type depend only on $\mathfrak{q}$, so we will
denote it by $\mathscr{B}_{\mathfrak{q}}$; we denote accordingly
$\mathcal{J}_{\mathfrak{q}}$ to the defining ideal of
$\mathscr{B}_{\mathfrak{q}}$. In addition, $(V,c)$ is realized as a Yetter-
Drinfeld module over $\Bbbk\mathbb{Z}^{\theta}$ in a canonical way: We set the
coaction on $V$ given by $\rho(x_{i})=\alpha_{i}\otimes x_{i}$ and the action
given by $\alpha_{i}\cdot x_{j}=q_{ij}\,x_{j}$, $i,j\in\mathbb{I}$. From here
we can deduce that $\mathscr{B}_{\mathfrak{q}}$ is
$\mathbb{Z}^{\theta}$-graded, where each $x_{i}$ has degree $\alpha_{i}$.

A pre-Nichols algebra of $V$ is a braided Hopf algebra $\mathscr{B}$ which is
the quotient of braided Hopf algebras of $T(V)$ by an
$\mathbb{N}_{0}$-homogeneous Hopf ideal $\mathcal{J}=\oplus_{n\geq
2}\mathcal{J}^{n}$. Thus $\mathcal{J}\subseteq\mathcal{J}(V)$ and there exist
canonical graded Hopf algebra epimorphisms

$T(V)\twoheadrightarrow\mathscr{B}\twoheadrightarrow\mathscr{B}(V)$

whose restriction to degree one is $\operatorname{id}_{V}$. The set of pre-
Nichols algebras of $V$ becomes a poset $\mathfrak{Pre}(V)$, where
$\mathscr{B}_{1}\leq\mathscr{B}_{2}$ if $\operatorname{id}_{V}$ induces an
epimorphism of braided Hopf algebras
$\mathscr{B}_{1}\twoheadrightarrow\mathscr{B}_{2}$. This poset has maximum and
minimum elements, i.e. $\mathscr{B}(V)$ and $T(V)$.

Assume that $\operatorname{GKdim}\mathscr{B}(V)<\infty$. The subset
$\mathfrak{Pre}_{\operatorname{fGK}}(V)$ of pre-Nichols algebras of $V$ with
finite Gelfand-Kirillov dimension is a subposet with maximum element
$\mathscr{B}(V)$. In case it admits a minimum $\widehat{\mathscr{B}}(V)$, we
will say that $\widehat{\mathscr{B}}(V)$ is the _eminent pre-Nichols algebra_
of $V$. The existence and computation of eminent pre-Nichols algebras
$\widehat{\mathscr{B}}(V)$ reduces the problem of finding the set of all pre-
Nichols algebras of $V$ with finite $\operatorname{GKdim}$ to the problem of
finding quotients of $\widehat{\mathscr{B}}(V)$. As we will recall in §2.3,
eminent pre-Nichols algebras exist for most $V$ of diagonal type.

###  2.2. Central extensions of braided Hopf algebras

As we want to study posets of pre-Nichols algebras, we have to deal with
extensions of connected Hopf algebras in ${}^{H}_{H}\mathcal{YD}$. Motivated
by [T, Theorem 3.2] and [A+, Proposition 3.6] we can state the following
correspondence between Hopf ideals and normal coideal subalgebras in
${}^{H}_{H}\mathcal{YD}$.

######  Proposition 2.1.

Let $R$ be a connected Hopf algebra in ${}^{H}_{H}\mathcal{YD}$. The
assignements

$\displaystyle\mathtt{A}\mapsto\mathtt{I}(\mathtt{A}):=R/R\mathtt{A}^{+},$
$\displaystyle I\mapsto\mathtt{A}(I):={}^{\operatorname{co}R/I}R,$

give a bijection between the set of normal right coideal subalgebras
$\mathtt{A}$ of $R$ and the set of Hopf ideals $I$ of $R$.

###### Proof.

By [A+, Proposition 3.6 (c)] we have a bijection between the set of right
coideal subalgebras $\mathtt{A}$ of $R$ and the set of coideals $I$ of $R$
whose quotient map is of $R$-modules. Now $\mathtt{A}$ is normal if and only
if $\mathtt{I}(\mathtt{A})$ is an ideal: the proof is by direct computation,
analogous to that in [T, Proposition 1.4]. ∎

Recall that an _extension of braided Hopf algebras_ [AN, §2.5] is a sequence
of morphisms of braided Hopf algebras
$\Bbbk\rightarrow\mathtt{A}\overset{\iota}{\to}\mathtt{C}\overset{\pi}{\to}\mathtt{B}\rightarrow\Bbbk$
such that $\iota$ is injective, $\pi$ is surjective,
$\ker\pi=\mathtt{C}\iota(\mathtt{A}^{+})$ and
$\mathtt{A}=\mathtt{C}^{\,\operatorname{co}\pi}$. For the sake of simplicity
we just write
$\mathtt{A}\overset{\iota}{\hookrightarrow}\mathtt{C}\overset{\pi}{\twoheadrightarrow}\mathtt{B}$.
If $\mathtt{C}$ is connected then any surjective braided Hopf algebra morphism
$\mathtt{C}\overset{\pi}{\twoheadrightarrow}\mathtt{B}$ gives an extension by
choosing $\mathtt{A}=\mathtt{C}^{\,\operatorname{co}\pi}$, see [A+, 3.6].

We say that an extension is _central_ if $\mathtt{A}$ is contained in the
center of $\mathtt{C}$.

In case that $\mathtt{A}$, $\mathtt{B}$, $\mathtt{C}$ are
$\mathbb{N}_{0}^{\theta}$-graded with finite-dimensional homogeneous
components and the maps $\iota$, $\pi$ preserve the
$\mathbb{N}_{0}^{\theta}$-grading, the Hilbert series of these algebras
satisfy the equality
${\mathcal{H}}_{\mathtt{C}}={\mathcal{H}}_{\mathtt{A}}{\mathcal{H}}_{\mathtt{B}}$,
cf. [ACSa, Lemma 2.4].

Now we deal with central extensions of Hopf algebras in
${}^{H}_{H}\mathcal{YD}$ whose right hand side term is a Nichols algebra.

######  Proposition 2.2.

Let
$\mathcal{Z}{\hookrightarrow}\mathscr{B}\overset{\pi}{\twoheadrightarrow}\mathscr{B}(V)$
be a central extension of connected graded braided Hopf algebras, where
$\mathscr{B}=\oplus_{n\geq 0}\mathscr{B}^{n}$ is a pre-Nichols algebra of $V$,
i.e. $\mathscr{B}^{1}=V$.

  1. (i)

The assignment $I\mapsto\mathscr{B}I$ is a bijective correspondence between
graded Hopf ideals of $\mathcal{Z}$ and graded Hopf ideals of $\mathscr{B}$
generated in degree $\geq 2$.

  2. (ii)

Assume that $\mathscr{B}$, $\mathscr{B}(V)$ are
$\mathbb{N}_{0}^{\theta}$-graded and $\pi$ preserves the
$\mathbb{N}_{0}^{\theta}$-grading, so $\mathcal{Z}$ is also
$\mathbb{N}_{0}^{\theta}$-graded. Let $I$ be a
$\mathbb{N}_{0}^{\theta}$-graded Hopf ideal of $\mathcal{Z}$,
$\mathscr{B}^{\prime}:=\mathscr{B}/\mathscr{B}I$,
$\mathtt{Z}=\mathcal{Z}^{\,\operatorname{co}\mathcal{Z}/I}$. Then
${\mathcal{H}}_{\mathscr{B}}={\mathcal{H}}_{\mathscr{B}^{\prime}}{\mathcal{H}}_{\mathtt{Z}}$.

###### Proof.

(i) Let $A=\oplus_{n\geq 0}A^{n}$ be a graded right coideal subalgebra of
$\mathscr{B}$ such that $A^{1}=0$. We claim that $A^{n}\subset\mathcal{Z}$ for
all $n\geq 0$: The proof is by induction on $n$. For $n=0$, $A^{0}=\Bbbk
1\subset\mathcal{Z}$. Now assume that $A^{k}\subseteq\mathcal{Z}$ for all
$k\leq n$. For each $x\in A^{n+1}$,

$\displaystyle\Delta(x)-x\otimes 1-1\otimes
x\in\oplus_{i=1}^{n}A_{i}\otimes\mathscr{B}_{n+1-i}.$

Thus, by inductive hypothesis,
$(\pi\otimes\operatorname{id})\Delta(x)=\pi(x)\otimes 1+1\otimes x$, so

$\displaystyle\Delta(\pi(x))=(\pi\otimes\pi)\Delta(x)=\pi(x)\otimes
1+1\otimes\pi(x).$

Hence $\pi(x)$ is a primitive element of $\mathscr{B}(V)$ in degree $n+1\geq
2$, so $\pi(x)=0$. Thus
$x\in{}^{\operatorname{co}\pi}\mathscr{B}=\mathcal{Z}$, and the inductive step
follows. Therefore, the set of graded right coideal subalgebras of
$\mathscr{B}$ such that $A^{1}=0$ is the set of graded coideal subalgebras of
$\mathcal{Z}$, and all these coideal subalgebras are normal since
$\mathcal{Z}$ is central.

Using Proposition 2.1, we obtain a bijective correspondence between the
corresponding Hopf ideals; that is, between the graded Hopf ideals of
$\mathcal{Z}$ and the graded Hopf ideals of $\mathscr{B}$ generated in degree
$\geq 2$.

(ii) Notice that
${\mathcal{H}}_{\mathcal{Z}}={\mathcal{H}}_{\mathcal{Z}^{\prime}}{\mathcal{H}}_{\mathtt{Z}}$
and
$\mathcal{Z}^{\prime}{\hookrightarrow}\mathscr{B}^{\prime}\overset{\pi}{\twoheadrightarrow}\mathscr{B}(V)$
is also a central extension of connected graded braided Hopf algebras. From
this exact sequence and the one of $\mathscr{B}$ we have that
${\mathcal{H}}_{\mathscr{B}}={\mathcal{H}}_{\mathcal{Z}}{\mathcal{H}}_{\mathscr{B}(V)}$
and
${\mathcal{H}}_{\mathscr{B}^{\prime}}={\mathcal{H}}_{\mathcal{Z}/I}{\mathcal{H}}_{\mathscr{B}(V)}$.
Thus the statement follows by putting together the three equalities involving
Hilbert series. ∎

###  2.3. Distinguished pre-Nichols algebras

Let $(V,c^{\mathfrak{q}})$ be a braided vector space of diagonal type with
$\operatorname{GKdim}\mathscr{B}_{\mathfrak{q}}<\infty$. By [AnG], this means
that $\mathscr{B}_{\mathfrak{q}}$ has a PBW basis with a finite set of
homogeneous generators; in other words, $\mathfrak{q}$ belongs to the lists in
[H2]. The set $\varDelta_{+}^{\mathfrak{q}}$ of _positive roots_ of
$\mathfrak{q}$ consist of the $\mathbb{N}_{0}^{\theta}$-degrees of these
generators, which is independent of the chosen PBW basis [AA, H1]. The _set of
roots_ of $\mathfrak{q}$ is
$\varDelta^{\mathfrak{q}}:=\varDelta_{+}^{\mathfrak{q}}\cup(-\varDelta_{+}^{\mathfrak{q}})$.

Assume from now on that $|\varDelta_{+}^{\mathfrak{q}}|<\infty$. For
$\alpha=(a_{1},\cdots,a_{\theta}),\beta=(b_{1},\cdots,b_{\theta})\in\mathbb{N}_{0}^{\theta}$
set

$\displaystyle q_{\alpha\beta}$
$\displaystyle:=\prod_{i,j=1}^{\theta}q_{ij}^{a_{i}b_{j}},$ $\displaystyle
N_{\beta}:=\operatorname{ord}q_{\beta\beta}\in\mathbb{N}\cup\\{\infty\\}.$

A total order $\succ$ on $\varDelta_{+}^{\mathfrak{q}}$ is _convex_ if for all
$\alpha\succ\beta\in\varDelta_{+}^{\mathfrak{q}}$ such that
$\alpha+\beta\in\varDelta_{+}^{\mathfrak{q}}$ we have that
$\alpha\succ\alpha+\beta\succ\beta$. For each convex order
$\beta_{1}\succ\cdots\succ\beta_{M}$ there exists a PBW basis with set of
generators $x_{\beta}$, $\beta\in\varDelta_{+}^{\mathfrak{q}}$, with each
$x_{\beta}$ of degree $\beta$. More explicitly, the set

$\displaystyle x_{\beta_{1}}^{n_{1}}\cdots x_{\beta_{M}}^{n_{M}},$
$\displaystyle 0\leq n_{i}<N_{\beta_{i}},$

is a basis of $\mathscr{B}_{\mathfrak{q}}$, see e.g. [An2, HY]. Thus, the
Hilbert series of $\mathscr{B}_{\mathfrak{q}}$ is

$\displaystyle{\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}(t)=\left(\prod_{\beta\in\varDelta_{+}^{\mathfrak{q}}:N_{\beta}=\infty}\frac{1}{1-t^{\beta}}\right)\left(\prod_{\beta\in\varDelta_{+}^{\mathfrak{q}}:N_{\beta}<\infty}\frac{1-t^{N_{\beta}\beta}}{1-t^{\beta}}\right).$

Next we move to pre-Nichols algebras of diagonal type and connected Dynkin
diagram. Between the Nichols algebras $\mathscr{B}_{\mathfrak{q}}$ with finite
$\operatorname{GKdim}$, some of them are infinite-dimensional. By [C] for most
of these $\mathfrak{q}$ the Nichols algebra $\mathscr{B}_{\mathfrak{q}}$ is
the unique pre-Nichols algebra with finite $\operatorname{GKdim}$, and for the
remaining ones there exist exactly one proper pre-Nichols algebra
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ with finite $\operatorname{GKdim}$
(which is then eminent).

