# Nurowski’s conformal class of a maximally symmetric $(2,3,5)$-distribution
and its Ricci-flat representatives

Matthew Randall  Institute of Mathematical Sciences
ShanghaiTech University
393 Middle Huaxia Road
Shanghai, 201210
China<EMAIL_ADDRESS>
###### Abstract.

We show that the solutions to the second-order differential equation
associated to the generalised Chazy equation with parameters $k=2$ and $k=3$
naturally show up in the conformal rescaling that takes a representative
metric in Nurowski’s conformal class associated to a maximally symmetric
$(2,3,5)$-distribution (described locally by a certain function
$\varphi(x,q)=\frac{q^{2}}{H^{\prime\prime}(x)}$) to a Ricci-flat one.

###### 2010 Mathematics Subject Classification:

53A30, 58A15, 34A05, 34A34 (primary)

The article concerns the occurrence of the $k=2$ and $k=3$ generalised Chazy
equation in a geometric setting, closely connected to the occurrence of the
solutions of the generalised Chazy equation with parameters $k=\frac{2}{3}$
and $k=\frac{3}{2}$ respectively. We first discuss the set-up in which the
differential equations will appear. This concerns the theory of maximally non-
integrable rank 2 distribution $\mathcal{D}$ on a 5-manifold $M$. The
maximally non-integrable condition of $\mathcal{D}$ determines a filtration of
the tangent bundle $TM$ given by

$\mathcal{D}\subset[\mathcal{D},\mathcal{D}]\subset[\mathcal{D},[\mathcal{D},\mathcal{D}]]\cong
TM.$

The distribution $[\mathcal{D},\mathcal{D}]$ has rank 3 while the full tangent
space $TM$ has rank 5, hence such a geometry is also known as a
$(2,3,5)$-distribution. Let $M_{xyzpq}$ denote the 5-dimensional mixed order
jet space $J^{2,0}(\mathbb{R},\mathbb{R}^{2})\cong
J^{2}(\mathbb{R},\mathbb{R})\times\mathbb{R}$ with local coordinates given by
$(x,y,z,p,q)=(x,y,z,y^{\prime},y^{\prime\prime})$ (see also [14], [15]). Let
$\mathcal{D}_{\varphi(x,y,z,y^{\prime},y^{\prime\prime})}$ denote the
maximally non-integrable rank 2 distribution on $M_{xyzpq}$ associated to the
underdetermined differential equation
$z^{\prime}=\varphi(x,y,z,y^{\prime},y^{\prime\prime})$. This means that the
distribution is annihilated by the following three 1-forms

$\displaystyle\omega_{1}={\rm d}y-p{\rm d}x,\qquad\omega_{2}={\rm d}p-q{\rm
d}x,\qquad\omega_{3}={\rm d}z-\varphi(x,y,z,p,q){\rm d}x.$

In [10], it is shown how to associate canonically to such a
(2,3,5)-distribution a conformal class of metrics of split signature $(2,3)$
(henceforth known as Nurowski’s conformal structure or Nurowski’s conformal
metrics) such that the rank 2 distribution is isotropic with respect to any
metric in the conformal class. The method of equivalence [3], [5], [10], [13],
[9] gives the coframing for Nurowski’s metric. The 1-forms in the coframe
satisfy the structure equations

$\displaystyle{\rm d}\theta_{1}$
$\displaystyle=\theta_{1}\wedge(2\Omega_{1}+\Omega_{4})+\theta_{2}\wedge\Omega_{2}+\theta_{3}\wedge\theta_{4},$
$\displaystyle{\rm d}\theta_{2}$
$\displaystyle=\theta_{1}\wedge\Omega_{3}+\theta_{2}\wedge(\Omega_{1}+2\Omega_{4})+\theta_{3}\wedge\theta_{5},$
(0.1) $\displaystyle{\rm d}\theta_{3}$
$\displaystyle=\theta_{1}\wedge\Omega_{5}+\theta_{2}\wedge\Omega_{6}+\theta_{3}\wedge(\Omega_{1}+\Omega_{4})+\theta_{4}\wedge\theta_{5},$
$\displaystyle{\rm d}\theta_{4}$
$\displaystyle=\theta_{1}\wedge\Omega_{7}+\frac{4}{3}\theta_{3}\wedge\Omega_{6}+\theta_{4}\wedge\Omega_{1}+\theta_{5}\wedge\Omega_{2},$
$\displaystyle{\rm d}\theta_{5}$
$\displaystyle=\theta_{2}\wedge\Omega_{7}-\frac{4}{3}\theta_{3}\wedge\Omega_{5}+\theta_{4}\wedge\Omega_{3}+\theta_{5}\wedge\Omega_{4}.$

A representative metric in Nurowski’s conformal class [10] is given by

(0.2) $\displaystyle
g=2\theta_{1}\theta_{5}-2\theta_{2}\theta_{4}+\frac{4}{3}\theta_{3}\theta_{3}.$

When $g$ has vanishing Weyl tensor, the distribution is called maximally
symmetric and has split $G_{2}$ as its group of local symmetries.

The historic example is the case where $\varphi$ is given by
$\varphi(x,y,z,p,q)=q^{2}$. When $\varphi(x,y,z,p,q)=q^{2}$, we obtain the
Hilbert-Cartan distribution associated to the Hilbert-Cartan equation
$z^{\prime}=(y^{\prime\prime})^{2}$. When $\varphi(x,y,z,p,q)=q^{m}$, we
obtain the distribution associated to the equation
$z^{\prime}=(y^{\prime\prime})^{m}$. For such distributions, Nurowski’s metric
[10] given by (0.2) has vanishing Weyl tensor precisely when
$m\in\\{-1,\frac{1}{3},\frac{2}{3},2\\}$. In these cases the maximally
symmetric distributions are all locally diffeomorphic to the Hilbert-Cartan or
flat model obtained when $m=2$.

In this article, we consider distributions here of the form
$\varphi(x,y,z,p,q)=\frac{q^{2}}{H^{\prime\prime}(x)}$. The Weyl tensor
vanishes in the case where $H(x)$ satisfies the 6th-order ordinary
differential equation (ODE) known as Noth’s equation [3]. For such maximally
symmetric distributions we find the corresponding Ricci-flat representatives
in Nurowki’s conformal class. This involves solving a second-order
differential equation (see Proposition 35 of [14]) to find the conformal scale
in which the Ricci tensor of the conformally rescaled metric vanishes, which
turns out to be related to the solutions of Noth’s equation. The 6th-order ODE
can be reduced to the generalised Chazy equation with parameter
$k=\frac{3}{2}$ and its Legendre dual is another 6th-order ODE that can be
reduced to the generalised Chazy equation with parameter $k=\frac{2}{3}$. We
find the second-order differential equation that determines the conformal
scale for Ricci-flatness involves solutions of the generalised Chazy equation
with parameter $k=3$ and in the dual case $k=2$. This is the content of
Theorems 3.1 and 3.2. We also give few remarks concerning the case for other
parameters of $k$.

The aim of finding Ricci-flat representatives is motivated by the
consideration that in the Ricci-flat, conformally flat case, we might be able
to integrate the structure equations and reexpress them in Monge normal form
to obtain the Hilbert-Cartan distribution. This is possible for the
distributions of the form $\varphi(x,y,z,p,q)=q^{m}$, with
$m\in\\{-1,\frac{1}{3},\frac{2}{3},2\\}$, but would require further
investigations in the general setting.

The computations here are done using the indispensable DifferentialGeometry
package in Maple 2018.

##  1\. Deriving the equation for Ricci-flatness

We shall consider the rank 2 distribution $\mathcal{D}_{\varphi(x,q)}$ on
$M_{xyzpq}$ associated to the underdetermined differential equation
$z^{\prime}=\varphi(x,y^{\prime\prime})$ where
$\varphi(x,y^{\prime\prime})=\frac{(y^{\prime\prime})^{2}}{H^{\prime\prime}(x)}$
and $H^{\prime\prime}(x)$ is a non-zero function of $x$. This is to say that
the distribution $\mathcal{D}_{\varphi(x,q)}$ is annihilated by the three
1-forms

$\displaystyle\omega_{1}=$ $\displaystyle{\rm d}y-p{\rm d}x,$
$\displaystyle\omega_{2}=$ $\displaystyle{\rm d}p-q{\rm d}x,$
$\displaystyle\omega_{3}=$ $\displaystyle{\rm d}z-\varphi(x,q){\rm d}x,$

where $\varphi(x,q)=\frac{q^{2}}{H^{\prime\prime}(x)}$. These three 1-forms
are completed to a coframing on $M_{xyzpq}$ by the additional 1-forms

$\displaystyle\omega_{4}={\rm d}q-\frac{H^{(3)}}{H^{\prime\prime}}q{\rm
d}x,\hskip 12.0pt\omega_{5}=-\frac{H^{\prime\prime}}{2}{\rm d}x.$

Taking appropriate linear combinations, we let

$\displaystyle\theta_{1}=$
$\displaystyle\omega_{3}-\frac{2}{H^{\prime\prime}}q\omega_{2},\qquad\theta_{2}=\omega_{1},\qquad\theta_{3}=\left(\frac{2}{H^{\prime\prime}}\right)^{\frac{1}{3}}\omega_{2},$

with

$\theta_{4}=\left(\frac{2}{H^{\prime\prime}}\right)^{\frac{2}{3}}\omega_{4}+a_{41}\theta_{1}+a_{42}\theta_{2}+a_{43}\theta_{3}$

and

$\theta_{5}=\left(\frac{2}{H^{\prime\prime}}\right)^{\frac{2}{3}}\omega_{5}+a_{51}\theta_{1}+a_{52}\theta_{2}+a_{53}\theta_{3}.$

Imposing Cartan’s structure equations (Nurowski’s conformal class of a
maximally symmetric $(2,3,5)$-distribution and its Ricci-flat representatives)
on $(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5})$ then gives the
constraints $a_{51}=a_{53}=0$ and $a_{41}=a_{52}$, which we can set both to be
zero, and we also find
$a_{42}=\frac{1}{30}\frac{2^{\frac{2}{3}}(3H^{\prime\prime}H^{(4)}-5(H^{(3)})^{2})}{(H^{\prime\prime})^{\frac{8}{3}}}$
and
$a_{43}=-\frac{2^{\frac{1}{3}}H^{(3)}}{3(H^{\prime\prime})^{\frac{4}{3}}}$.
The metric
$g=2\theta_{1}\theta_{5}-2\theta_{2}\theta_{4}+\frac{4}{3}\theta_{3}\theta_{3}$
is conformally flat, i.e. the metric $g$ has vanishing Weyl tensor if and only
if $H(x)$ is a solution to the 6th-order nonlinear differential equation

(1.1) $\displaystyle
10(H^{\prime\prime})^{3}H^{(6)}-70(H^{\prime\prime})^{2}H^{(3)}H^{(5)}-49(H^{\prime\prime})^{2}(H^{(4)})^{2}+280H^{\prime\prime}(H^{(3)})^{2}H^{(4)}-175(H^{(3)})^{4}=0.$

This equation is called Noth’s equation [3]. In this case the distribution of
the form $\mathcal{D}_{\varphi(x,q)}$ is maximally symmetric and in the paper
we will concern ourselves with the problem of finding Ricci-flat
representatives in the conformal class of metrics associated to this
distribution.

The explicit form of the metric given by the distribution
$\mathcal{D}_{\varphi(x,q)}$ is as follows. If we replace
$H^{\prime\prime}(x)=e^{\int\frac{2}{3}P(x){\rm d}x}$, we find that equation
(1.1) reduces to the $k=\frac{3}{2}$ generalised Chazy equation

$P^{\prime\prime\prime}-2PP^{\prime\prime}+3P^{\prime
2}-\frac{4}{36-(\frac{3}{2})^{2}}(6P^{\prime}-P^{2})^{2}=0,$

and we find that the conformally rescaled metric
$\tilde{g}=2^{-\frac{2}{3}}(H^{\prime\prime})^{\frac{2}{3}}g$ has the form

$\displaystyle\tilde{g}=-\frac{2}{15}(P^{\prime}-\frac{4}{9}P^{2})\omega_{1}\omega_{1}+\frac{4}{9}P\omega_{1}\omega_{2}+\frac{4}{3}\omega_{2}\omega_{2}+2\omega_{3}\omega_{5}-2\omega_{1}\omega_{4}-4qe^{\int-\frac{2}{3}P{\rm
d}x}\omega_{2}\omega_{5}.$

We can reexpress this metric as

$\displaystyle\tilde{g}$
$\displaystyle=-\frac{2}{15}(P^{\prime}-\frac{1}{6}P^{2})\omega_{1}\omega_{1}+\frac{4}{3}\left(\frac{P}{6}\omega_{1}+\omega_{2}\right)\left(\frac{P}{6}\omega_{1}+\omega_{2}\right)+2\omega_{3}\omega_{5}-2\omega_{1}\omega_{4}-4qe^{\int-\frac{2}{3}P{\rm
d}x}\omega_{2}\omega_{5}.$

By defining the new coframes

$\displaystyle\tilde{\omega}_{3}$ $\displaystyle=e^{\int\frac{2P}{3}{\rm
d}x}\omega_{3},$ $\displaystyle\tilde{\omega}_{5}$
$\displaystyle=e^{-\int\frac{2P}{3}{\rm d}x}\omega_{5},$

and making the further substitution $Q=P^{2}-6P^{\prime}$, we get the
following cosmetic improvement for $\tilde{g}$:

$\displaystyle\tilde{g}$
$\displaystyle=\frac{1}{45}Q\omega_{1}\omega_{1}+\frac{4}{3}\left(\frac{P}{6}\omega_{1}+\omega_{2}\right)\left(\frac{P}{6}\omega_{1}+\omega_{2}\right)+2\tilde{\omega}_{3}\tilde{\omega}_{5}-2\omega_{1}\omega_{4}-4q\omega_{2}\tilde{\omega}_{5}.$

From this we can rescale the metric $\tilde{g}$ further by a conformal factor
$\Omega$ to obtain a Ricci-flat representative. When ${\rm
Ric}(\Omega^{2}\tilde{g})=0$, we say that $\Omega^{2}\tilde{g}$ is a Ricci-
flat representative of Nurowski’s conformal class. We find that
$\Omega^{2}\tilde{g}$ is Ricci-flat when $\Omega$ satisfies the second-order
differential equation

$\Omega^{\prime\prime}\Omega-2(\Omega^{\prime})^{2}-\frac{2}{3}P\Omega\Omega^{\prime}-\frac{1}{18}P^{2}\Omega^{2}-\frac{1}{30}Q\Omega^{2}=0.$

We make the substitution $\Omega=\frac{1}{\rho}e^{-\frac{1}{3}\int P{\rm d}x}$
to obtain

(1.2) $\rho^{\prime\prime}-\frac{1}{45}Q\rho=0,$

where $\rho(x)$ is to be determined.

The function $H(x)$ is related to another function $F(\tilde{x})$ by a
Legendre transformation [3], [11]. We say that $F(\tilde{x})$ is the Legendre
dual of $H(x)$ determined by the relation $H(x)+F(\tilde{x})=x\tilde{x}$. This
implies $\tilde{x}=H^{\prime}(x)$ with ${\rm d}\tilde{x}=H^{\prime\prime}{\rm
d}x$ and $H^{\prime\prime}=\frac{1}{F_{\tilde{x}\tilde{x}}}$. We can make use
of this transformation to write ${\rm d}x=F_{\tilde{x}\tilde{x}}{\rm
d}\tilde{x}$. The Legendre dual of the distribution
$\mathcal{D}_{\varphi(x,q)}$ is therefore given by the annihilator of the
three 1-forms

$\displaystyle\omega_{1}=$ $\displaystyle{\rm d}y-pF_{\tilde{x}\tilde{x}}{\rm
d}\tilde{x},$ $\displaystyle\omega_{2}=$ $\displaystyle{\rm
d}p-qF_{\tilde{x}\tilde{x}}{\rm d}\tilde{x},$ $\displaystyle\omega_{3}=$
$\displaystyle{\rm d}z-q^{2}(F_{\tilde{x}\tilde{x}})^{2}{\rm d}\tilde{x}$

on the mixed jet space with local coordinates $(\tilde{x},y,z,p,q)$.
Relabelling $\tilde{x}$ with $x$, we have

$\displaystyle\omega_{1}=$ $\displaystyle{\rm d}y-pF^{\prime\prime}{\rm d}x,$
$\displaystyle\omega_{2}=$ $\displaystyle{\rm d}p-qF^{\prime\prime}{\rm d}x,$
$\displaystyle\omega_{3}=$ $\displaystyle{\rm
d}z-q^{2}(F^{\prime\prime})^{2}{\rm d}x.$

Here $F$ now becomes a function of $x$. These three 1-forms are completed to a
coframing on $M$ with local coordinates $(x,y,z,p,q)$ by the additional
1-forms

$\displaystyle\omega_{4}={\rm
d}q+\frac{F^{\prime\prime\prime}}{F^{\prime\prime}}q{\rm d}x,\hskip
12.0pt\omega_{5}=-\frac{1}{2}{\rm d}x.$

(These are the Legendre transformed 1-forms $\omega_{4}$ and $\omega_{5}$).
Similar as before, we consider the linear combinations