Thus we can restrict to the problem of determining pre-Nichols algebras of
finite $\operatorname{GKdim}$ when $\dim\mathscr{B}_{\mathfrak{q}}<\infty$.
This is equivalent to the fact that $N_{\beta}<\infty$ for all
$\beta\in\varDelta_{+}^{\mathfrak{q}}$. In this case, there exists a pre-
Nichols algebra with finite $\operatorname{GKdim}$, called the _distinguished
pre-Nichols algebra_ $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ [An3]. This pre-
Nichols algebra is the quotient
$\widetilde{\mathscr{B}}_{\mathfrak{q}}=T(V)/\mathcal{I}_{\mathfrak{q}}$ by
the ideal $\mathcal{I}_{\mathfrak{q}}$ generated by the set of defining
relations in [An1, Theorem 3.1] adding a few extra relations and removing
relations of the form $x_{\alpha}^{N_{\alpha}}$ for
$\alpha\in\mathfrak{O}^{\mathfrak{q}}_{+}$. Here,

$\displaystyle\mathfrak{O}^{\mathfrak{q}}:=\left\\{\alpha\in\varDelta^{\mathfrak{q}}:q_{\alpha\beta}q_{\beta\alpha}\in\\{q_{\alpha\alpha}^{n}:n\in\mathbb{Z}\\}\text{
for all }\beta\in\mathbb{Z}^{\theta}\right\\}$

is the set of roots of Cartan type, cf. [An1, AAR], and
$\mathfrak{O}^{\mathfrak{q}}_{+}:=\mathfrak{O}^{\mathfrak{q}}\cap\mathbb{N}_{0}^{\theta}$.
By [An3] the set

$\displaystyle x_{\beta_{1}}^{n_{1}}\cdots x_{\beta_{M}}^{n_{M}},$
$\displaystyle 0\leq n_{i}<\widetilde{N}_{\beta_{i}},$

is a basis of $\widetilde{\mathscr{B}}_{\mathfrak{q}}$, where
$\widetilde{N}_{\beta}:=\begin{cases}\infty,&\beta\in\mathfrak{O}^{\mathfrak{q}}_{+},\\\
N_{\beta},&\beta\notin\mathfrak{O}^{\mathfrak{q}}_{+}.\end{cases}$ Thus, the
Hilbert series of $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is

$\displaystyle{\mathcal{H}}_{\widetilde{\mathscr{B}}_{\mathfrak{q}}}(t)=\left(\prod_{\beta\in\mathfrak{O}^{\mathfrak{q}}_{+}}\frac{1}{1-t^{\beta}}\right)\left(\prod_{\beta\in\varDelta_{+}^{\mathfrak{q}}-\mathfrak{O}^{\mathfrak{q}}_{+}}\frac{1-t^{N_{\beta}\beta}}{1-t^{\beta}}\right).$

Let $\mathcal{Z}_{\mathfrak{q}}$ be the subalgebra of
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$ generated by
$z_{\alpha}:=x_{\alpha}^{N_{\alpha}}$,
$\alpha\in\mathfrak{O}^{\mathfrak{q}}_{+}$. By [An3] there exists an extension
of braided Hopf algebras

(2.1)
$\displaystyle\mathcal{Z}_{\mathfrak{q}}{\hookrightarrow}\widetilde{\mathscr{B}}_{\mathfrak{q}}\overset{\pi}{\twoheadrightarrow}\mathscr{B}_{\mathfrak{q}},$

i.e.
$\mathcal{Z}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}^{\operatorname{co}\pi}$.
Moreover $\mathcal{Z}_{\mathfrak{q}}$ is a $q$-central Hopf subalgebra of
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$, which is a $q$-polynomial algebra in
variables $z_{\alpha}$, $\alpha\in\mathfrak{O}^{\mathfrak{q}}_{+}$. Now set

$\displaystyle\underline{\beta}$
$\displaystyle:=N_{\beta}\beta,\,\beta\in\mathfrak{O}^{\mathfrak{q}},$
$\displaystyle\underline{\mathfrak{O}}^{\mathfrak{q}}$
$\displaystyle:=\\{\underline{\beta}:\beta\in\mathfrak{O}^{\mathfrak{q}}\\},$
$\displaystyle\underline{\mathfrak{O}}^{\mathfrak{q}}_{+}$
$\displaystyle:=\mathfrak{O}^{\mathfrak{q}}\cap\mathbb{N}_{0}^{\theta}.$

By [AAR, Theorem 3.7], $\underline{\mathfrak{O}}^{\mathfrak{q}}$ is a root
system (in the classical sense), with basis

$\displaystyle\varPi^{\mathfrak{q}}:=\\{\underline{\gamma}\in\underline{\mathfrak{O}}^{\mathfrak{q}}_{+}:\underline{\gamma}\neq\underline{\alpha}+\underline{\beta}\text{
for all
}\underline{\alpha},\underline{\beta}\in\underline{\mathfrak{O}}^{\mathfrak{q}}_{+}\\}.$

With the notation above, the Hilbert series of
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$ can be also written as

(2.2) $\displaystyle{\mathcal{H}}_{\widetilde{\mathscr{B}}_{\mathfrak{q}}}(t)$
$\displaystyle={\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}(t){\mathcal{H}}_{\mathcal{Z}_{\mathfrak{q}}}(t)={\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}(t)\left(\prod_{\underline{\beta}\in\underline{\mathfrak{O}}^{\mathfrak{q}}_{+}}\frac{1}{1-t^{\underline{\beta}}}\right).$

######  Example 2.3.

Fix $A=(a_{ij})$ a finite Cartan matrix, $(d_{i})\in\mathbb{N}^{\theta}$
minimal such that $(d_{i}a_{ij})$ is symmetric and $q\in\Bbbk$ is a root of
unity of order $N$ coprime with all $a_{ij}$’s. Let $\mathfrak{q}=(q_{ij})$,
where $q_{ij}=q^{d_{i}a_{ij}}$. In this case $\mathfrak{q}$ is of _Cartan
type_ and $\widetilde{\mathscr{B}}_{\mathfrak{q}}\simeq
U_{q}^{+}(\mathfrak{g})$, where $\mathfrak{g}$ is the (finite-dimensional)
semisimple Lie algebra with $A$ as Cartan matrix. Moreover

$\displaystyle N_{\beta}$ $\displaystyle=N\text{ for all
}\beta\in\Delta^{\mathfrak{q}}_{+},$
$\displaystyle\mathfrak{O}^{\mathfrak{q}}_{+}$
$\displaystyle=\Delta^{\mathfrak{q}}_{+},$ $\displaystyle\text{thus
}\underline{\mathfrak{O}}^{\mathfrak{q}}_{+}$
$\displaystyle=\\{N\beta:\beta\in\Delta^{\mathfrak{q}}_{+}\\}.$

By [DKP] $\mathcal{Z}_{\mathfrak{q}}$ is the algebra of functions of the
unipotent algebraic group with Lie algebra $\mathfrak{n}_{+}$ (the positive
part of $\mathfrak{g}$, for a fixed Borel subalgebra).

From the Hilbert series we check that
$\operatorname{GKdim}\widetilde{\mathscr{B}}_{\mathfrak{q}}=|\mathfrak{O}_{+}^{\mathfrak{q}}|<\infty$.
Thus we may wonder if $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is the eminent
pre-Nichols algebra of $\mathfrak{q}$. This is mostly the case. More
precisely:

######  Theorem 2.4.

[ASa, ACSa, ACSa2] Let $(V,\mathfrak{q})$ be a braided vector space of
diagonal type such that $\dim\mathscr{B}_{\mathfrak{q}}<\infty$ and the Dynkin
diagram is connected. Then the distinguished pre-Nichols algebra
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$ is eminent, except in the following
cases:

  1. (A)

Cartan $A_{\theta}$ or $D_{\theta}$ with $q=-1$,

  2. (B)

$A_{2}$ with $q\in\mathbb{G}_{3}^{\prime}$,

  3. (C)

${\bf A}_{3}(q|\\{2\\})$ or ${\bf A}_{3}(q|\\{1,2,3\\})$, with
$q\in\mathbb{G}_{\infty}$,

  4. (D)

$\mathfrak{g}(2,3)$ with any of the following Dynkin diagram

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If $\mathfrak{q}$ is as in (A), it is not even known whether the eminent pre-
Nichols algebra exists. But for the other cases, there is an answer: the
eminent pre-Nichols algebra is a $q$-central extension of the distinguished
pre-Nichols algebra, as we will describe below.

######  Theorem 2.5.

Let $(V,\mathfrak{q})$ be a braided vector space of diagonal type such that
$\dim\mathscr{B}_{\mathfrak{q}}<\infty$.

  1. (a)

[ASa] If $\mathfrak{q}$ is of type $A_{2}$ with $q\in\mathbb{G}_{3}^{\prime}$,
then the eminent pre-Nichols algebra of $\mathfrak{q}$ is

$\displaystyle\widehat{\mathscr{B}}_{\mathfrak{q}}=T(V)/\langle
x_{1112},x_{2112},x_{2221},x_{1221}\rangle,$

and $\operatorname{GKdim}\widehat{\mathscr{B}}_{\mathfrak{q}}=5$. Let
$Z_{\mathfrak{q}}$ be the subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$
generated by $x_{112}$, $x_{221}$. There is a $\mathbb{N}_{0}^{2}$-homogeneous
$q$-central extension of braided Hopf algebras
$Z_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}$,
and $Z_{\mathfrak{q}}$ is a $q$-polynomial algebra in variables $x_{112}$ and
$x_{221}$.

  2. (b)

[ACSa] If $\mathfrak{q}$ is of type ${\bf A}_{3}(q|\\{2\\})$ with
$q\in\mathbb{G}_{N}^{\prime}$, then

$\displaystyle\widehat{\mathscr{B}}_{\mathfrak{q}}=T(V)/\langle
x_{2}^{2},x_{13},x_{112},x_{332}\rangle$

is the eminent pre-Nichols algebra of $\mathfrak{q}$, and
$\operatorname{GKdim}\widehat{\mathscr{B}}_{\mathfrak{q}}=3$. Let
$Z_{\mathfrak{q}}$ be the subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$
generated by $x_{u}:=[x_{123},x_{2}]_{c}$. There is a
$\mathbb{N}_{0}^{3}$-homogeneous $q$-central extension of braided Hopf
algebras
$Z_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}$,
and $Z_{\mathfrak{q}}$ is a $q$-polynomial algebra in $x_{u}$.

  3. (c)

[ACSa] If $\mathfrak{q}$ is of type ${\bf A}_{3}(q|\\{1,2,3\\})$, with
$q\in\mathbb{G}_{N}^{\prime}$, then

$\displaystyle\widehat{\mathscr{B}}_{\mathfrak{q}}=T(V)/\langle
x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{213},[x_{123},x_{2}]_{c}\rangle$

is the eminent pre-Nichols algebra of $\mathfrak{q}$, and
$\operatorname{GKdim}\widehat{\mathscr{B}}_{\mathfrak{q}}=3$. Let
$Z_{\mathfrak{q}}$ be the subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$
generated by $x_{13}$. There is a $\mathbb{N}_{0}^{3}$-homogeneous $q$-central
extension of braided Hopf algebras
$Z_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}$,
and $Z_{\mathfrak{q}}$ is a $q$-polynomial algebra in $x_{13}$.