$\displaystyle\theta_{1}=\omega_{3}-2F^{\prime\prime}q\omega_{2},\qquad\theta_{2}=\omega_{1},\qquad\theta_{3}=(2F^{\prime\prime})^{\frac{1}{3}}\omega_{2},$

with

$\theta_{4}=(2F^{\prime\prime})^{\frac{2}{3}}\omega_{4}+b_{41}\theta_{1}+b_{42}\theta_{2}+b_{43}\theta_{3}$

and

$\theta_{5}=(2F^{\prime\prime})^{\frac{2}{3}}\omega_{5}+b_{51}\theta_{1}+b_{52}\theta_{2}+b_{53}\theta_{3}.$

Imposing Cartan’s structure equations (Nurowski’s conformal class of a
maximally symmetric $(2,3,5)$-distribution and its Ricci-flat representatives)
on $(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5})$ again gives
$b_{51}=b_{53}=0$ and $b_{41}=b_{52}$, which we set to be zero. We also obtain
$b_{42}=-\frac{1}{30}\frac{2^{\frac{2}{3}}(3F^{\prime\prime}F^{(4)}-4(F^{(3)})^{2})}{(F^{\prime\prime})^{\frac{10}{3}}}$
and $b_{43}=\frac{2^{\frac{1}{3}}F^{(3)}}{3(F^{\prime\prime})^{\frac{5}{3}}}$.
A representative metric of Nurowski’s conformal class is again given by (0.2).
The condition that the metric $g$ is conformally flat, i.e. the metric $g$ has
vanishing Weyl tensor, occurs when $F(x)$ is a solution to the nonlinear
differential equation

(1.3) $\displaystyle
10(F^{\prime\prime})^{3}F^{(6)}-80(F^{\prime\prime})^{2}F^{(3)}F^{(5)}-51(F^{\prime\prime})^{2}(F^{(4)})^{2}+336F^{\prime\prime}(F^{(3)})^{2}F^{(4)}-224(F^{(3)})^{4}=0.$

If we replace $F^{\prime\prime}(x)=e^{\int\frac{1}{2}P(x){\rm d}x}$, we find
that the conformally rescaled metric
$\tilde{g}=2^{\frac{1}{3}}(F^{\prime\prime})^{-\frac{2}{3}}g$ has the form

(1.4) $\displaystyle\tilde{g}=\frac{1}{30}(6P^{\prime}-P^{2})e^{\int-P{\rm
d}x}\omega_{1}\omega_{1}-\frac{2}{3}Pe^{-\int\frac{1}{2}P{\rm
d}x}\omega_{1}\omega_{2}+\frac{8}{3}\omega_{2}\omega_{2}+4\omega_{3}\omega_{5}-4\omega_{1}\omega_{4}-8qe^{\int\frac{1}{2}P{\rm
d}x}\omega_{2}\omega_{5}.$

Here equation (1.3) is reduced to the generalised Chazy equation

$P^{\prime\prime\prime}-2PP^{\prime\prime}+3P^{\prime
2}-\frac{4}{36-(\frac{2}{3})^{2}}(6P^{\prime}-P^{2})^{2}=0$

for $P(x)$ with parameter $k=\frac{2}{3}$. From the form of the metric
$\tilde{g}$ we can locally rescale the metric again by a conformal factor to
obtain Ricci-flat representatives.

We find that the Ricci tensor of $\Omega^{2}\tilde{g}$ is zero when $\Omega$
satisfies

$40\Omega^{\prime\prime}\Omega-80(\Omega^{\prime})^{2}-6\Omega^{2}P^{\prime}+\Omega^{2}P^{2}=0.$

If we make the substitution $\Omega=\frac{1}{\nu}$, then we obtain the
differential equation

(1.5) $\nu^{\prime\prime}-\frac{1}{40}Q\nu=0$

where $Q=P^{2}-6P^{\prime}$ and $\nu$ is to be determined. From the form the
metric $\tilde{g}$ in (1.4), we can also define new coframes by

$\displaystyle\tilde{\omega}_{1}$ $\displaystyle=e^{-\int\frac{P}{2}{\rm
d}x}\omega_{1}=\frac{{\rm d}y}{F^{\prime\prime}}-p{\rm d}x,$
$\displaystyle\tilde{\omega}_{2}$ $\displaystyle=\omega_{2}={\rm
d}p-qF^{\prime\prime}{\rm d}x,$ $\displaystyle\tilde{\omega}_{3}$
$\displaystyle=e^{-\int\frac{P}{2}{\rm d}x}\omega_{3}=\frac{{\rm
d}z}{F^{\prime\prime}}-q^{2}F^{\prime\prime}{\rm d}x,$
$\displaystyle\tilde{\omega}_{4}$ $\displaystyle=e^{\int\frac{P}{2}{\rm
d}x}\omega_{4}=F^{\prime\prime}{\rm d}q+qF^{\prime\prime\prime}{\rm d}x,$
$\displaystyle\tilde{\omega}_{5}$ $\displaystyle=e^{\int\frac{P}{2}{\rm
d}x}\omega_{5}=-\frac{F^{\prime\prime}}{2}{\rm d}x.$

We have used that $e^{-\int\frac{P}{2}{\rm d}x}=\frac{1}{F^{\prime\prime}}$.
Also replacing $6P^{\prime}-P^{2}=-Q$, this gives the cosmetic improvement for
$\tilde{g}$:

$\displaystyle\tilde{g}=-\frac{Q}{30}\tilde{\omega}_{1}\tilde{\omega}_{1}-\frac{2P}{3}\tilde{\omega}_{1}\tilde{\omega}_{2}+\frac{8}{3}\tilde{\omega}_{2}\tilde{\omega}_{2}+4\tilde{\omega}_{3}\tilde{\omega}_{5}-4\tilde{\omega}_{1}\tilde{\omega}_{4}-8q\tilde{\omega}_{2}\tilde{\omega}_{5}.$

We now investigate the solutions to (1.2) and (1.5). They are given by
Theorems 3.1 and 3.2. We first review some results about the solutions to the
generalised Chazy equation.

##  2\. Generalised Chazy equation

The generalised Chazy equation with parameter $k$ is given by

$\displaystyle y^{\prime\prime\prime}-2yy^{\prime\prime}+3y^{\prime
2}-\frac{4}{36-k^{2}}(6y^{\prime}-y^{2})^{2}=0$

and Chazy’s equation

$\displaystyle y^{\prime\prime\prime}-2yy^{\prime\prime}+3y^{\prime 2}=0$

is obtained in the limit as $k$ tends to infinity. The generalised Chazy
equation was introduced in [6], [7] and studied more recently in [8], [1], [2]
and [4]. The generalised Chazy equation with parameters $k=\frac{2}{3}$,
$\frac{3}{2}$, $2$ and $3$ was also further investigated in [12]. The solution
to the generalised Chazy equation is given by the following (see also [4] and
[12]). Let

$\displaystyle w_{1}$ $\displaystyle=-\frac{1}{2}\frac{{\rm d}}{{\rm
d}x}\log\frac{s^{\prime}}{s(s-1)},$ $\displaystyle w_{2}$
$\displaystyle=-\frac{1}{2}\frac{{\rm d}}{{\rm
d}x}\log\frac{s^{\prime}}{s-1},$ $\displaystyle w_{3}$
$\displaystyle=-\frac{1}{2}\frac{{\rm d}}{{\rm d}x}\log\frac{s^{\prime}}{s},$

where $s=s(\alpha,\beta,\gamma,x)$ is a solution to the Schwarzian
differential equation

(2.1) $\\{s,x\\}+\frac{1}{2}(s^{\prime})^{2}V=0$

and

$\\{s,x\\}=\frac{{\rm d}}{{\rm
d}x}\left(\frac{s^{\prime\prime}}{s^{\prime}}\right)-\frac{1}{2}\left(\frac{s^{\prime\prime}}{s^{\prime}}\right)^{2}$

is the Schwarzian derivative with the potential $V$ given by

(2.2)
$V=\frac{1-\beta^{2}}{s^{2}}+\frac{1-\gamma^{2}}{(s-1)^{2}}+\frac{\beta^{2}+\gamma^{2}-\alpha^{2}-1}{s(s-1)}.$

The combination $y=-2w_{1}-2w_{2}-2w_{3}$ solves the generalised Chazy
equation when

$(\alpha,\beta,\gamma)=\left(\frac{1}{3},\frac{1}{3},\frac{2}{k}\right)\text{~{}or~{}}\left(\frac{2}{k},\frac{2}{k},\frac{2}{k}\right).$

The combination $y=-w_{1}-2w_{2}-3w_{3}$ solves the generalised Chazy equation
when

$(\alpha,\beta,\gamma)=\left(\frac{1}{k},\frac{1}{3},\frac{1}{2}\right)\text{~{}or~{}}\left(\frac{1}{k},\frac{2}{k},\frac{1}{2}\right)\text{~{}or~{}}\left(\frac{1}{k},\frac{1}{3},\frac{3}{k}\right),$

with permutations of $w_{1}$, $w_{2}$ and $w_{3}$ in $y$ corresponding to
permutations of the values $\alpha$, $\beta$ and $\gamma$ in
$(\alpha,\beta,\gamma)$. The combination $y=-w_{1}-w_{2}-4w_{3}$ solves the
generalised Chazy equation whenever

$(\alpha,\beta,\gamma)=\left(\frac{1}{k},\frac{1}{k},\frac{4}{k}\right)\text{~{}or~{}}\left(\frac{1}{k},\frac{1}{k},\frac{2}{3}\right),$

again permuting $w_{1}$, $w_{2}$ and $w_{3}$ in $y$ corresponds to permuting
the values $\alpha$, $\beta$, $\gamma$ in $(\alpha,\beta,\gamma)$. Following
[1], the functions $w_{1}$, $w_{2}$ and $w_{3}$ satisfy the following system
of differential equations:

$\displaystyle w_{1}^{\prime}=w_{2}w_{3}-w_{1}(w_{2}+w_{3})+\tau^{2},$ (2.3)
$\displaystyle w_{2}^{\prime}=w_{3}w_{1}-w_{2}(w_{3}+w_{1})+\tau^{2},$
$\displaystyle w_{3}^{\prime}=w_{1}w_{2}-w_{3}(w_{1}+w_{2})+\tau^{2},$

where

$\tau^{2}=\alpha^{2}(w_{1}-w_{2})(w_{3}-w_{1})+\beta^{2}(w_{2}-w_{3})(w_{1}-w_{2})+\gamma^{2}(w_{3}-w_{1})(w_{2}-w_{3}).$

The second-order differential equation associated to the generalised Chazy
equation with parameter $k$ is given by

(2.4) $\displaystyle u_{ss}+\frac{1}{4}Vu=0$

with the same potential $V$ as given in (2.2). This corresponds to the general
solution of the Schwarzian differential equation (2.1) after exchanging
dependent and independent variables [8]. In this case $x=\frac{u_{2}}{u_{1}}$
where $u_{1}$ and $u_{2}$ are linearly independent solutions to (2.4). Using
the further substitution
$u(s)=(s-1)^{\frac{1-\gamma}{2}}s^{\frac{1-\beta}{2}}z(s)$, the equation (2.4)
can be brought to the hypergeometric differential equation

$\displaystyle s(1-s)z_{ss}+(c-(a+b+1)s)z_{s}-abz=0$

with

$a=\frac{1}{2}(1-\alpha-\beta-\gamma),\qquad
b=\frac{1}{2}(1+\alpha-\beta-\gamma),\qquad c=1-\beta.$

From the differential equations (2), we can recover $s$ by
$s=\frac{w_{1}-w_{3}}{w_{2}-w_{3}}$. From this we deduce
$s^{\prime}=2(w_{1}-w_{2})s$ and we also obtain the relation ${\rm
d}s=2(w_{1}-w_{2})s{\rm d}x$.

##  3\. Main results: Solving the equations for Ricci-flatness

In this section we give the general solution to the differential equation
(1.2) where $Q=P^{2}-6P^{\prime}$ and $P$ is a solution of the $k=\frac{3}{2}$
generalised Chazy equation in Theorem 3.1 and the general solution to the
differential equation (1.5) where again $Q=P^{2}-6P^{\prime}$ and $P$ is a
solution of the $k=\frac{2}{3}$ generalised Chazy equation in Theorem 3.2. We
first prove the following theorem

######  Theorem 3.1.

The solution to the differential equation

$\rho^{\prime\prime}-\frac{1}{45}Q\rho=0,$

where $Q=P^{2}-6P^{\prime}$ and $P$ is a solution to the $k=\frac{3}{2}$
generalised Chazy equation, is given by $\rho=\frac{u}{v}$ where $v$ is the
solution to the second-order differential equation associated to the
$k=\frac{3}{2}$ generalised Chazy equation and $u$ is a solution to the
second-order differential equation associated to the $k=3$ generalised Chazy
equation.

###### Proof.

To prove the claim, we consider the second-order differential equation of the
form $v_{ss}+\frac{1}{4}Vv=0$ associated to the generalised Chazy equation
with parameter $k=\frac{3}{2}$, where $V$ is the function given by

$\displaystyle
V=\frac{1-\beta^{2}}{s^{2}}+\frac{1-\gamma^{2}}{(s-1)^{2}}+\frac{\beta^{2}+\gamma^{2}-\alpha^{2}-1}{s(s-1)}.$

We find that $v=v(s(x))$ as a function of $x$ satisfies

$\displaystyle
v_{xx}-2(w_{1}-w_{2}-w_{3})v_{x}-((\alpha^{2}-1)w_{1}^{2}+(\beta^{2}-1)w_{2}^{2}+(\gamma^{2}-1)w_{3}^{2})v$
$\displaystyle+((\alpha^{2}+\beta^{2}-\gamma^{2}-1)w_{1}w_{2}+(\alpha^{2}-\beta^{2}+\gamma^{2}-1)w_{1}w_{3}-(\alpha^{2}-\beta^{2}-\gamma^{2}+1)w_{2}w_{3})v=0.$

We have used that

$\displaystyle\frac{{\rm d}}{{\rm
d}s}=\frac{(w_{2}-w_{3})}{2(w_{1}-w_{2})(w_{1}-w_{3})}\frac{{\rm d}}{{\rm
d}x}$

and the differential equations (2). Furthermore, the Wronskian
$W=v_{1}(v_{2})_{s}-v_{2}(v_{1})_{s}$ of the solutions to this differential
equation satisfies $W_{s}=0$, so $W=c_{0}$ and we have

$\displaystyle v_{1}^{2}=2c_{0}(w_{1}-w_{2})s$

from the consideration that $s^{\prime}=2(w_{1}-w_{2})s=\frac{v_{1}^{2}}{W}$,
and furthermore we obtain from the differential equation that the Wronskian
$W=\frac{v(s(x))^{2}}{2(w_{1}-w_{2})s(x)}$ satisfies, that

(3.1) $\displaystyle v_{x}-v(w_{1}-w_{2}-w_{3})=0.$

This equation implies the differential equation for $v$ above, by using the
fact that $w_{i}$ satisfy the differential equations (2).

Upon making the substitution $\rho=\frac{u(x)}{v(x)}$ into equation (1.2), and
using equation (3.1), we obtain a differential equation for $u(x)$ remaining.
This differential equation for $u(x)$ turns out to be of the form

$\displaystyle
u_{xx}-2(w_{1}-w_{2}-w_{3})u_{x}-((\tilde{\alpha}^{2}-1)w_{1}^{2}+(\tilde{\beta}^{2}-1)w_{2}^{2}+(\tilde{\gamma}^{2}-1)w_{3}^{2})u$
$\displaystyle+((\tilde{\alpha}^{2}+\tilde{\beta}^{2}-\tilde{\gamma}^{2}-1)w_{1}w_{2}+(\tilde{\alpha}^{2}-\tilde{\beta}^{2}+\tilde{\gamma}^{2}-1)w_{1}w_{3}-(\tilde{\alpha}^{2}-\tilde{\beta}^{2}-\tilde{\gamma}^{2}+1)w_{2}w_{3})u=0,$

which is the same differential equation for $v$ with different constants
$\tilde{\alpha}$, $\tilde{\beta}$, $\tilde{\gamma}$. This is the differential
equation associated to the generalised Chazy equation with parameter $k=3$. We
see this automatically when we compute the values with $Q=P^{2}-6P^{\prime}$
where $P$ is the solution of the generalised Chazy equation with parameter
$k=\frac{3}{2}$. Specialising to the case where $k=\frac{3}{2}$, we obtain the
following:

For the solutions given by $P=-2w_{1}-2w_{2}-2w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{4}{3},\frac{4}{3},\frac{4}{3})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{2}{3},\frac{2}{3})$.
When $(\alpha,\beta,\gamma)=(\frac{4}{3},\frac{1}{3},\frac{1}{3})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{1}{3},\frac{1}{3})$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{3},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$.
When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{4}{3},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{3},\frac{2}{3},\frac{1}{2})$.
When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{3},2)$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{3},\frac{1}{3},1)$.

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{2}{3},\frac{2}{3})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{1}{3},\frac{1}{3})$.
When $(\alpha,\beta,\gamma)=(\frac{8}{3},\frac{2}{3},\frac{2}{3})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{4}{3},\frac{1}{3},\frac{1}{3})$.

The values $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ are precisely the
ones that show up in the solutions of the $k=3$ generalised Chazy equation.
See [12] for the list of $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ when
$k=3$.