  4. (d)

[ACSa2] If $\mathfrak{q}$ is of type $\mathfrak{g}(2,3)$ with diagram
$\textstyle{\overset{-1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi}$$\textstyle{\overset{-1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\xi}$$\textstyle{\overset{-1}{\circ}}$, then set

$\displaystyle x_{u}$ $\displaystyle:=[[x_{12},x_{123}]_{c},x_{2}]_{c},$
$\displaystyle x_{v}$ $\displaystyle:=[[x_{123},x_{23}]_{c},x_{2}]_{c},$

Then the eminent pre-Nichols algebra is

$\displaystyle\widehat{\mathscr{B}}_{\mathfrak{q}}=$ $\displaystyle
T(V)/\langle
x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{13},[x_{1},x_{u}]_{c},[x_{1},x_{v}]_{c},[x_{u},x_{3}]_{c},[x_{v},x_{3}]_{c}\rangle,$

and $\operatorname{GKdim}\widehat{\mathscr{B}}_{\mathfrak{q}}=6$. Let
$Z_{\mathfrak{q}}$ be the subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$
generated by $x_{u}$ and $x_{v}$. There is a $\mathbb{N}_{0}^{3}$-homogeneous
$q$-central extension of braided Hopf algebras
$Z_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}$,
and $Z_{\mathfrak{q}}$ is a $q$-polynomial algebra in variables $x_{u}$,
$x_{v}$.

  5. (e)

[ACSa2] If $\mathfrak{q}$ is of type $\mathfrak{g}(2,3)$ with diagram
$\textstyle{\overset{-1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{2}}$$\textstyle{\overset{\xi}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$$\scriptstyle{\xi}$$\textstyle{\overset{-1}{\circ}}$, then set

$\displaystyle x_{u}$ $\displaystyle:=[[x_{123},x_{2}]_{c},x_{2}]_{c},$
$\displaystyle x_{v}$ $\displaystyle:=[x_{123},x_{12^{2}3}]_{c},$

Then the eminent pre-Nichols algebra is

$\displaystyle\widehat{\mathscr{B}}_{\mathfrak{q}}=$ $\displaystyle
T(V)/\langle
x_{1}^{2},x_{3}^{2},x_{13},[x_{223},x_{23}]_{c},x_{221},x_{2223},[x_{v},x_{3}]_{c},[x_{12^{3}3^{2}},x_{2}]_{c},[x_{12^{3}3^{2}},x_{3}]_{c}\rangle.$

and $\operatorname{GKdim}\widehat{\mathscr{B}}_{\mathfrak{q}}=6$. Let
$Z_{\mathfrak{q}}$ be the subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$
generated by $x_{u}$ and $x_{v}$. There is a $\mathbb{N}_{0}^{3}$-homogeneous
$q$-central extension of braided Hopf algebras
$Z_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}$,
and $Z_{\mathfrak{q}}$ is a $q$-polynomial algebra in variables $x_{u}$,
$x_{v}$.

We denote by $\mathfrak{Pre}(\mathfrak{q})$ and
$\mathfrak{Pre}_{\operatorname{fGK}}(\mathfrak{q})$ the corresponding posets
of pre-Nichols algebras when $V$ is of diagonal type with matrix
$\mathfrak{q}$. Let $\mathfrak{Pre}^{\operatorname{gr}}(\mathfrak{q})$ be the
subposet of those pre-Nichols algebras which are
$\mathbb{N}_{0}^{\theta}$-graded, and
$\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})=\mathfrak{Pre}_{\operatorname{fGK}}(\mathfrak{q})\cap\mathfrak{Pre}^{\operatorname{gr}}(\mathfrak{q})$,
i.e. the subposet of those pre-Nichols algebras with finite
$\operatorname{GKdim}$ which are $\mathbb{N}_{0}^{\theta}$-graded. The main
result of this work is the characterization of
$\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$ for
all those cases where the eminent pre-Nichols algebra of all connected
components of $\mathfrak{q}$ is known.

######  Remark 2.6.

The main reason behind the restriction to the subposet of
$\mathbb{N}_{0}^{\theta}$-graded pre-Nichols algebras is that this is the set
of all pre-Nichols algebras that can be realised in the category of Yetter-
Drinfeld modules for any principal realization of $V$ over a group $\Gamma$.
Recall that a principal realization of a braided vector space of diagonal type
means that there exists a basis $(x_{i})$ of $V$, elements $g_{i}\in
Z(\Gamma)$ and $\chi_{i}\in\widehat{\Gamma}$ such that the coaction of $x_{i}$
is given by $g_{i}$ and $g\cdot x_{i}=\chi_{i}(g)x_{i}$ for all $g\in\Gamma$,
so $q_{ij}=\chi_{j}(g_{i})$ for all $i,j\in\mathbb{I}$.

For example, let $\mathfrak{q}=(q_{ij})$ be such that $\widetilde{q}_{ij}=1$,
$q_{ii}\in\mathbb{G}_{N_{i}}^{\prime}$, for all $i\neq j\in\mathbb{I}$, where
$N_{i}\in\mathbb{N}_{0}$. The distinguished pre-Nichols algebra is the so
called quantum plane, $\widehat{\mathscr{B}}_{\mathfrak{q}}=T(V)/\langle
x_{ij}|i<j\in\mathbb{I}\rangle$. Fix $i\neq j$ such that $N_{i}=N_{j}$ and
$a\in\Bbbk^{\times}$. As $x_{i}^{N_{i}},x_{j}^{N_{j}}$ are primitive elements
of the same degree, the quotient

$\displaystyle\mathscr{B}=T(V)/\langle
x_{k\ell},\,k<\ell\in\mathbb{I};\,x_{i}^{N_{i}}-a\,x_{j}^{N_{j}}\rangle=\widehat{\mathscr{B}}_{\mathfrak{q}}/\langle
x_{i}^{N_{i}}-a\,x_{j}^{N_{j}}\rangle$

is a pre-Nichols algebra of $\mathfrak{q}$ of finite $\operatorname{GKdim}$.
Fix also a principal realization over a group $\Gamma$: If either
$g_{i}^{N_{i}}\neq g_{j}^{N_{j}}$ or else
$\chi_{i}^{N_{i}}\neq\chi_{j}^{N_{j}}$, then $\mathscr{B}$ is not an object in
${}^{\Bbbk\Gamma}_{\Bbbk\Gamma}\mathcal{YD}$.

###  2.4. Twist equivalence and pre-Nichols algebras

Recall that two braiding matrices $\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}}$
and $\mathfrak{p}=(p_{ij})_{i,j\in\mathbb{I}}$ are _twist-equivalent_ if
$q_{ii}=p_{ii}$ and $q_{ij}q_{ji}=p_{ij}p_{ji}$ for all $i\neq
j\in\mathbb{I}$, cf. [AS2, Definition 3.8]. We write
$\mathfrak{q}\sim\mathfrak{p}$. Notice that two matrices are twist-equivalent
if and only if their Dynkin diagrams coincide.

Let $(V,c)$ and $(W,d)$ be the braided vector spaces of diagonal type with
matrices $\mathfrak{q}$, $\mathfrak{p}$, respectively. We want to relate Hopf
ideals of $T(V)$ and $T(W)$.

######  Proposition 2.7.

Let $\mathfrak{q}=(q_{ij})_{i,j\in\mathbb{I}}$ and
$\mathfrak{p}=(p_{ij})_{i,j\in\mathbb{I}}$ be twist-equivalent matrices.

  1. (i)

There exists an isomorphism of posets
$\Psi:\mathfrak{Pre}^{\operatorname{gr}}(\mathfrak{q})\to\mathfrak{Pre}^{\operatorname{gr}}(\mathfrak{p})$
preserving the Hilbert series.

  2. (ii)

$\Psi$ restricts to an isomorphism
$\Psi:\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})\to\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{p})$.

###### Proof.

(i) In [AS2, Proposition 3.9] the authors introduce a linear isomorphism
$\psi:\mathscr{B}_{\mathfrak{q}}\to\mathscr{B}_{\mathfrak{p}}$, which is a
coalgebra isomorphism: Let us recall more details about this isomorphism. In
loc. cit. the authors take the group cocycle

$\displaystyle\sigma$
$\displaystyle:\mathbb{Z}^{\theta}\times\mathbb{Z}^{\theta}\to\Bbbk^{\times},$
$\displaystyle\sigma(g_{i},g_{j})$
$\displaystyle=\begin{cases}p_{ij}q_{ij}^{-1},&i\leq j\\\ 1&i>j.\end{cases}$

For any Hopf algebra
$R\in{}^{\Bbbk\mathbb{Z}^{\theta}}_{\Bbbk\mathbb{Z}^{\theta}}\mathcal{YD}$,
this group cocycle induces, up to projection, a Hopf cocycle
$\sigma:R\\#\Bbbk\mathbb{Z}^{\theta}\otimes
R\\#\Bbbk\mathbb{Z}^{\theta}\to\Bbbk$ and we consider the Hopf algebra
$(R\\#\Bbbk\mathbb{Z}^{\theta})_{\sigma}$: The coalgebra structure does not
change and the canonical inclusion
$\Bbbk\mathbb{Z}^{\theta}\hookrightarrow(R\\#\Bbbk\mathbb{Z}^{\theta})_{\sigma}$
and projection
$(R\\#\Bbbk\mathbb{Z}^{\theta})_{\sigma}\twoheadrightarrow\Bbbk\mathbb{Z}^{\theta}$
are still Hopf algebra maps. Thus $(R\\#\Bbbk\mathbb{Z}^{\theta})_{\sigma}$
decomposes as $(R\\#\Bbbk\mathbb{Z}^{\theta})_{\sigma}\simeq
R_{\sigma}\\#\Bbbk\mathbb{Z}^{\theta}$. As stated in [AS2, Lemma 2.12], the
assignment $R\mapsto R_{\sigma}$ takes Hopf algebras in
${}^{\Bbbk\mathbb{Z}^{\theta}}_{\Bbbk\mathbb{Z}^{\theta}}\mathcal{YD}$ to Hopf
algebras in
${}^{\Bbbk\mathbb{Z}^{\theta}}_{\Bbbk\mathbb{Z}^{\theta}}\mathcal{YD}$, where
the coalgebra structure keeps unchanged, so it restricts to graded Hopf
algebras: If $R=\oplus_{n\geq 0}R^{n}$, then $R_{\sigma}=\oplus_{n\geq
0}R_{\sigma}^{n}$, with $R_{\sigma}^{n}=R^{n}$ as vector spaces.

Let $V$, $W$ be as above: as usual we consider
$V,W\in{}^{\Bbbk\mathbb{Z}^{\theta}}_{\Bbbk\mathbb{Z}^{\theta}}\mathcal{YD}$.
Working as in [AS2, Proposition 3.9 & Remark 3.10] we check that
$T(V)_{\sigma}=T(W)$. Any $\mathbb{Z}^{\theta}$-graded pre-Nichols algebra
$\mathscr{B}$ of $V$ is a Hopf alegebra in
${}^{\Bbbk\mathbb{Z}^{\theta}}_{\Bbbk\mathbb{Z}^{\theta}}\mathcal{YD}$: the
Hopf algebra projection $\pi:T(V)\twoheadrightarrow\mathscr{B}$ gives a Hopf
algebra projection
$\pi:T(W)=T(V)_{\sigma}\twoheadrightarrow\mathscr{B}_{\sigma}$, which
preserves the $\mathbb{N}_{0}$-graded components. Thus, we have a map
$\Psi:\mathfrak{Pre}^{\operatorname{gr}}(\mathfrak{q})\to\mathfrak{Pre}^{\operatorname{gr}}(\mathfrak{p})$,
$\Psi(\mathscr{B})=\mathscr{B}_{\sigma}$. Moreover, $\Psi$ is a map of posets,
which has an inverse map given by $\sigma^{-1}$.

(ii) As $\Psi$ preserves the Hilbert series,
$\operatorname{GKdim}\mathscr{B}_{\sigma}=\operatorname{GKdim}\mathscr{B}$ by
[KL, Lemma 6.1]. ∎

Next we want to reduce to the case in which $\mathcal{Z}_{\mathfrak{p}}$ is
central (more than skew central). We can do this reduction up to twist-
equivalence.

######  Lemma 2.8.

Let $\mathfrak{q}$ be a matrix such that
$\dim\mathscr{B}_{\mathfrak{q}}<\infty$. There exists
$\mathfrak{p}\sim\mathfrak{q}$ such that $\mathcal{Z}_{\mathfrak{p}}$ is a
central subalgebra of $\widetilde{\mathscr{B}}_{\mathfrak{p}}$.

###### Proof.

By [AAY, Remark 4.4] we need to find a matrix $\mathfrak{p}\sim\mathfrak{q}$
such that $p_{\alpha_{i}\beta}^{N_{\beta}}=1$ for all $i\in\mathbb{I}$ and all
$\beta\in\varPi_{\mathfrak{p}}$, and it is enough to check it just for one
matrix $\mathfrak{q}$ in each Weyl equivalence class.