∎

The determination of solutions to equation (1.5) is similar to that of Theorem
3.1. We prove the following

######  Theorem 3.2.

The solution to the differential equation

(3.2) $\nu^{\prime\prime}-\frac{1}{40}Q\nu=0,$

where $Q=P^{2}-6P^{\prime}$ and $P$ is a solution of the $k=\frac{2}{3}$
generalised Chazy equation, is given by $\nu=\frac{u}{v}$, where $v$ is a
solution to the second-order differential equation associated to the
$k=\frac{2}{3}$ generalised Chazy equation and $u$ is a solution to the
second-order differential equation associated to the $k=2$ generalised Chazy
equation.

###### Proof.

The proof of the claim is similar to the proof of the previous theorem. From
the differential equation of the form $v_{ss}+\frac{1}{4}Vv=0$ associated to
the $k=\frac{2}{3}$ generalised Chazy equation, where $V$ is the function
given by

$\displaystyle
V=\frac{1-\beta^{2}}{s^{2}}+\frac{1-\gamma^{2}}{(s-1)^{2}}+\frac{\beta^{2}+\gamma^{2}-\alpha^{2}-1}{s(s-1)},$

we find that $v=v(s(x))$ as a function of $x$ satisfies

$\displaystyle
v_{xx}-2(w_{1}-w_{2}-w_{3})v_{x}-((\alpha^{2}-1)w_{1}^{2}+(\beta^{2}-1)w_{2}^{2}+(\gamma^{2}-1)w_{3}^{2})v$
$\displaystyle+((\alpha^{2}+\beta^{2}-\gamma^{2}-1)w_{1}w_{2}+(\alpha^{2}-\beta^{2}+\gamma^{2}-1)w_{1}w_{3}-(\alpha^{2}-\beta^{2}-\gamma^{2}+1)w_{2}w_{3})v=0.$

Like in the proof of Theorem 3.1, it can also be deduced that (3.1) holds for
$v$, i.e.

(3.3) $\displaystyle v_{x}-v(w_{1}-w_{2}-w_{3})=0,$

which again implies the differential equation for $v$ above, by using the fact
that $w_{i}$ satisfy the differential equations (2).

Upon making the substitution $\nu=\frac{u(x)}{v(x)}$ into equation (3.2), and
using equation (3.3), we obtain a differential equation for $u(x)$ remaining.
The differential equation for $u(x)$ is again

(3.4) $\displaystyle
u_{xx}-2(w_{1}-w_{2}-w_{3})u_{x}-((\tilde{\alpha}^{2}-1)w_{1}^{2}+(\tilde{\beta}^{2}-1)w_{2}^{2}+(\tilde{\gamma}^{2}-1)w_{3}^{2})u$
$\displaystyle+((\tilde{\alpha}^{2}+\tilde{\beta}^{2}-\tilde{\gamma}^{2}-1)w_{1}w_{2}+(\tilde{\alpha}^{2}-\tilde{\beta}^{2}+\tilde{\gamma}^{2}-1)w_{1}w_{3}-(\tilde{\alpha}^{2}-\tilde{\beta}^{2}-\tilde{\gamma}^{2}+1)w_{2}w_{3})u=0,$

which is the same differential equation for $v$ but with different constants
$\tilde{\alpha}$, $\tilde{\beta}$, $\tilde{\gamma}$. The equation (3.4)
corresponds to the second-order differential equation associated to the $k=2$
generalised Chazy equation. To see this, we shall compute these constants when
$Q=P^{2}-6P^{\prime}$ and $P$ is the solution of the generalised Chazy
equation with parameter $k=\frac{2}{3}$. Specialising to the case where
$k=\frac{2}{3}$, we obtain the following:

For the solutions given by $P=-2w_{1}-2w_{2}-2w_{3}$, when
$(\alpha,\beta,\gamma)=(3,3,3)$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(1,1,1)$. When
$(\alpha,\beta,\gamma)=(3,\frac{1}{3},\frac{1}{3})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(1,\frac{1}{3},\frac{1}{3})$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{3}{2},\frac{1}{3},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{2},\frac{1}{3},\frac{1}{2})$.
When $(\alpha,\beta,\gamma)=(\frac{3}{2},3,\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{2},1,\frac{1}{2})$.
When $(\alpha,\beta,\gamma)=(\frac{3}{2},\frac{1}{3},\frac{9}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{1}{2},\frac{1}{3},\frac{3}{2})$.

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{3}{2},\frac{3}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(\frac{2}{3},\frac{1}{2},\frac{1}{2})$.
When $(\alpha,\beta,\gamma)=(6,\frac{3}{2},\frac{3}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})=(2,\frac{1}{2},\frac{1}{2})$.

The values $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ are precisely the
ones that show up in the solutions of the $k=2$ generalised Chazy equation.
See also [12] for the list of $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$
when $k=2$.

∎

##  4\. Solution to the equation for Ricci-flatness for general Chazy
parameter

More generally, when $P$ is a solution to the generalised Chazy equation with
parameter $k$, the metric $g$ is no longer conformally flat but we can still
find the conformal scale for which the Ricci tensor vanishes.

In the case of (1.2) with solutions given by $\nu=\frac{u}{v}$ where $v$ is
the second-order differential equation associated to the generalised Chazy
equation with parameter $k$, we find that $u$ is a solution to the second-
order differential equation associated to the generalised Chazy equation with
parameter $\tilde{k}$ with

(4.1) $\frac{45}{\tilde{k}^{2}}-\frac{9}{k^{2}}=1.$

The values $(\alpha,\beta,\gamma)$ appearing in $V$ in the differential
equation $v_{ss}+\frac{1}{4}Vv=0$ are related to the values
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ appearing in $V$ in the
differential equation $u_{ss}+\frac{1}{4}Vu=0$ by the following. For the
solutions given by $P=-2w_{1}-2w_{2}-2w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{2}{k},\frac{2}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle\frac{45}{4}\tilde{\alpha}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt\frac{45}{4}\tilde{\beta}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt\frac{45}{4}\tilde{\gamma}^{2}-\left(\frac{3}{k}\right)^{2}=1.$

When $(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{1}{3},\frac{1}{3})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle\frac{45}{4}\tilde{\alpha}^{2}-\left(\frac{3}{k}\right)^{2}=1$

and $\tilde{\beta}=\frac{1}{3}$, $\tilde{\gamma}=\frac{1}{3}$. Here and
subsequently, we shall consider the positive square root that gives positive
$\tilde{\alpha}$, $\tilde{\beta}$ and $\tilde{\gamma}$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 45\tilde{\alpha}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt\tilde{\beta}=\frac{1}{3},\hskip 12.0pt\tilde{\gamma}=\frac{1}{2}.$

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{2}{k},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 45\tilde{\alpha}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt\frac{45}{4}\tilde{\beta}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt\tilde{\gamma}=\frac{1}{2}.$

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{3}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 45\tilde{\alpha}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt\tilde{\beta}=\frac{1}{3},\hskip
12.0pt5\tilde{\gamma}-\left(\frac{3}{k}\right)^{2}=1.$

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{4}{k},\frac{1}{k},\frac{1}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle\frac{45}{16}\tilde{\alpha}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt45\tilde{\beta}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt45\tilde{\gamma}^{2}-\left(\frac{3}{k}\right)^{2}=1.$

When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{k},\frac{1}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle\tilde{\alpha}=\frac{2}{3},\hskip
12.0pt45\tilde{\beta}^{2}-\left(\frac{3}{k}\right)^{2}=1,\hskip
12.0pt45\tilde{\gamma}^{2}-\left(\frac{3}{k}\right)^{2}=1.$

In all cases the appropriate substitution of $\tilde{\alpha}$, $\tilde{\beta}$
and $\tilde{\gamma}$ in terms of the Chazy parameter $\tilde{k}$ gives
equation (4.1), so it can be seen that the equation for $u$ is the second-
order differential equation associated to the generalised Chazy equation with
parameter $\tilde{k}$, related to $k$ by (4.1). The further substitution
$k=\frac{3}{m}$ and $\tilde{k}=\frac{3}{\tilde{m}}$ into (4.1) gives

$5\tilde{m}^{2}-m^{2}=1,$

which has integer solutions when considered as a negative Pell equation. For
integer solutions $m$ and $\tilde{m}$ we obtain

$\displaystyle m$
$\displaystyle=\pm(\frac{1}{2}(2+\sqrt{5})^{2n+1}+\frac{1}{2}(2-\sqrt{5})^{2n+1}),$
$\displaystyle\tilde{m}$
$\displaystyle=\pm(\frac{\sqrt{5}}{10}(2+\sqrt{5})^{2n+1}-\frac{\sqrt{5}}{10}(2-\sqrt{5})^{2n+1}).$

They take on values $(m,\tilde{m})=(2,1)$, $(38,17)$, $(682,305)$,
$(12238,5473)$ and so on for $n\in\mathbb{N}\cup\\{0\\}$. They also give the
corresponding pairs of Chazy parameters $(k,\tilde{k})=(\frac{3}{2},3)$,
$(\frac{3}{38},\frac{3}{17})$ and so on, with the fundamental solution ($n=0$)
agreeing with the result of Theorem 3.1 in the conformally flat case.