If $\mathfrak{q}$ belongs to the one-parameter families, then the existence of
$\mathfrak{p}$ follows by [AAY, Appendix A]. The remaining cases are treated
case-by-case. ∎

Putting together Proposition 2.7 and Lemma 2.8 we can restrict to matrices
$\mathfrak{q}$ such that $\mathcal{Z}_{\mathfrak{q}}$ is a central Hopf
subalgebra. If so, then $\mathcal{Z}_{\mathfrak{q}}$ is the algebra of
functions of an algebraic group. Taking graded duals in the central extension
in (2.1) we get a new extension of braided graded connected Hopf algebras

(2.3)
$\displaystyle\mathscr{B}_{\mathfrak{q}^{t}}{\hookrightarrow}\mathcal{L}_{\mathfrak{q}}\overset{\pi}{\twoheadrightarrow}\mathcal{Z}_{\mathfrak{q}}^{d}.$

Here
$\mathcal{L}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}^{t}}^{d}$ is
called the _Lusztig algebra_ of $\mathfrak{q}$, cf. [AAR]. Notice that
$\mathcal{Z}_{\mathfrak{q}}^{d}$ is cocommutative, thus it is (isomorphic to)
the enveloping algebra of a finite-dimensional nilpotent Lie algebra
$\mathfrak{n}_{\mathfrak{q}}$: By [AAR, Theorem 1.1],
$\mathfrak{n}_{\mathfrak{q}}$ is the positive part of the semisimple Lie
algebra with root system $\underline{\mathfrak{O}}^{\mathfrak{q}}$.

##  3\. Hopf ideals of algebras of functions of unipotent groups

According with the previous section we have to understand the coalgebra
structure of the algebra of functions of the unipotent algebraic group whose
Lie algebra is the positive part $\mathfrak{n}_{+}$ of a semisimple Lie
algebra $\mathfrak{g}$ with root system $\varDelta$, that we denote
$\mathcal{Z}_{\varDelta}$. Let $\vartheta$ be the rank of $\varDelta$,
$\mathbb{I}:=\\{1,\cdots,\vartheta\\}$.

As an algebra,
$\mathcal{Z}_{\varDelta}=\Bbbk[z_{\alpha}|\alpha\in\varDelta_{+}]$. Thus the
set $M(\mathcal{Z}_{\varDelta})$ of monomials in variables $z_{\alpha}$,
$\alpha\in\varDelta_{+}$, is a basis of $\mathcal{Z}_{\varDelta}$; we will say
that a monomial $m\in M(\mathcal{Z}_{\varDelta})$ _appears_ in an element
$w\in\mathcal{Z}_{\varDelta}$ if the coefficient of $m$ in the expression of
$w$ as a linear combination of the elements of $M(\mathcal{Z}_{\varDelta})$ is
not zero. We also fix $M(\mathcal{Z}_{\varDelta})\otimes
M(\mathcal{Z}_{\varDelta})$ and $M(\mathcal{Z}_{\varDelta})\otimes
M(\mathcal{Z}_{\varDelta})\otimes M(\mathcal{Z}_{\varDelta})$ as bases of
$\mathcal{Z}_{\varDelta}\otimes\mathcal{Z}_{\varDelta}$ and
$\mathcal{Z}_{\varDelta}\otimes\mathcal{Z}_{\varDelta}\otimes\mathcal{Z}_{\varDelta}$,
and say that $m\otimes m^{\prime}$, respectively $m\otimes m^{\prime}\otimes
m^{\prime\prime}$, appears in a term, correspondingly.

Let $Q_{+}$ be the lattice of positive roots, i.e. if
$\\{\alpha_{i}|i\in\mathbb{I}\\}$ is the set of simple roots of
$\varDelta_{+}$, then $Q_{+}=\sum_{i=1}^{\vartheta}\mathbb{Z}\alpha_{i}$. Then
$\mathcal{Z}_{\varDelta}$ is a $Q_{+}$-graded Hopf algebra, with each
$z_{\alpha}$ of degree $\alpha$.

Now we relate $Z_{\varDelta}$ and $U(\mathfrak{n}_{+})$. Although this
relation may be well known, we give a proof since the involved tools are
useful in the forthcoming proofs.

######  Lemma 3.1.

  1. (1)

The $Q_{+}$-graded dual of $\mathcal{Z}_{\varDelta}$ is (isomorphic to)
$U(\mathfrak{n}_{+})$, the enveloping algebra of the nilpotent Lie algebra
$\mathfrak{n}_{+}$.

  2. (2)

The subspace $\mathcal{P}(\mathcal{Z}_{\varDelta})$ of primitive elements of
$\mathcal{Z}_{\varDelta}$ has basis $\\{z_{\alpha_{i}}\,|\,i\in\mathbb{I}\\}$.

###### Proof.

For (1), let $U=\oplus_{\beta\in Q_{+}}U_{\beta}$ be the graded dual of
$\mathcal{Z}_{\varDelta}$, with $U_{\beta}$ the component of degree $\beta$.
As $\mathcal{Z}_{\varDelta}$ is connected (as algebra) and commutative, $U$ is
connected (as coalgebra), and cocommutative. Thus $U$ is the enveloping
algebra of its primitive elements.

The subspace $\mathcal{P}(U)$ of primitive elements of $U$ is the space of
derivations $\mathcal{Z}_{\varDelta}\to\Bbbk$, which is canonically identified
with $\mathfrak{n}_{+}$.

For (2), we proceed dually to the previous statement:
$\mathcal{P}(\mathcal{Z}_{\varDelta})$ is the space of derivations
$\partial:U\to\Bbbk$. This space has dimension $\leq\vartheta$, since $U$ is
generated by $e_{i}$, $i\in\mathbb{I}$, as algebra, so any derivation
$\partial:U\to\Bbbk$ is univocally determined by the values $\partial(e_{i})$,
$i\in\mathbb{I}$. On the other hand, each $z_{\alpha_{i}}$ is primitive, so
$\dim\mathcal{P}(\mathcal{Z}_{\varDelta})\geq\vartheta$. From these two
statements, $\dim\mathcal{P}(\mathcal{Z}_{\varDelta})=\vartheta$, and
$\mathcal{P}(\mathcal{Z}_{\varDelta})$ has basis
$\\{z_{\alpha_{i}}\,|\,i\in\mathbb{I}\\}$. ∎

######  Remark 3.2.

For each $\beta\in\varDelta_{+}$ fix a non zero element
$\xi_{\beta}\in\mathfrak{n}_{+}$ of degree $\beta$. Then
$\\{\xi_{\beta}|\beta\in\varDelta_{+}\\}$ is a basis of $\mathfrak{n}_{+}$;
moreover, if $\beta,\gamma\in\varDelta_{+}$ are such that
$\beta+\gamma\in\varDelta_{+}$, then there exists ${d}_{(\beta|\gamma)}\neq 0$
such that $[\xi_{\beta},\xi_{\gamma}]={d}_{(\beta|\gamma)}\xi_{\beta+\gamma}$,
and if $\beta+\gamma\notin\varDelta_{+}$, then $[\xi_{\beta},\xi_{\gamma}]=0$.

Next we introduce the family of subsets of $\varDelta_{+}$ which will
parametrize the graded Hopf ideals of $\mathcal{Z}_{\varDelta}$.

######  Definition 3.3.

Let $A,B\subseteq\varDelta_{+}$.

  1. (i)

We say that $A$ is _compatible_ if for all $\gamma\in A$ and all pair
$\alpha,\beta\in\varDelta_{+}$ such that $\gamma=\alpha+\beta$, then either
$\alpha\in A$ or $\beta\in A$.

  2. (ii)

We say that $B$ is _closed by sums_ if for all $\alpha,\beta\in B$, either
$\alpha+\beta\notin\varDelta_{+}$ or $\alpha+\beta\in B$.

######  Remark 3.4.

$A$ is compatible if and only if $\varDelta_{+}-A$ is closed by sums.

We denote by $\mathcal{P}_{c}(\varDelta_{+})$ the set of all subsets of
$\varDelta_{+}$ that are closed by sums.

######  Remark 3.5.

$\mathcal{P}_{c}(\varDelta_{+})$ is a subposet of $\mathcal{P}(\varDelta_{+})$
with maximum $\varDelta_{+}$ and minimum $\emptyset$, as both trivial subsets
are closed by sums.

To deal with subsets of roots as above we need first a statement on sums of
roots.

######  Lemma 3.6.

Let $m\geq 3$. If $\alpha,\gamma_{1},\cdots,\gamma_{m}\in\varDelta_{+}$ are
such that $\alpha=\sum_{i=1}^{m}\gamma_{i}$, then there exist
$j<k\in\mathbb{I}_{m}$ such that $\gamma_{j}+\gamma_{k}\in\varDelta_{+}$.

###### Proof.

By induction on $\operatorname{ht}(\alpha)\geq 3$. If
$\operatorname{ht}(\alpha)=3$, then either $\alpha=2\alpha_{r}+\alpha_{s}$,
$r\neq s$ or else $\alpha=\alpha_{r}+\alpha_{s}+\alpha_{t}$: In both cases
$m=3$ and the $\gamma_{i}$’s are simple roots: in the first case
$\alpha_{r}+\alpha_{s}\in\varDelta_{+}$, while in the second
$\alpha_{r}+\alpha_{s}\in\varDelta_{+}$ up to permute the sub-indices.

Now assume that the statement holds for roots of height $\leq h$ and take
$\alpha,\gamma_{1},\cdots,\gamma_{m}\in\varDelta_{+}$ such that
$\operatorname{ht}(\alpha)=h+1$, $\alpha=\sum_{i=1}^{m}\gamma_{i}$. As the
Cartan matrix of $\varDelta$ is finite, there exists $\ell\in\mathbb{I}$ such
that $\alpha_{\ell}^{\vee}(\alpha)>0$, hence
$\operatorname{ht}(s_{\ell}(\alpha))<\operatorname{ht}(\alpha)$. We have two
possibilities:

  * •

$\gamma_{i}\neq\alpha_{\ell}$ for all $i\in\mathbb{I}_{m}$. Thus
$s_{\ell}(\gamma_{i}),s_{\ell}(\alpha)\in\varDelta_{+}$, and
$s_{\ell}(\alpha)=\sum_{i=1}^{m}s_{\ell}(\gamma_{i})$. Applying inductive
hypothesis for $s_{\ell}(\alpha)$, there exist $j<k\in\mathbb{I}_{m}$ such
that $s_{\ell}(\gamma_{j})+s_{\ell}(\gamma_{k})\in\varDelta_{+}$. As
$s_{\ell}(\gamma_{j})+s_{\ell}(\gamma_{k})=s_{\ell}(\gamma_{j}+\gamma_{k})$
and $\gamma_{j}+\gamma_{k}\neq\alpha_{\ell}$ since
$\operatorname{ht}(\gamma_{j}+\gamma_{k})\geq 2$, we have that
$\gamma_{j}+\gamma_{k}\in\varDelta_{+}$.

  * •

There exists $i$ such that $\gamma_{i}=\alpha_{\ell}$. Up to permute the
indices we may assume that $\gamma_{m}=\alpha_{\ell}$. We know that
$\alpha-k\alpha_{\ell}\in\varDelta_{+}$ for all $0\leq
k\leq\alpha_{\ell}^{\vee}(\alpha)$; in particular,
$\alpha-\alpha_{\ell}\in\varDelta_{+}$ and
$\alpha-\alpha_{\ell}=\sum_{i=1}^{m-1}\gamma_{i}$. If $m=3$, then
$\gamma_{1}+\gamma_{2}=\alpha-\alpha_{\ell}\in\varDelta_{+}$. If $m>3$, then
we apply inductive hypothesis for $\alpha-\alpha_{\ell}\in\varDelta_{+}$.

In any case, there exist $j<k\in\mathbb{I}_{m}$ such that
$\gamma_{j}+\gamma_{k}\in\varDelta_{+}$. ∎

We write a slightly different version of compatibility which will be useful in
the forthcoming results.

######  Proposition 3.7.

A subset $A$ is compatible if and only if for all $\alpha\in A$ and
$\gamma_{i}\in\varDelta_{+}$ such that $\alpha=\sum_{i=1}^{n}\gamma_{i}$, then
there exists $i$ such that $\gamma_{i}\in A$.

###### Proof.

$(\Leftarrow)$ The case $n=2$ is exactly the definition of compatibility.

$(\Rightarrow)$ By induction on $n$: If $n=2$, then it holds by definition. If
the statement holds for sums of less than $n$ positive roots and
$\alpha=\sum_{i=1}^{n}\gamma_{i}$, $n\geq 3$, with
$\gamma_{i}\in\varDelta_{+}$ then we can apply Lemma 3.6: up to permutation we
may assume that $\gamma^{\prime}:=\gamma_{n-1}+\gamma_{n}\in\varDelta_{+}$.
But $\alpha=\sum_{i=1}^{n-2}\gamma_{i}+\gamma^{\prime}$ and we can apply
inductive hypothesis: either $\gamma_{i}\in A$ for some $i\leq n-2$ (in which
case we are done) or else $\gamma^{\prime}\in A$, which says that either
$\gamma_{n-1}\in A$ or $\gamma_{n}\in A$ by definition of compatibility. ∎

Now we check that subsets closed by sums classify Lie subalgebras of
$\mathfrak{n}_{+}$. More explicitly:

######  Proposition 3.8.

There exists a bijective correspondence between subsets closed by sums and
$Q_{+}$-graded Lie subalgebras of $\mathfrak{n}_{+}$ given by
$B\longmapsto\mathfrak{n}(B):=\oplus_{\beta\in B}\Bbbk\xi_{\beta}$.

###### Proof.

First we check that the map is well defined, i.e. each $\mathfrak{n}(B)$ is a
Lie subalgebra. This follows by Remark 3.2. Indeed, for each pair
$\beta,\gamma\in B$, either $\beta+\gamma\notin\varDelta_{+}$, in which case
$[\xi_{\beta},\xi_{\gamma}]=0$, or else $\beta+\gamma\in B$ and
$[\xi_{\beta},\xi_{\gamma}]={d}_{(\beta|\gamma)}\xi_{\beta+\gamma}\in\mathfrak{n}(B)$.

The map is injective by definition, thus it remains to check that it is
surjective. Let $\mathfrak{n}\subseteq\mathfrak{n}_{+}$ be a $Q_{+}$-graded
Lie subalgebra. As $\mathfrak{n}$ is $Q_{+}$-graded,
$\mathfrak{n}=\oplus_{\beta\in B}\Bbbk\xi_{\beta}$, where $B$ is the subset of
non-trivial homogeneous components. We have to check that $B$ is closed by
sums, which follows again by Remark 3.2 as $\mathfrak{n}$ is a Lie subalgebra.
∎

Next we introduce a family of ideals and quotients of
$\mathcal{Z}_{\varDelta}$ indexed by $\mathcal{P}(\varDelta_{+})$. For each
$B\subseteq\varDelta_{+}$, then we set

(3.1) $\displaystyle I(B)$ $\displaystyle:=\langle
x_{\beta}|\beta\in\varDelta_{+}-B\rangle,$ $\displaystyle Z(B)$
$\displaystyle:=\mathcal{Z}_{\varDelta}/I(B).$

By definition, $Z(B)$ is a polynomial ring in variables (the images of)
$z_{\beta}$, $\beta\in B$.

We are mostly interested in those ideals attached to
$B\in\mathcal{P}_{c}(\varDelta_{+})$ as we will see next.

######  Lemma 3.9.

If $B\in\mathcal{P}_{c}(\varDelta_{+})$, then $I(B)$ is a $Q_{+}$-graded Hopf
ideal.

###### Proof.

Let $A=\varDelta_{+}-B$. It is enough to prove that $\Delta(z_{\alpha})\in
I(B)\otimes\mathcal{Z}_{\varDelta}+\mathcal{Z}_{\varDelta}\otimes I(B)$ for
all $\alpha\in A$. Let $\alpha\in A$: $\Delta(z_{\alpha})-z_{\alpha}\otimes
1-1\otimes z_{\alpha}$ is a linear combination of terms

$\displaystyle z_{\gamma_{1}}\cdots z_{\gamma_{k}}\otimes
z_{\gamma_{k+1}}\cdots z_{\gamma_{m}},$ $\displaystyle 1\leq
k<m,\,\gamma_{i}\in\varDelta_{+},\,\sum_{i=1}^{m}\gamma_{i}=\alpha.$

By Proposition 3.7, for each term $z_{\gamma_{1}}\cdots z_{\gamma_{k}}\otimes
z_{\gamma_{k+1}}\cdots z_{\gamma_{m}}$ there exists $i$ such that
$\gamma_{i}\in A$, thus it belongs to
$I(B)\otimes\mathcal{Z}_{\varDelta}+\mathcal{Z}_{\varDelta}\otimes I(B)$. ∎

We will use the previous result to give a parametrization of graded Hopf
ideals of $\mathcal{Z}_{\varDelta}$.

######  Theorem 3.10.

There exists an anti-isomorphism of posets between
$\mathcal{P}_{c}(\varDelta_{+})$ and the set of $Q_{+}$-graded Hopf ideals of
$\mathcal{Z}_{\varDelta}$ given by

$B\in\mathcal{P}_{c}(\varDelta_{+})\longmapsto Z(B).$

###### Proof.

By Lemma 3.9 the map is well defined. Moreover, the map is an anti-morphism of
posets, and is injective since if $B\neq
B^{\prime}\in\mathcal{P}_{c}(\varDelta_{+})$, then we may assume that there
exists $\alpha\in B^{\prime}-B$, in which case $z_{\alpha}\in
I(B)-I(B^{\prime})$.

Thus, it remains to prove that the map is surjective. For this, we first
notice the following. For each $B\in\mathcal{P}_{c}(\varDelta_{+})$, as $I(B)$
is a Hopf ideal, $I(B)^{\perp}\subseteq U(\mathfrak{n}_{+})$111As before, here
we take the $Q^{+}$-graded dual of $\mathcal{Z}_{\varDelta}$. is a Hopf
subalgebra of $U(\mathfrak{n}_{+})$ by [R, Proposition 5.2.5]. Thus
$I(B)^{\perp}=U(\mathfrak{n})$ for some Lie subalgebra $\mathfrak{n}$ of
$\mathfrak{n}_{+}$. By Proposition 3.8,
$\mathfrak{n}=\mathfrak{n}(B^{\prime})$ for some subset
$B^{\prime}\subseteq\varDelta_{+}$ closed by sums: clearly, $B=B^{\prime}$.
Now take $I$ a $Q_{+}$-graded Hopf ideal of $\mathcal{Z}_{\varDelta}$: again,
the subspace $I^{\perp}\subseteq U(\mathfrak{n}_{+})$ is a Hopf subalgebra of
$U(\mathfrak{n}_{+})$, so $I^{\perp}=U(\mathfrak{n}(C))$ for some subset
$C\subseteq\varDelta_{+}$ closed by sums. But by the previous argument,
$U(\mathfrak{n}(C))=I(B)^{\perp}$ for $B=C$, hence $I=I(B)$. ∎

##  4\. Posets of pre-Nichols algebras of diagonal type

Here we proceed to describe the poset of all graded pre-Nichols algebras of
$\mathfrak{q}$ with finite $\operatorname{GKdim}$. First we assume that the
diagram of $\mathfrak{q}$ is connected and consider two cases: whether the
eminent pre-Nichols algebra is or is not the distinguished one. Later on we
give an approach towards the non-connected case: we have an obstruction to
attack the general case coming from those cases where the eminent pre-Nichols
algebra is not known.

###  4.1. Eminent pre-Nichols algebras which are not distinguished

Throughout this section, $\mathfrak{q}$ will denote a braiding matrix of one
of the following types: Cartan $A_{2}$ with $q\in\mathbb{G}_{3}^{\prime}$,
${\bf A}_{3}(q|\\{2\\})$ with $q\in\mathbb{G}_{\infty}$, ${\bf
A}_{3}(q|\\{1,2,3\\})$ with $q\in\mathbb{G}_{\infty}$, $\mathfrak{g}(2,3)$
with diagram $d_{1}$ or $\mathfrak{g}(2,3)$ with diagram $d_{2}$. Therefore,
the eminent pre-Nichols algebra $\widehat{\mathscr{B}}_{\mathfrak{q}}$ is not
the distinguished pre-Nichols algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$
as described in Theorem 2.5, and we can consider three subalgebras of
coinvariants, associated to the non-trivial canonical projections
$\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\widetilde{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\mathscr{B}_{\mathfrak{q}}$:

(4.1) $\displaystyle\widehat{\mathcal{Z}}_{\mathfrak{q}}$
$\displaystyle=\widehat{\mathscr{B}}_{\mathfrak{q}}^{\operatorname{co}\mathscr{B}_{\mathfrak{q}}},$
$\displaystyle\mathcal{Z}_{\mathfrak{q}}$
$\displaystyle=\widetilde{\mathscr{B}}_{\mathfrak{q}}^{\operatorname{co}\mathscr{B}_{\mathfrak{q}}},$
$\displaystyle Z_{\mathfrak{q}}$
$\displaystyle=\widehat{\mathscr{B}}_{\mathfrak{q}}^{\operatorname{co}\widetilde{\mathscr{B}}_{\mathfrak{q}}}$

By Theorem 2.5, $Z_{\mathfrak{q}}$ is a skew central Hopf subalgebra of
$\widehat{\mathscr{B}}_{\mathfrak{q}}$, and by [An3]
$\mathcal{Z}_{\mathfrak{q}}$ is a skew central Hopf subalgebra of
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$, and both are skew polynomial
algebras: $\underline{\mathfrak{O}^{\mathfrak{q}}}$ is the set of degrees of
the generators of $\mathcal{Z}_{\mathfrak{q}}$. We will see that
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is also polynomial algebras whose
generators are obtained by joining the generators of
$\mathcal{Z}_{\mathfrak{q}}$ and $Z_{\mathfrak{q}}$, and at the same time a
skew central Hopf subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$. To
this end we introduce the set
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ by extending
$\underline{\mathfrak{O}^{\mathfrak{q}}_{+}}$ with the degrees of the
generators of $Z_{\mathfrak{q}}$. Explicitly,

(4.2) $\displaystyle\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$
$\displaystyle:=\begin{cases}\\{1^{3},1^{3}2^{3},2^{3},\mathbf{1^{2}2},\mathbf{12^{2}}\\},&\text{type
}A_{2},q\in\mathbb{G}_{3}^{\prime},\\\
\\{1^{N},3^{N},\mathbf{12^{2}3}\\},&\text{type }{\bf
A}_{3}(q|\\{2\\}),q\in\mathbb{G}_{\infty},\\\
\\{1^{N}2^{N},2^{N}3^{N},\mathbf{13}\\},&\text{type }{\bf
A}_{3}(q|\\{1,2,3\\}),q\in\mathbb{G}_{\infty},\\\
\\{1^{3}2^{3},1^{3}2^{6}3^{3},2^{3}3^{3},1^{6}2^{6}3^{6},\mathbf{1^{2}2^{3}3},\mathbf{12^{3}3^{2}}\\},&\text{type
}\mathfrak{g}(2,3)\text{ with diagram }d_{1},\\\
\\{1^{3}2^{3}3^{3},1^{3}2^{6}3^{3},2^{3},2^{6}3^{6},\mathbf{12^{3}3},\mathbf{1^{2}2^{3}3^{2}}\\},&\text{type
}\mathfrak{g}(2,3)\text{ with diagram }d_{2},\end{cases}$

where bold degrees are those of the generators of $Z_{\mathfrak{q}}$. By
[ACSa, Lemma 2.4],

$\displaystyle\left.\begin{aligned}
{\mathcal{H}}_{\widehat{\mathscr{B}}_{\mathfrak{q}}}&={\mathcal{H}}_{Z_{\mathfrak{q}}}{\mathcal{H}}_{\widetilde{\mathscr{B}}_{\mathfrak{q}}}={\mathcal{H}}_{\widehat{\mathcal{Z}}_{\mathfrak{q}}}{\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}\\\
{\mathcal{H}}_{\widetilde{\mathscr{B}}_{\mathfrak{q}}}&={\mathcal{H}}_{\mathcal{Z}_{\mathfrak{q}}}{\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}\end{aligned}\right\\}$
$\displaystyle\implies$
$\displaystyle{\mathcal{H}}_{\widehat{\mathcal{Z}}_{\mathfrak{q}}}$
$\displaystyle={\mathcal{H}}_{Z_{\mathfrak{q}}}{\mathcal{H}}_{\mathcal{Z}_{\mathfrak{q}}}=\prod_{\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}}\tfrac{1}{1-t^{\beta}}.$

######  Proposition 4.1.

  1. (i)

$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a skew central Hopf subalgebra of
$\widehat{\mathscr{B}}_{\mathfrak{q}}$, and is a skew polynomials ring whose
generators are homogeneous:
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is the set of their
degrees.

  2. (ii)

For each subset
$B\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ closed by
sums, there exists a $\mathbb{Z}^{\theta}$-graded pre-Nichols algebra
$\mathscr{B}(\mathfrak{q},B)$ with Hilbert series

$\displaystyle{\mathcal{H}}_{\mathscr{B}(\mathfrak{q},B)}(t)$
$\displaystyle={\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}(t)\left(\prod_{\alpha\in
B}\tfrac{1}{1-t^{\alpha}}\right).$

  3. (iii)

The map
$\mathcal{P}_{c}(\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}})\to\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$,
$B\mapsto\mathscr{B}(\mathfrak{q},B)$ is an anti-isomorphism of posets.