In the case of (1.5) with solutions given by $\nu=\frac{u}{v}$ where $v$ is
the second-order differential equation associated to the generalised Chazy
equation with parameter $k$, we find that $u$ is a solution to the second-
order differential equation associated to the generalised Chazy equation with
parameter $\tilde{k}$ with

(4.2) $\frac{40}{\tilde{k}^{2}}-\frac{4}{k^{2}}=1.$

In this case we obtain the relationship between the values
$(\alpha,\beta,\gamma)$ and $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ as
follows. For $P=-2w_{1}-2w_{2}-2w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{2}{k},\frac{2}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 10\tilde{\alpha}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt10\tilde{\beta}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt10\tilde{\gamma}^{2}-\left(\frac{2}{k}\right)^{2}=1.$

Considering integer solutions $\alpha$ and $\tilde{\alpha}$ to the negative
Pell equation $10\tilde{\alpha}^{2}-\alpha^{2}=1$ (and also $\beta$,
$\tilde{\beta}$ and $\gamma$, $\tilde{\gamma}$ respectively), we find

$\displaystyle\alpha$
$\displaystyle=\pm(\frac{1}{2}(3+\sqrt{10})^{2n+1}+\frac{1}{2}(3-\sqrt{10})^{2n+1}),$
$\displaystyle\tilde{\alpha}$
$\displaystyle=\pm(\frac{\sqrt{10}}{20}(3+\sqrt{10})^{2n+1}-\frac{\sqrt{10}}{20}(3-\sqrt{10})^{2n+1}),$

where $n\in\mathbb{Z}$. Positive integer solutions are given by
$(\alpha,\tilde{\alpha})=(3,1)$, $(117,37)$, $(4443,1405)$, $(168717,53353)$
and so on for $n\in\mathbb{N}\cup\\{0\\}$. They give the relationship between
the pairs of Chazy parameters $k=\frac{2}{\alpha}$ and
$\tilde{k}=\frac{2}{\tilde{\alpha}}$, with $(k,\tilde{k})=(\frac{2}{3},2)$,
$(\frac{2}{117},\frac{2}{37})$ and so on for $n\in\mathbb{N}\cup\\{0\\}$. For
these parameters the associated hypergeometric functions are algebraic. Again
the fundamental solution ($n=0$) agrees with the result of Theorem 3.2 in the
conformally flat case.

The determination of the other values of $(\alpha,\beta,\gamma)$ and
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ are as follows. For the same
$P$, when $(\alpha,\beta,\gamma)=(\frac{2}{k},\frac{1}{3},\frac{1}{3})$, we
find $(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 10\tilde{\alpha}^{2}-\left(\frac{2}{k}\right)^{2}=1$

and $\tilde{\beta}=\frac{1}{3}$, $\tilde{\gamma}=\frac{1}{3}$.

For the solutions given by $P=-w_{1}-2w_{2}-3w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 40\tilde{\alpha}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt\tilde{\beta}=\frac{1}{3},\hskip 12.0pt\tilde{\gamma}=\frac{1}{2}.$

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{2}{k},\frac{1}{2})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 40\tilde{\alpha}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt10\tilde{\beta}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt\tilde{\gamma}=\frac{1}{2}.$

When $(\alpha,\beta,\gamma)=(\frac{1}{k},\frac{1}{3},\frac{3}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle 40\tilde{\alpha}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt\tilde{\beta}=\frac{1}{3},\hskip
12.0pt\frac{40}{9}\tilde{\gamma}-\left(\frac{2}{k}\right)^{2}=1.$

Finally for the solutions given by $P=-4w_{1}-w_{2}-w_{3}$, when
$(\alpha,\beta,\gamma)=(\frac{4}{k},\frac{1}{k},\frac{1}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle\frac{5}{2}\tilde{\alpha}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt40\tilde{\beta}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt40\tilde{\gamma}^{2}-\left(\frac{2}{k}\right)^{2}=1.$

When $(\alpha,\beta,\gamma)=(\frac{2}{3},\frac{1}{k},\frac{1}{k})$, we find
$(\tilde{\alpha},\tilde{\beta},\tilde{\gamma})$ with

$\displaystyle\tilde{\alpha}=\frac{2}{3},\hskip
12.0pt40\tilde{\beta}^{2}-\left(\frac{2}{k}\right)^{2}=1,\hskip
12.0pt40\tilde{\gamma}^{2}-\left(\frac{2}{k}\right)^{2}=1.$

In all cases the appropriate substitution of $\tilde{\alpha}$, $\tilde{\beta}$
and $\tilde{\gamma}$ in terms of the Chazy parameter $\tilde{k}$ gives
equation (4.2), and therefore the equation for $u$ is the second-order
differential equation associated to the generalised Chazy equation with
parameter $\tilde{k}$, related to $k$ by (4.2). Altogether, with the exception
of the parameters $k=\frac{3}{2}$ and $k=\frac{2}{3}$ as mentioned above, they
give Ricci-flat but non-conformally flat examples of Nurowski’s metric.

## References

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# Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

Weicai Wu a, Shouchuan Zhang b and Yao-Zhong Zhang c,d
$a$. Department of Mathematics, Zhejiang University
Hangzhou 310007, P.R. China
$b$. Department of Mathematics, Hunan University
Changsha 410082, P.R. China
$c$. School of Mathematics and Physics, The University of Queensland
Brisbane 4072, Australia
$d$. CAS Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics
Chinese Academy of Sciences, Beijing 100190, China
Emails<EMAIL_ADDRESS>(SZ<EMAIL_ADDRESS>(YZZ)

###### Abstract

It is shown that if $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal
type with $\dim V>1$, then $\dim(\mathfrak{B}(V))=\infty$ $($resp.
$\dim(\mathfrak{L}(V))=\infty$$)$ $($ resp. $\dim(\mathfrak{L}^{-}(V))=\infty$
$)$ if and only if $\Delta(\mathfrak{B}(V))$ is an arithmetic root system and
the quantum numbers (i.e. the fixed parameters ) of generalized Dynkin
diagrams of $V$ are of finite order.

2010 Mathematics Subject Classification: 16W30, 22E60
Keywords: Nichols algebra, Nichols Lie algebra, Nichols braided Lie algebra.

##  1 Introduction

The question of finite-dimensionality of Nichols algebras dominates an
important part of the recent developments in the theory of (pointed) Hopf
algebras(see e.g. [AHS08, AS10, An11, He05, He06a, He06b, WZZ15a, WZZ15b,
ZWTZ]. The interest in this problem comes from the lifting method by
Andruskiewitsch and Schneider to classify finite dimensional (Gelfand-
Kirillov) pointed Hopf algebras, which are generalizations of quantized
enveloping algebras of semi-simple Lie algebras.

The classification of arithmetic root systems is obtained in [He05] and
[He06a]. It is proved in [WZZ15b] that Nichols algebra $\mathfrak{B}(V)$ is
finite-dimensional if and only if Nichols braided Lie algebra
$\mathfrak{L}(V)$ is finite-dimensional. It is shown in [ZWTZ] that
$\dim(\mathfrak{L}^{-}(V))=\infty$ if $\Delta(\mathfrak{B}(V))$ is not an
arithmetic root system.

The main motivation for the subject of this paper and papers [He05, He06a] is
the following question of Andruskiewitsch [A02].

Question 5.9. Given a braided vector space $V$ of diagonal type, decide when
$\mathfrak{B}(V)$ is finite dimensional. If so, compute $\dim\mathfrak{B}(V)$,
and give a nice presentation by generators and relations.

Papers [He05] and [He06a] solved the first part of this problem under the
following the additional property in [He05, Section 2.2]:

(P) the height of a PBW generator ( i.e hard super-letter ) of
$\mathbb{Z}^{n}$ -degree $e$ is finite if and only if $2\leq{\rm
ord}(\chi(e,e))<\infty$, and in this case it coincides with ${\rm
ord}(\chi(e,e))$.

In this paper we show that property (P) holds in arithmetic root systems (see
Theorem 4.1 ). Consequently, this paper solves the first part of problem 5.9.

In order to study infinite dimensional Lie algebras V. Kac and R. Moody
independently introduced Kac-Moody algebras in 1960s ( [Ka85, Wa02] ).

In this paper we focus on when a Nichols algebra and a Nichols (braided) Lie
algebra have finite dimension. Such finite-dimensional Nichols algebras play a
fundamental role in various subjects such as pointed Hopf algebras and
logarithmic quantum field theories. There exists a large number of examples of
infinite dimensional Lie algebras in Nichols Lie algebras.