###### Proof.

We prove the three statements for each case. First assume that $\mathfrak{q}$
is of Cartan type $A_{2}$ with $q\in\mathbb{G}_{3}^{\prime}$: by [ASa, Lemma
4.10], $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a central Hopf subalgebra
and, as an algebra, is a skew polynomial ring in variables $x_{1}^{3}$,
$x_{2}^{3}$, $x_{112}$, $x_{221}$ and $x_{12}^{3}$, so (i) holds. For (ii),
notice that
$\alpha,\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ are
such that
$\alpha+\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ iff
$\alpha+\beta=1^{3}2^{3}$ and either $\\{\alpha,\beta\\}=\\{1^{3},2^{3}\\}$ or
else $\\{\alpha,\beta\\}=\\{1^{2}2,12^{2}\\}$. By [ASa, Lemma 4.10],
$x_{1}^{3}$, $x_{2}^{3}$, $x_{112}$, $x_{221}$ are primitive and the following
formula holds in $\widehat{\mathscr{B}}_{\mathfrak{q}}$:

(4.3) $\displaystyle\Delta(x_{12}^{3})$ $\displaystyle=x_{12}^{3}\otimes
1+1\otimes x_{12}^{3}+(1-q^{2})q_{21}^{3}x_{1}^{3}\otimes
x_{2}^{3}+(q^{2}-q)x_{112}\otimes x_{221}.$

Thus, if $B\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is
closed by sums, then
$\mathscr{B}(\mathfrak{q},B)=\widehat{\mathscr{B}}_{\mathfrak{q}}/\langle
x_{\alpha},\alpha\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}-B\rangle$
is a pre-Nichols algebra with the desired Hilbert series. Reciprocally, if
$\mathscr{B}$ is a $\mathbb{Z}^{\theta}$-graded pre-Nichols algebra, then
either $\mathscr{B}=\widehat{\mathscr{B}}_{\mathfrak{q}}$ or else one of the
primitive elements $x_{1}^{3}$, $x_{2}^{3}$, $x_{112}$, $x_{221}$ is zero,
since the subspace of primitive elements is spanned by these primitive
elements (not in degree one) and $V$. Let $T_{1}$ be the set of degrees of
$\\{x_{1}^{3},x_{2}^{3},x_{112},x_{221}\\}$ of those elements annihilating in
$\mathscr{B}$. If $T_{1}\cap\\{x_{1}^{3},x_{2}^{3}\\}=\emptyset$ or
$T_{1}\cap\\{x_{112},x_{221}\\}=\emptyset$ then $x_{12}^{3}$ cannot be zero in
$\mathscr{B}$, otherwise $x_{12}^{3}$ is primitive and this element may be
zero in $\mathscr{B}$. Thus set $T=T_{1}$ if $x_{12}^{3}\neq 0$ in
$\mathscr{B}$, or $T=T_{1}\cup\\{1^{3}2^{3}\\}$ if $x_{12}^{3}=0$ in
$\mathscr{B}$. Then $B:=T^{c}$ is closed by sums, with
$\mathscr{B}=\mathscr{B}(\mathfrak{q},B)$, and hence (iii) follows.

Next assume that $\mathfrak{q}$ is of type ${\bf A}_{3}(q|\\{2\\})$ with
$q\in\mathbb{G}_{\infty}$. By the proof of [ACSa, Proposition 5.5],
$x_{12^{2}3}$ is skew central. Using the relations verified in that proof,
also $x_{1}^{N}$ and $x_{3}^{N}$ are skew central, and the three elements are
primitive. Thus $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a skew polynomial
algebra with generators $x_{12^{2}3}$, $x_{1}^{N}$ and $x_{3}^{N}$, and (i)
follows. Now every subset of
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is closed by sums
since
$\alpha+\beta\notin\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ for
all $\alpha,\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$.
Thus the proof of (ii) and (iii) are straightforward. The case in which
$\mathfrak{q}$ is of type ${\bf A}_{3}(q|\\{1,2,3\\})$ with
$q\in\mathbb{G}_{\infty}$ follows analogously taking into account the proof of
[ACSa, Proposition 5.6].

If $\mathfrak{q}$ is of type $\mathfrak{g}(2,3)$ with diagram $d_{1}$, then

$\displaystyle x_{u}$ $\displaystyle=[[x_{12},x_{123}]_{c},x_{2}]_{c}$ and
$\displaystyle x_{v}$ $\displaystyle=[[x_{123},x_{23}]_{c},x_{2}]_{c}$

are primitive in $\widehat{\mathscr{B}}_{\mathfrak{q}}$ and skew central by
[ACSa2, Proposition 4.2]. Also,

$\displaystyle[x_{i},x_{23}^{3}]_{c}$ $\displaystyle=[x_{i},x_{12}^{3}]_{c}=0$
$\displaystyle\text{ for all }i\in\\{1,2,3\\}$

Indeed, the proof of that proposition says that
$[x_{1},x_{23}^{3}]_{c}=[x_{3},x_{12}^{3}]_{c}=0$, and the other relations
follow from the quantum Serre relations. Thus $x_{12}^{3}$, $x_{23}^{3}$ are
skew-central in $\widehat{\mathscr{B}}_{\mathfrak{q}}$. As these elements are
primitive in $\widetilde{\mathscr{B}}_{\mathfrak{q}}$,

$\displaystyle\Delta(x_{ij}^{3})$ $\displaystyle\in x_{ij}^{3}\otimes
1+1\otimes x_{ij}^{3}+\langle
x_{u},x_{v}\rangle\otimes\widehat{\mathscr{B}}_{\mathfrak{q}}+\widehat{\mathscr{B}}_{\mathfrak{q}}\otimes\langle
x_{u},x_{v}\rangle.$

Taking into account the $\mathbb{Z}^{\theta}$-degree we check that
$x_{12}^{3}$ and $x_{23}^{3}$ are primitive in
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ as well.

Now we claim that
$B=\\{x_{u}^{a_{1}}x_{v}^{a_{2}}x_{12}^{3a_{3}}x_{12^{2}3}^{3a_{4}}x_{23}^{3a_{5}}x_{123}^{6a_{6}}|a_{i}\in\mathbb{N}_{0}\\}$
is a basis of $\widehat{\mathcal{Z}}_{\mathfrak{q}}$. Indeed these elements
belong to $\widehat{\mathcal{Z}}_{\mathfrak{q}}$, are linearly independent by
[ACSa2, Proposition 4.2] and then they must generate
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ because of the expression of
${\mathcal{H}}_{\widehat{\mathcal{Z}}_{\mathfrak{q}}}$ above. As
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a normal subalgebra of
$\widehat{\mathscr{B}}_{\mathfrak{q}}$, we have that
$\operatorname{ad}_{c}x_{i}(x_{12^{2}3}^{3})\in\widehat{\mathcal{Z}}_{\mathfrak{q}}$
for $i=1,2,3$. But we can check that there are no elements of degrees
$1^{4}2^{6}3^{3}$, $1^{3}2^{7}3^{3}$, $1^{3}2^{6}3^{4}$ in $B$, so
$\operatorname{ad}_{c}x_{i}(x_{12^{2}3}^{3})=0$ for all $i=1,2,3$. A similar
argument shows that $\operatorname{ad}_{c}x_{i}(x_{123}^{6})=0$ for all
$i=1,2,3$, so $x_{12^{2}3}^{3}$ and $x_{123}^{6}$ are skew central. It means
that $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is skew central, and also a
coideal subalgebra on both sides, so it is a central Hopf subalgebra. As
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a skew polynomial algebra with
generators $x_{12}^{3}$, $x_{12^{2}3}^{3}$, $x_{23}^{3}$, $x_{123}^{6}$,
$x_{u}$ and $x_{v}$, (i) follows.

Now $x_{123}^{6}$ is skew primitive since the unique pairs of elements
$(b_{1},b_{2})\in B\times B$ such that the sums of their degrees is
$1^{6}2^{6}3^{6}$ are $(x_{123}^{6},1)$ and $(1,x_{123}^{6})$. Now we compute
$\Delta(x_{12^{2}3}^{3})$. By direct computation,

$\displaystyle\Delta(x_{12^{2}3})$ $\displaystyle=x_{12^{2}3}\otimes
1+1\otimes x_{12^{2}3}+3\zeta q_{23}x_{12}\otimes
x_{23}+(1-\zeta^{2})x_{123}\otimes x_{2}+3x_{1}\otimes x_{23}x_{2}.$

Using the degree again, the possible non-trivial summands of
$\Delta(x_{12^{2}3}^{3})$ are $x_{12}^{3}\otimes x_{23}^{3}$ and $x_{u}\otimes
x_{v}$. Hence we just look at the corresponding degrees and use that
$x_{u}=[x_{12},x_{12^{2}3}]_{c}$ and $x_{v}=[x_{12^{2}3},x_{23}]_{c}$ to check
the following identity:

(4.4) $\displaystyle\Delta(x_{12^{2}3}^{3})$
$\displaystyle=x_{12^{2}3}^{3}\otimes 1+1\otimes
x_{12^{2}3}^{3}-27q_{21}^{3}q_{31}^{3}x_{12}^{3}\otimes x_{23}^{3}+3\zeta
q_{21}^{2}q_{31}^{2}q_{32}x_{u}\otimes x_{v}.$

Next we see that
$A\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is
compatible if and only if either $1^{3}2^{6}3^{3}\notin A$ or else
$1^{3}2^{6}3^{3}\in A$ and $\\{1^{3}2^{3},2^{3}3^{3}\\}\cap
A,\\{1^{2}2^{3}3,12^{3}3^{2}\\}\cap A\neq\emptyset$, and a subset is
compatible if and only of the complement is closed by sums, as in Remark 3.4.
Thus (ii) and (iii) follow by an argument analogous to the Cartan case
$A_{2}$.

Finally, assume that $\mathfrak{q}$ is of type $\mathfrak{g}(2,3)$ with
diagram $d_{2}$. The proof of this case is analogous to the one above but
using [ACSa2, Proposition 4.3] and its proof. Indeed, set

$\displaystyle x_{u}$ $\displaystyle=[x_{12^{2}3},x_{2}]_{c}$ and
$\displaystyle x_{v}$ $\displaystyle=[x_{123},x_{12^{2}3}]_{c}.$

By loc. cit. $x_{u}$ and $x_{v}$ are primitive in
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ and skew central. In addition,
$x_{2}^{3}$ is primitive and skew central by direct computation, and
$x_{23}^{6}$ and $x_{123}^{3}$ are also primitive and skew-central in
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ since they are primitive and skew
central in $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ and we take into account
the $\mathbb{Z}^{\theta}$-degree as above. Next we observe that
$B=\\{x_{u}^{a_{1}}x_{v}^{a_{2}}x_{123}^{3a_{3}}x_{12^{2}3}^{3a_{4}}x_{2}^{3a_{5}}x_{23}^{6a_{6}}|a_{i}\in\mathbb{N}_{0}\\}$
is a basis of $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ by looking at the
expression of the Hilbert series
${\mathcal{H}}_{\widehat{\mathcal{Z}}_{\mathfrak{q}}}$ as above, so
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a central Hopf subalgebra. As
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a skew polynomial algebra with
generators $x_{2}^{3}$, $x_{12^{2}3}^{3}$, $x_{123}^{3}$, $x_{23}^{6}$,
$x_{u}$ and $x_{v}$, (i) follows.

We need now an explicit expression of $\Delta(x_{12^{2}3}^{3})$. To do so we
compute first

$\displaystyle\Delta(x_{12^{2}3})$ $\displaystyle=x_{12^{2}3}\otimes
1+1\otimes x_{12^{2}3}-q_{23}(1-\zeta^{2})x_{12}\otimes x_{23}$
$\displaystyle\qquad+(1-\zeta^{2})x_{123}\otimes x_{2}-3\zeta^{2}x_{1}\otimes
x_{23}x_{2}.$

Working as in (4.4) we obtain the following:

(4.5) $\displaystyle\Delta(x_{12^{2}3}^{3})$
$\displaystyle=x_{12^{2}3}^{3}\otimes 1+1\otimes
x_{12^{2}3}^{3}+3q_{21}^{3}q_{23}^{3}\zeta(1-\zeta)x_{123}^{3}\otimes
x_{2}^{3}+q_{21}^{2}q_{23}^{2}(\zeta-1)x_{v}\otimes x_{u}.$

Now, $A\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is
compatible if and only if either $1^{3}2^{6}3^{3}\notin A$ or else
$1^{3}2^{6}3^{3}\in A$ and $\\{1^{3}2^{3}3^{3},2^{3}\\}\cap
A,\\{1^{2}2^{3}3^{2},12^{3}3\\}\cap A\neq\emptyset$, so (ii) and (iii) follow
as above. ∎

We finish this subsection by identifying the algebraic groups of types Cartan
$A_{2}$ and $\mathfrak{g}(2,3)$ (for both diagrams). Assume that
$\mathfrak{q}$ is such that $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a
central Hopf subalgebra (that we can assume up to twist the braiding by Lemma
2.8).