The main results of this paper are as follows. If $\mathfrak{B}(V)$ is
connected Nichols algebra of diagonal type with $\dim V>1$, then
$\dim(\mathfrak{B}(V))=\infty$ $($resp. $\dim(\mathfrak{L}(V))=\infty$$)$ $($
resp. $\dim(\mathfrak{L}^{-}(V))=\infty$ $)$ if and only if
$\Delta(\mathfrak{B}(V))$ is an arithmetic root system and the quantum numbers
(i.e. the fixed parameters ) of generalized Dynkin diagrams of $V$ are of
finite order.

##  Preliminaries

Throughout this paper braided vector space $V$ is of diagonal type with basis
$x_{1},x_{2},\cdots,x_{n}$ and braiding $C(x_{i}\otimes
x_{j})=p_{ij}x_{j}\otimes x_{i}$ without special announcement. Nichols algebra
$\mathfrak{B}(V)$ over braided vector space $V$ has two gradations. A
gradation is the length such that $\mid u\mid=r$ of length of $u$ for
$u=x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}$. Another gradation is
$\mathbb{Z}^{n}$-graded such that ${\rm deg}(x_{i})=e_{i}$ for $1\leq i\leq
n$, where $E=\\{e_{1},e_{2},\cdots,e_{n}\\}$ is a basis of $\mathbb{Z}^{n}.$

Define linear map $p$ from $\mathfrak{B}(V)\otimes\mathfrak{B}(V)$ to $F$ such
that $p(u\otimes v)=\chi(deg(u),deg(v)),$ for any homogeneous element
$u,v\in\mathfrak{B}(V).$ For convenience, $p(u\otimes v)$ is denoted by
$p_{uv}$. Let $\widetilde{p}_{uv}:=p_{uv}p_{vu}.$

Let $A=:\\{x_{1},x_{2},\cdots,x_{n}\\}$ a alphabet; $A^{*}$ denotes the set of
all of words in $A$; $A^{+}=:A^{*}\setminus 1$. Define
$x_{1}<x_{2}<\cdots<x_{n}$ and the order on $A^{*}$ is the lexicographic
orderings. The concepts of words refer [Lo83].

######  Definition 1.1.

[Kh99, Def. 1] A word $u$ is called a Lyndon word if $\mid u\mid=1$ or $\mid
u\mid\geq 2$, and for each representation $u=u_{1}u_{2}$, where $u_{1}$and
$u_{2}$ are nonempty words, the inequality $u<u_{2}u_{1}$ holds.

Any word $u\in A^{*}$ has a unique decomposition into the product of a non-
increasing sequence of Lyndon words by [Lo83, Th.5.1.5], If $u$ is a Lyndon
word with $\mid u\mid>1$, then there uniquely exist two Lyndon words $v$ and
$w$ such that $u=vw$ and $v$ is shortest (see [Lo83, Pro. 5.1.3] and [He07])
(the composition is called the Shirshov decomposition of $u$).

######  Definition 1.2.

We inductively define a linear map $[\ \ ]$ from $A^{+}$ to $\mathfrak{B}(V)$
as follows:

(1) $[u]=:u$ when $u$ is a letter;

(2) $[u]=:[w][v]-p_{wv}[v][w]$ when $u$ is a Lyndon word with $\mid u\mid>1$
and $u=vw$ is a Shirshov decomposition;

(3) $[u]=:[[[l_{1},l_{2}],l_{3}]\cdots l_{n}]$, when $u=l_{1}l_{2}\cdots
l_{n}$ is a Shirshov decomposition.

$[u]$ is called a nonassociative word for any $u\in A^{+}$. $[u]$ is called a
super-letter if $u$ is a Lyndon word. A multiplication of super-letters is
called a super-word.

######  Definition 1.3.

[Kh99, Def. 6] A super-letter $[u]$ is said to be hard if it is not a linear
combination of super-words with length $\mid u\mid$ in greater super-letters
than $[u]$.

######  Definition 1.4.

[Kh99, Def 7] or [He07, before Th. 10] We said that the height of a super-
letter $[u]$ with length $d$ equals a natural number $h$ if $h$ is least with
the following properties:

(1) $p_{uu}$ is a $t$-primitive root of unity, and $h=t$;

(2) super-word $[u]^{h}$ is a linear combination of super-words of length $hd$
in greater super-letters than $[u].$

If the number $h$ with above properties does not exist then we say that the
height of $[u]$ is infinite.

Let ${\rm ord}(p_{uu})$ denote the order of $p_{uu}$ with respect to
multiplication.

$D(V)=D=:\\{[u]\mid[u]\hbox{ is a hard super-letter }\\}$,we know $h_{u}>1$
for $\ \forall\ [u]\in D$.

$P(V)=P=:\\{[u_{1}]^{k_{1}}[u_{2}]^{k_{2}}\cdots[u_{s}]^{k_{s}}\ |\ [u_{i}]\in
D,k_{i},s\in\mathbb{N}_{0};0\leq k_{i}<h_{u_{i}};1\leq i\leq
s;u_{s}<u_{s-1}<\cdots<u_{1}\\}$

If $[u]\in D$ and ${\rm ord}(p_{u,u})=m>1$ with $h_{u}=\infty$, then $[u]$ is
called an $m$-infinity element.

######  Theorem 1.5.

[Kh99, Th. 2] or [He05, Th. 1.4.6], $P$ is a basis of $\mathfrak{B}(V)$.

Let $G$ be a nonempty set; $R$ a nonempty subset of $G\times G$, and
$\circ:R\rightarrow G$ a map of sets. The pair $(G,\circ)$ is called a
groupoid if it satisfies the following conditions.

${\rm GD_{1}}$ If $(x,y)\in R$ then each of the three elements $x,y,x\circ y$
is uniquely determined by the other two.

${\rm GD_{2}}$ If $(x,y),(y,z)\in R$ then $(x\circ y,z),(x,y\circ z)\in R$ and
$(x\circ y)\circ z=x\circ(y\circ z).$

${\rm GD_{3}}$ If $(x,y),(x\circ y,z)\in R$ then $(y,z),(x,y\circ z)\in R$ and
$(x\circ y)\circ z=x\circ(y\circ z)$.

${\rm GD_{4}}$ If $(y,z),(x,y\circ z)\in R$ then $(x,y),(x\circ y,z)\in R$ and
$(x\circ y)\circ z=x\circ(y\circ z)$.

${\rm GD_{5}}$ For all $x\in G$ there exist unique elements $e,f,y\in G$ such
that $(e,x),(x,f),(y,x)\in R$, $e\circ x=x\circ f=x,$ and $y\circ x=f$.

${\rm GD_{6}}$ If $e\circ e=e,f\circ f=f$ for certain $e,f\in G$ then there
exists $x\in G$ such that $e\circ x=x\circ f=x.$

Let $E$ and $F$ be bases of $\mathbb{Z}^{n}$ and $\chi$ a bicharacter on
$\mathbb{Z}^{n}.$ Define $m(e^{\prime},e^{\prime\prime}):={\rm
min}\\{m\in\mathbb{N}_{0}\mid\chi(e^{\prime},e^{\prime})^{m}\chi(e^{\prime},e^{\prime\prime})\chi(e^{\prime\prime},e^{\prime})=1\
{\rm or}\ \chi(e^{\prime},e^{\prime})^{m+1}=1,$
$\chi(e^{\prime},e^{\prime})\not=1\\}$ for $e^{\prime},e^{\prime\prime}\in F$
with $e^{\prime}\not=e^{\prime\prime}$ and $m(e^{\prime},e^{\prime}):=-2$.
$s_{e^{\prime},F}(e^{\prime\prime}):=e^{\prime\prime}+m(e^{\prime},e^{\prime\prime})e^{\prime}$
for $e^{\prime},e^{\prime\prime}\in F.$

Let $\widetilde{W}$ denote the groupoid consisting of all pairs $(T,E)$, where
$T\in{\rm Aut}(\mathbb{Z}^{n})$ and $E$ is a basis of $\mathbb{Z}^{n}$, and
the composition $(T_{1},E_{1})\circ(T_{2},E_{2})$ is defined (and is then
equal to $(T_{1}T_{2},E_{2}))$ if and only if $T_{2}(E_{2})=E_{1}$ (i.e.
$R=\\{(T_{1},E_{1})\times(T_{2},E_{2})\mid T_{2}(E_{2})=E_{1}\\}$).

Let $E$ be a basis of $\mathbb{Z}^{n}$ and $\chi$ a bicharacter on
$\mathbb{Z}^{n}.$ Define $W_{\chi,E}$ as the smallest subgroupoid of
$\widetilde{W}$ which contains $({\rm id},E)$, and if $({\rm id},F)\in
W_{\chi,E}$ for a basis $F$ of $\mathbb{Z}^{n}$ and $f\in F$, then
$(s_{f,F},F),({\rm id},s_{f,F}(F))\in W_{\chi,E}$ whenever $s_{f,F}$ is
defined. It is called the Weyl groupoid associated to the pair $(\chi,E)$.