Let $\mathfrak{n}_{+}$ be the positive part of the Lie algebra
$\mathfrak{g}=\mathfrak{sl}_{5}$, which is a 10-dimensional nilpotent algebra
with generators $e_{ij}$, $1\leq i\leq j\leq 4$, where

$\displaystyle e_{ii}$ $\displaystyle=e_{i},$ $\displaystyle e_{ij}$
$\displaystyle=[e_{ik},e_{k+1\,j}],$ $\displaystyle i\leq k<j.$

Notice that $e_{ij}$ has degree $\alpha_{ij}$ for all $i\leq j$.

Let $Z_{\varDelta}$ be the corresponding algebraic group with Lie algebra
$\mathfrak{n}_{+}$. The subset

$\displaystyle B=\\{1,123,1234,234,4\\}\subseteq\varDelta_{+}$

is closed by sums. The associated quotient Hopf algebra $Z(B)$ in Theorem 3.10
is a polynomial ring in variables $z_{\beta}$, $\beta\in B$, where all
$z_{\beta}\neq z_{1234}$ are primitive, and

$\displaystyle\Delta(z_{1234})$ $\displaystyle=z_{1234}\otimes 1+1\otimes
z_{1234}+z_{1}\otimes z_{234}+z_{123}\otimes z_{4}.$

######  Lemma 4.2.

Assume that $\mathfrak{q}$ is such that $\widehat{\mathcal{Z}}_{\mathfrak{q}}$
is a central Hopf subalgebra.

  1. (i)

If $\mathfrak{q}$ is of Cartan type $A_{2}$ and $q\in\mathbb{G}_{3}^{\prime}$,
then $\widehat{\mathcal{Z}}_{\mathfrak{q}}\simeq Z(B)$ as Hopf algebras.

  2. (ii)

If $\mathfrak{q}$ is of type ${\bf A}_{3}(q|\\{2\\})$ or ${\bf
A}_{3}(q|\\{1,2,3\\})$ with $q\in\mathbb{G}_{\infty}$, then
$\widehat{\mathcal{Z}}_{\mathfrak{q}}\simeq\Bbbk[z_{1},z_{2},z_{3}]$ as Hopf
algebras.

  3. (iii)

If $\mathfrak{q}$ is of type $\mathfrak{g}(2,3)$ and diagram $d_{1}$ or
$d_{2}$, then $\widehat{\mathcal{Z}}_{\mathfrak{q}}\simeq Z(B)\times\Bbbk[z]$
as Hopf algebras.

###### Proof.

It follows case-by-case, using the coproduct formulas (4.3), (4.4) and (4.5)
in the proof of Proposition 4.1. ∎

###  4.2. The connected case

Let $\mathfrak{q}$ be a braiding of diagonal type such that
$\dim\mathscr{B}_{\mathfrak{q}}<\infty$, the Dynkin diagram is connected and
$\mathfrak{q}$ is not of Cartan type $A_{\theta}$, $D_{\theta}$ with $q=-1$.

Those cases where the eminent pre-Nichols algebra
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ is not the distinguished pre-Nichols
algebra $\widetilde{\mathscr{B}}_{\mathfrak{q}}$ were treated in §4.1, so
assume for a while that
$\widehat{\mathscr{B}}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}$.
Thus
$\widehat{\mathcal{Z}}_{\mathfrak{q}}=\widehat{\mathscr{B}}_{\mathfrak{q}}^{\operatorname{co}\mathscr{B}_{\mathfrak{q}}}=\mathcal{Z}_{\mathfrak{q}}$
is a skew central Hopf subalgebra of $\widehat{\mathscr{B}}_{\mathfrak{q}}$,
and an skew polynomial algebra such that
$\underline{\mathfrak{O}^{\mathfrak{q}}_{+}}$ is the set of degrees of the
generators of $\mathcal{Z}_{\mathfrak{q}}$. Hence we set
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}=\underline{\mathfrak{O}^{\mathfrak{q}}_{+}}$
for this case: we have already defined
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ for the other five
kinds of braidings in (4.2), so in any case
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is the set of degrees
of the generators of $\widehat{\mathcal{Z}}_{\mathfrak{q}}$.

We will extend Proposition 4.1 to any $\mathfrak{q}$ with connected Dynkin
diagram. We deal first with the existence of pre-Nichols algebras. For each
$\underline{\beta}\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$
let $z_{\underline{\beta}}$ be the corresponding generator of degree
$\underline{\beta}$: if $\underline{\beta}=N_{\beta}\beta$ for some Cartan
root $\beta$, then $z_{\underline{\beta}}:=x_{\beta}^{N_{\beta}}$, otherwise
$z_{\underline{\beta}}$ is the extra relation of such degree. For example, for
Cartan type $A_{2}$ and $q\in\mathbb{G}_{3}^{\prime}$, $z_{1^{2}2}=x_{112}$
and $z_{12^{2}}=x_{221}$.

######  Lemma 4.3.

Let $\mathfrak{q}$ be a matrix with connected Dynkin diagram such that
$\dim\mathscr{B}_{\mathfrak{q}}<\infty$ and is not of Cartan type $A_{n}$,
$D_{n}$ with $q=-1$.

For each $B\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$
closed by sums, the quotient

$\displaystyle\mathscr{B}(\mathfrak{q},B):=\widehat{\mathscr{B}}_{\mathfrak{q}}/\langle
z_{\underline{\beta}}|\underline{\beta}\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}-B\rangle$

is a $\mathbb{Z}^{\theta}$-graded pre-Nichols algebra
$\mathscr{B}(\mathfrak{q},B)$ with Hilbert series

(4.6) $\displaystyle{\mathcal{H}}_{\mathscr{B}(\mathfrak{q},B)}(t)$
$\displaystyle={\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}(t)\left(\prod_{\underline{\beta}\in
B}\tfrac{1}{1-t^{\underline{\beta}}}\right).$

###### Proof.

As mentioned above, it suffices to deal with the case
$\widehat{\mathscr{B}}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}$
as in Theorem 2.4, since the statement holds for other $\mathfrak{q}$ by
Proposition 4.1 (ii). Here,
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}=\varDelta_{+}$ is the
set of positive roots of the semisimple Lie algebra attached to
$\widetilde{\mathscr{B}}_{\mathfrak{q}}$ in [AAR]. Moreover, we can assume
that $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is central up to change
$\mathfrak{q}$ by a twist equivalent braiding matrix, see Proposition 2.7 and
Lemma 2.8, in which case
$\widehat{\mathcal{Z}}_{\mathfrak{q}}=\mathcal{Z}_{\varDelta}$.

Let $B\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ be a
subset closed by sums. By Remark 3.4, $A=B^{c}$ is compatible, so
$I(B)=\langle z_{\underline{\beta}}|\underline{\beta}\in A\rangle$ is an
$\mathbb{N}_{0}^{\theta}$-graded Hopf ideal of $\mathcal{Z}_{\varDelta}$ by
Lemma 3.9. Thus $\widehat{\mathscr{B}}_{\mathfrak{q}}I(B)$ is a Hopf ideal of
$\widehat{\mathscr{B}}_{\mathfrak{q}}$. By Proposition 2.2 (ii),
${\mathcal{H}}_{\widetilde{\mathscr{B}}_{\mathfrak{q}}}(t)={\mathcal{H}}_{\mathscr{B}(\mathfrak{q},B)}(t){\mathcal{H}}_{\mathtt{Z}}(t)$,
where $\mathtt{Z}$ is the subalgebra of coinvariants of the projection
$\mathcal{Z}_{\mathfrak{q}}\twoheadrightarrow\mathcal{Z}_{\mathfrak{q}}/I(B)$.
As the subalgebra $Z$ generated by $z_{\underline{\beta}}$,
$\underline{\beta}\in A$, is simultaneously a coideal subalgebra and
polynomial ring in these variables such that $\langle Z^{+}\rangle=I(B)$, we
have that $Z=\mathtt{Z}$ by Proposition 2.1, so
${\mathcal{H}}_{\mathtt{Z}}(t)=\left(\prod_{\alpha\in
A}\tfrac{1}{1-t^{\alpha}}\right)$. As

$\displaystyle{\mathcal{H}}_{\widetilde{\mathscr{B}}_{\mathfrak{q}}}(t)={\mathcal{H}}_{\mathscr{B}_{\mathfrak{q}}}(t)\left(\prod_{\alpha\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}}\tfrac{1}{1-t^{\alpha}}\right)$

by (2.2), we obtain the expression (4.6) for
${\mathcal{H}}_{\mathscr{B}(\mathfrak{q},B)}(t)$. ∎

######  Remark 4.4.

Let $B=\\{\underline{\gamma}_{1},\cdots,\underline{\gamma}_{L}\\}$ be a
numeration of $B$. Then the set

$\displaystyle x_{\beta_{1}}^{n_{1}}\cdots
x_{\beta_{M}}^{n_{M}}z_{\underline{\gamma}_{1}}^{p_{1}}\cdots
z_{\underline{\gamma}_{L}}^{p_{L}},$ $\displaystyle 0\leq
n_{i}<N_{\beta_{i}},\,0\leq p_{j}<\infty,$

is a basis of $\mathscr{B}(\mathfrak{q},B)$.

Now we state a characterization of the poset of
$\mathbb{N}_{0}^{\theta}$-graded pre-Nichols algebras.

######  Theorem 4.5.

Let $\mathfrak{q}$ a matrix with connected Dynkin diagram such that
$\dim\mathscr{B}_{\mathfrak{q}}<\infty$ and is not of Cartan type $A_{n}$,
$D_{n}$ with $q=-1$. The map

$\displaystyle\mathcal{P}_{c}(\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}})\to\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q}),$
$\displaystyle B\mapsto\mathscr{B}(\mathfrak{q},B),$

is an anti-isomorphism of posets.

###### Proof.

As in the proof of Lemma 4.3, we can restrict to those $\mathfrak{q}$ such
that
$\widehat{\mathscr{B}}_{\mathfrak{q}}=\widetilde{\mathscr{B}}_{\mathfrak{q}}$
since the remaining cases were treated in Proposition 4.1 (iii), and we can
use Proposition 2.7 and Lemma 2.8 to assume that $\mathcal{Z}_{\mathfrak{q}}$
is central up to change $\mathfrak{q}$ by a twist equivalent matrix.

By Lemma 4.3, the map above is injective: if $B\neq B^{\prime}$ are two
different sets closed by sums, then
${\mathcal{H}}_{\mathscr{B}(\mathfrak{q},B)}(t)\neq{\mathcal{H}}_{\mathscr{B}(\mathfrak{q},B^{\prime})}(t)$.
Also, it is a anti morphism of posets.

On the other hand, we will see that the map is also surjective. Let
$\mathscr{B}\in\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$:

  * •

By definition, there exists an $\mathbb{N}_{0}^{\theta}$-graded Hopf ideal
$\mathcal{I}$ such that
$\mathscr{B}\simeq\widehat{\mathscr{B}}_{\mathfrak{q}}/\mathcal{I}$.

  * •

By Proposition 2.2 (i) there exists a graded Hopf ideal $I$ of
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ such that
$\mathcal{I}=\widehat{\mathscr{B}}_{\mathfrak{q}}I$.

  * •

By Theorem 3.10 there exists a set
$B\subseteq\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}=\varDelta_{+}$
closed by sums such that $I=I(B)$.

All in all, we have that

$\displaystyle\mathscr{B}$
$\displaystyle=\widehat{\mathscr{B}}_{\mathfrak{q}}/\widehat{\mathscr{B}}_{\mathfrak{q}}I(B)=\widehat{\mathscr{B}}_{\mathfrak{q}}/\langle
z_{\underline{\beta}}|\underline{\beta}\in
A\rangle=\mathscr{B}(\mathfrak{q},B),$

so the map is also surjective, and the statement follows. ∎

######  Example 4.6.

Fix $\zeta\in\mathbb{G}_{3}^{\prime}$. Let $\mathfrak{q}$ be a braiding matrix
of type $\mathfrak{g}(2,3)$ with Dynkin diagram

$\displaystyle d_{3}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
11.88884pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overset{-1}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.04025pt\raise 6.75554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39444pt\hbox{$\scriptstyle{\xi^{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.77776pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overset{-\xi^{2}}{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
77.73882pt\raise 6.75554pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39444pt\hbox{$\scriptstyle{\xi^{2}}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
99.50829pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overset{-1}{\circ}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Then $\widetilde{\mathscr{B}}_{\mathfrak{q}}=\widehat{\mathscr{B}}$ is
presented by generators and relations

$\displaystyle x_{13},$ $\displaystyle x_{2221},$ $\displaystyle x_{2223},$
$\displaystyle x_{1}^{2},$ $\displaystyle x_{3}^{2},=0,$
$\displaystyle[x_{1},x_{223}]_{c}+q_{23}x_{12^{2}3}-(1-\zeta^{2})x_{2}x_{123}.$

Also,
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}=\\{1^{2}2^{3},1^{3}2^{6}3^{3},2^{3}3^{3},2^{6}\\}$
is (isomorphic to) the set of positive roots associated to a Lie algebra of
type $A_{2}\times A_{1}$, and $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is the
subalgebra generated by

$\displaystyle z_{1}$ $\displaystyle=x_{12}^{3},$ $\displaystyle z_{12}$
$\displaystyle=x_{12^{2}3}^{3},$ $\displaystyle z_{2}$
$\displaystyle=x_{23}^{3},$ $\displaystyle z_{3}$ $\displaystyle=x_{2}^{6}.$

The poset
$\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$ is the
following:

When we move from left to right, the $\operatorname{GKdim}$ goes down from 4
to 0.

We can observe here that the poset of graded pre-Nichols algebras is not
preserved by Weyl equivalence: if $\mathfrak{q}$ is also of type
$\mathfrak{g}(2,3)$ but with Dynkin diagram $d_{1}$ or $d_{2}$, then
$\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$ has 50
elements, since this is the number of subsets of
$\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ closed by sums.

###  4.3. The non-connected case

Here we take an arbitraty matrix $\mathfrak{q}$ (whose diagram is not
necessarily connected). Following the spirit of [ASa, §3] we set

$\displaystyle\mathbb{I}^{\pm}$ $\displaystyle:=\\{i\in\mathbb{I}:q_{ii}=\pm
1,\widetilde{q}_{ij}=1\text{ for all }j\neq i\\},$
$\displaystyle\mathbb{I}^{(c)}$
$\displaystyle:=\mathbb{I}-\left(\mathbb{I}^{+}\cup\mathbb{I}^{-}\right).$

that is, $\mathbb{I}^{\pm}$ contains all the connected components of an
isolated vertex labelled with $\pm 1$ while $\mathbb{I}^{(c)}$ is the union of
those points labelled with $q_{ii}\neq\pm 1$ and those connected components
with at least two vertices. Let $C_{1},\cdots,C_{d}$ be the partition of
$\mathbb{I}^{(c)}$ in connected components: i.e.
$\mathbb{I}^{(c)}=\cup_{\ell=1}^{d}C_{d}$, where the diagram of each
$\mathfrak{q}^{(\ell)}:=(q_{ij})_{i,j\in C_{\ell}}$ is connected and
$\widetilde{q}_{ij}=1$ if $i\in C_{\ell_{1}}$, $j\in C_{\ell_{2}}$,
$\ell_{1}\neq\ell_{2}$.

Let $\mathscr{B}$ be an $\mathbb{N}_{0}$-graded pre-Nichols algebra of
$\mathfrak{q}$. For each $1\leq\ell\leq d$, let $\mathscr{B}^{(\ell)}$ be the
subalgebra of $\mathscr{B}$ generated by $x_{i}$, $i\in C_{\ell}$, which is a
pre-Nichols algebra of $\mathfrak{q}^{(\ell)}$. Similarly, let
$\mathscr{B}^{\pm}$ be the subalgebra of $\mathscr{B}$ generated by $x_{i}$,
$i\in\mathbb{I}^{\pm}$, and
$\mathfrak{q}^{(\pm)}:=(q_{ij})_{i,j\in\mathbb{I}^{\pm}}$.

######  Lemma 4.7.

Let $1\leq\ell\leq d$, $i\in C_{\ell}$, $j\in\mathbb{I}-C_{\ell}$ such that
$q_{jj}\neq 1$. If $\operatorname{GKdim}\mathscr{B}<\infty$, then $x_{ij}=0$
in $\mathscr{B}$.

###### Proof.

By hypothesis, $i$ and $j$ are not connected, so $x_{ij}$ is primitive in
$\mathscr{B}$. Also, $\operatorname{ord}q_{ii}\geq 2$ since
$C_{\ell}\subseteq\mathbb{I}^{(c)}$, and also $\operatorname{ord}q_{jj}\geq 2$
since $q_{jj}\neq 1$. If either $\operatorname{ord}q_{ii}>2$ or
$\operatorname{ord}q_{jj}>2$, then $x_{ij}=0$ by [ASa, Proposition 3.2].
Otherwise $q_{ii}=q_{jj}=-1$ and there exists $k\neq i,j$ such that
$\widetilde{q}_{ik}\neq 1=\widetilde{q}_{jk}$. Suppose that $x_{ij}\neq 0$ in
$\mathscr{B}$. Set $y_{1}=x_{i}$, $y_{2}=x_{k}$, $y_{3}=x_{ij}$ and $W:=\Bbbk
y_{1}+\Bbbk y_{2}+\Bbbk y_{3}$ is a three-dimensional subspace of
$\mathcal{P}(\mathscr{B})$. Working as in the proof of [ASa, Proposition 3.2],
$\operatorname{GKdim}\mathscr{B}(W)<\infty$ where $W$ is of diagonal type with
matrix $\mathfrak{p}=(p_{rs})_{1\leq r,s\leq 3}$. By direct computation,
$p_{33}=1$ and $\widetilde{p}_{23}=\widetilde{q}_{ik}\neq 1$, so we get a
contradiction with [AAH, Proposition 4.16]. Hence $x_{ij}=0$ in $\mathscr{B}$.
∎

Assume that $\mathbb{I}^{+}=\mathbb{I}^{-}=\emptyset$ and each $C_{\ell}$ is
not of Cartan type $A_{n}$, $D_{n}$ with $q=-1$ (thus we know the eminent pre-
Nichols algebra of each $\mathfrak{q}^{(\ell)}$) and
$\operatorname{GKdim}\mathscr{B}<\infty$. Then
$\operatorname{GKdim}\mathscr{B}^{(\ell)}<\infty$, so $\mathscr{B}^{(\ell)}$
is a quotient of $\widehat{\mathscr{B}}_{\mathfrak{q}^{(\ell)}}$. Set

(4.7) $\displaystyle\widehat{\mathscr{B}}_{\mathfrak{q}}$
$\displaystyle:=\bigotimes_{1\leq\ell\leq
d}\widehat{\mathscr{B}}_{\mathfrak{q}^{(\ell)}},$
$\displaystyle\widehat{\mathcal{Z}}_{\mathfrak{q}}$
$\displaystyle:=\bigotimes_{1\leq\ell\leq
d}\widehat{\mathcal{Z}}_{\mathfrak{q}^{(\ell)}},$
$\displaystyle\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$
$\displaystyle=\bigcup_{1\leq\ell\leq
d}\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}^{(\ell)}}_{+}},$

extending the definitions we have made from the connected to the non-connected
case. Hence we have an extension of $\mathbb{N}_{0}$-graded Hopf algebras
$\widehat{\mathcal{Z}}_{\mathfrak{q}}\hookrightarrow\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\mathscr{B}_{\mathfrak{q}}$,
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is a polynomial ring in variables
$z_{\underline{\beta}}$,
$\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ of degree
$\underline{\beta}$, and we may wonder if
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ is an eminent pre-Nichols algebra of
$\mathfrak{q}$. We will see that this is the case, and then prove that the
poset $\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$
splits as the product of the posets
$\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q}^{(\ell)})$.

######  Theorem 4.8.

Let $\mathfrak{q}$ be such that $\mathbb{I}^{+}=\mathbb{I}^{-}=\emptyset$.

  1. (i)

If $\mathscr{B}$ is an $\mathbb{N}_{0}$-graded pre-Nichols algebra of
$\mathfrak{q}$ such that $\operatorname{GKdim}\mathscr{B}<\infty$, then

$\displaystyle\mathscr{B}\simeq\otimes_{\ell=1}^{d}\mathscr{B}^{(\ell)},$

and $\widehat{\mathscr{B}}_{\mathfrak{q}}$ is the eminent pre-Nichols algebra
of $\mathfrak{q}$.

  2. (ii)

There exists an anti-isomorphism of posets

$\displaystyle\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})\simeq\prod_{\ell=1}^{d}\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q}^{(\ell)})\simeq\mathcal{P}_{c}(\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}).$

###### Proof.

(i) Notice that
$\mathscr{B}^{\prime}=\otimes_{\ell=1}^{d}\mathscr{B}^{(\ell)}$ is an
$\mathbb{N}_{0}^{\theta}$-graded Hopf algebra with defining relations those
defining each $\mathscr{B}^{(\ell)}$ together with $x_{ij}=0$ for $i\in
C_{k}$, $j\in C_{\ell}$, $k\neq\ell$. Thus Lemma 4.7 says that there exists a
surjective map $\mathscr{B}^{\prime}\twoheadrightarrow\mathscr{B}$ of
$\mathbb{N}_{0}$-graded Hopf algebras, which is the identity on $V$, i.e. a
map of pre-Nichols algebras of $\mathfrak{q}$. As we also have a map
$\widehat{\mathscr{B}}_{\mathfrak{q}}\twoheadrightarrow\mathscr{B}^{\prime}$,
the composition of both map gives a map of pre-Nichols algebras. As
$\mathscr{B}$ is arbitrary with finite $\operatorname{GKdim}$,
$\widehat{\mathscr{B}}_{\mathfrak{q}}$ is eminent.

Hence we move to the description of $\mathbb{N}_{0}^{\theta}$-graded quotients
of $\widehat{\mathscr{B}}_{\mathfrak{q}}$. As before, we can use Proposition
2.7 and Lemma 2.8 to assume that $\widehat{\mathcal{Z}}_{\mathfrak{q}}$ is
central. By Proposition 2.2 we have to compute all the Hopf ideals of
$\widehat{\mathcal{Z}}_{\mathfrak{q}}$, which in turn (by taking graded duals)
are classified by $\mathbb{N}_{0}^{\theta}$-graded Lie subalgebras of the
nilpotent Lie algebra $\mathfrak{n}_{\mathfrak{q}}=\prod_{1\leq\ell\leq
d}\mathfrak{n}_{\mathfrak{q}^{(\ell)}}$: any of these Lie subalgebras of
$\mathfrak{n}_{\mathfrak{q}}$ is the product of
$\mathbb{N}_{0}^{\theta}$-graded Lie subalgebras of each
$\mathfrak{n}_{\mathfrak{q}^{(\ell)}}$, and these are classified by subsets
$B^{(\ell)}$ of $\varDelta_{+}^{\mathfrak{q}^{(\ell)}}$ closed by sums,
applying Theorem 3.10 and Proposition 4.1 (depending on $\ell$).

Coming back to $\widehat{\mathcal{Z}}_{\mathfrak{q}}$, we quotient this Hopf
algebra by $z_{\underline{\beta}}$ for those $\underline{\beta}\notin
B:=\cup_{1\leq\ell\leq d}B^{(\ell)}$. We obtain then the pre-Nichols algebra

$\mathscr{B}(\mathfrak{q},B):=\widehat{\mathscr{B}}_{\mathfrak{q}}/\langle
z_{\underline{\beta}}|\underline{\beta}\notin
B\rangle\simeq\bigotimes_{1\leq\ell\leq d}\mathscr{B}(q^{(\ell)},B^{(\ell)}),$

and any
$\mathscr{B}\in\mathfrak{Pre}_{\operatorname{fGK}}^{\operatorname{gr}}(\mathfrak{q})$
is of this shape, with
$\mathscr{B}^{(\ell)}\simeq\mathscr{B}(q^{(\ell)},B^{(\ell)})$, $1\leq\ell\leq
d$.

Now (ii) follows from (i) and the fact that a subset
$B\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}}_{+}}$ is closed by sums
if and only if each
$B^{(\ell)}=B\cap\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}^{(\ell)}}_{+}}$
is closed by sum, since the sum $\alpha+\beta$ of two roots
$\alpha\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}^{(k)}}_{+}}$,
$\beta\in\underline{\widehat{\mathfrak{O}}^{\mathfrak{q}^{(\ell)}}_{+}}$,
$k\neq\ell$, is not a root. ∎

######  Remark 4.9.

Putting together Theorems 4.8, 4.5 and [C] we get a full description of the
poset of all $\mathbb{N}_{0}^{\theta}$-graded pre-Nichols algebras with finite
$\operatorname{GKdim}$ when all connected components are not points with label
$\pm 1$ neither of Cartan type $A_{n}$, $D_{n}$ and $q=-1$.

######  Remark 4.10.

Pre-Nichols algebras with finite $\operatorname{GKdim}$ for
$\mathfrak{q}=\mathfrak{q}^{+}$ were classified in [A1, §3.4]. We do not
include them in Theorem 4.8 since we do not know at the moment how to control
the interaction between $\mathscr{B}^{+}$ and the pre-Nichols algebras
$\mathscr{B}^{-}$, $\mathscr{B}^{(\ell)}$.

At the same time, we do not have at the moment a description of all pre-
Nichols algebras with finite $\operatorname{GKdim}$ for
$\mathfrak{q}=\mathfrak{q}^{-}$ and for connected components of Cartan type
$A_{\theta}$, $D_{\theta}$. Anyway, by Lemma 4.7 we may wonder that the poset
of $\mathfrak{q}$ decomposes as the product of the posets of pre-Nichols of
different connected components and that of $\mathfrak{q}^{-}$.

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