The groupoid $W_{\chi,E}$ is called full, if $s_{f,F}$ is well-defined for all
bases $F$ of $\mathbb{Z}^{n}$ with $({\rm id},F)\in W_{\chi,E}$ and for all
$f\in F$.

A triple $(\Delta,\chi,E)$ is called an arithmetic root system if $W_{\chi,E}$
is full and finite wirh $\Delta:=\cup\\{F\mid({\rm id},F)\in W_{\chi,E}\\}.$

$\Delta^{+}(\mathfrak{B}(V)):=\\{\deg(u)\mid[u]\in D\\}$ and
$\Delta(\mathfrak{B}(V)):=\Delta^{+}(\mathfrak{B}(V))\cup\Delta^{-}(\mathfrak{B}(V))$.
Let $\mathfrak{L}(V)$ denote the braided Lie algebras generated by $V$ in
$\mathfrak{B}(V)$ under Lie operations $[x,y]=yx-p_{yx}xy$, for any
homogeneous elements $x,y\in\mathfrak{B}(V)$. $(\mathfrak{L}(V),[\ ])$ is
called Nichols braided Lie algebra of $V$. Let $\mathfrak{L}^{-}(V)$ denote
the Lie algebras generated by $V$ in $\mathfrak{B}(V)$ under Lie operations
$[x,y]^{-}=yx-xy$, for any homogeneous elements $x,y\in\mathfrak{B}(V)$.
$(\mathfrak{L}^{-}(V),[\ ]^{-})$ is called Nichols Lie algebra of $V$.

The dual $\mathfrak{B}(V^{*})$ of Nichols algebra $\mathfrak{B}(V)$ of rank
$n$ in [He05, Section 1.3]. Let $y_{i}$ be a dual basis of $x_{i}$.
$\delta(y_{i})=g_{i}^{-1}\otimes y_{i}$, $g_{i}\cdot y_{j}=p_{ij}^{-1}y_{j}$
and $\Delta(y_{i})=g_{i}^{-1}\otimes y_{i}+y_{i}\otimes 1.$ There exists a
bilinear map $<\cdot,\cdot>$:
$(\mathfrak{B}(V^{*})\\#FG)\times\mathfrak{B}(V)$ $\longrightarrow$
$\mathfrak{B}(V)$ such that

$<y_{i},uv>=<y_{i},u>v+g_{i}^{-1}.u<y_{i},v>$ and
$<y_{i},<y_{j},u>>=<y_{i}y_{j},u>$

for any $u,v\in\mathfrak{B}(V)$. Furthermore, for any
$u\in\oplus_{i=1}^{\infty}\mathfrak{B}(V)_{(i)}$, one has that $u=0$ if and
only if $<y_{i},u>=0$ for any $1\leq i\leq n.$

We have the braided Jacobi identity as follows:

$\displaystyle[[u,v],w]=[u,[v,w]]+p_{vw}^{-1}[[u,w],v]+(p_{wv}-p_{vw}^{-1})v\cdot[u,w].$
(1) $\displaystyle[u,v\cdot w]=p_{wu}[uv]\cdot w+v\cdot[uw].$ (2)

######  Theorem 1.6.

([He05, Th. 2.5.3]) If $\chi$ is a bicharacter on $\mathbb{Z}^{n}$ and
$(\Delta,\chi,E)$ is an arithmetic root system then for the braided vector
space $V$ of diagonal type with $\dim V=n$ and with structure constants
$q_{ij}:=\chi(e_{i},e_{j})$ one has $\Delta(\mathfrak{B}(V))=\Delta$.
Conversely, if $V$ is a braided vector space of diagonal type such that
$\Delta^{+}(\mathfrak{B}(V))$ is finite then
$(\Delta(\mathfrak{B}(V)),\chi,E)$ is an arithmetic root system, where
$E=\\{e_{1},e_{2},\cdots,e_{n}\\}$ and $\chi(e_{i},e_{j})=q_{ij}$ for $1\leq
i,j\leq n.$

If $(\Delta,\chi,E)$ is an arithmetic root system and a hyperplane $H$ with
$0\in H$ (i.e. $\dim H=n-1$), then there exists
$E_{H}:=\\{f_{1},f_{2},\cdots,f_{l}\\}$ such that $(\Delta\cap
H,\chi_{\Gamma\times\Gamma},E_{H})$ is an arithmetic root subsystem with
$\Gamma=\mathbb{R}(\Delta\cap H)\cap\mathbb{Z}^{n},$ written
$\Delta(\chi,f_{1},f_{2},\cdots,f_{l})=(\Delta\cap
H,\chi_{\Gamma\times\Gamma},E_{H})$ in short (see [He05, Pro. 2.7.1]).

Throughout, $\mathbb{Z}=:\\{x\mid x\hbox{ is an integer}\\}.$
$\mathbb{R}=:\\{x\mid x\hbox{ is a real number}\\}$. $\mathbb{N}_{0}=:\\{x\mid
x\in\mathbb{Z},x\geq 0\\}.$ $\mathbb{N}=:\\{x\mid x\in\mathbb{Z},x>0\\}$.
$\mathbb{R}_{+}=:\\{x\geq 0\mid x\hbox{ is a real number}\\}$. $F$ denotes the
base field, which is an algebraic closed field with characteristic zero.
$F^{*}=F\backslash\\{0\\}$. $\mathbb{S}_{n}$ denotes symmetric group,
$n\in\mathbb{N}$.

##  2 $m$-infinity elements

In this section we show that there does not exist any $m$-infinity elements in
arithmetic root systems.

######  Lemma 2.1.

Assume that
$\widetilde{f_{1}},\widetilde{f_{2}},\cdots,\widetilde{f_{\widetilde{l}}}$ are
linearly independent in arithmetic root system $(\Delta,\chi,E)$ with
$\widetilde{H}:=\mathbb{R}(\widetilde{f_{1}},\widetilde{f_{2}},\widetilde{f_{3}},\cdots,\widetilde{f_{\widetilde{l}}})$.
If

$\displaystyle\Delta\cap\widetilde{H}\subseteq\mathbb{R}_{+}(\widetilde{f_{1}},\widetilde{f_{2}},\cdots,\widetilde{f_{\widetilde{l}}})\cup-\mathbb{R}_{+}(\widetilde{f_{1}},\widetilde{f_{2}},\cdots,\widetilde{f_{\widetilde{l}}}),$
(3)

then there exists a hyperplane $H$ with $0\in H$ and
$E_{H}=\\{f_{1},f_{2},\cdots,f_{l}\\}$ with $l=\widetilde{l}$, such that
$\Delta\cap H=\Delta\cap\widetilde{H}$ and $(\Delta\cap
H,\chi_{\Gamma\times\Gamma},E_{H})$ is an arithmetic root system with
$\Gamma=\mathbb{R}(\Delta\cap H)\cap\mathbb{Z}^{n}.$

Proof. It is clear that there exists $H_{1}$ with $H_{1}\cap\Delta=\emptyset$
such that $H=\widetilde{H}\oplus H_{1}$ with $\dim H=n-1.$ By [He05, Pro.
2.7.1], there exists $E_{H}=\\{f_{1},f_{2},\cdots,f_{l}\\}$ such that
$(\Delta\cap H,\chi_{\Gamma\times\Gamma},E_{H})$ is an arithmetic root system
with $\Gamma=\mathbb{R}(\Delta\cap H)\cap\mathbb{Z}^{n}$ and

$\displaystyle E_{H}\subseteq\Delta\cap
H=\Delta\cap\widetilde{H}\subseteq\mathbb{R}E_{H}.$ (4)

Considering (3) and (4), we have $l=\widetilde{l}$. $\Box$

######  Lemma 2.2.

If $(\Delta(\mathfrak{B}(V)),\chi,E)$ is an arithmetic root system with
$[v],[w]\in D$ and ${\rm deg}(v)\not={\rm deg}(w),$ then there exists an
arithmetic root subsystem $\Delta(\chi;f_{1},f_{2})$ of
$\Delta(\mathfrak{B}(V))$ such that ${\rm deg}(v)$ and ${\rm deg}(w)$ are in
the subsystem. Furthermore, if $[u]=[[v],[w]]$ is the Shirshov decomposition
of $[u]\in D$ then the subsystem $\Delta(\chi;f_{1},f_{2})$ is connected and
${\rm deg}(u)$ are in the subsystem.

Proof. We show this by the following steps.

(i) Let $v_{1}=v$ and $v_{2}=w$. If
$\deg(v)-k\deg(w)\notin\mathbb{N}\cdot\Delta(\mathfrak{B}(V))$ for any
$k\in\mathbb{N}$, let $\widetilde{f_{1}}:={\rm deg}(v_{1})$ and
$\widetilde{f_{2}}:={\rm deg}(v_{2})$. If there exists a $k_{1}\in\mathbb{N}$
such that
$\deg(v_{1})-k_{1}\deg(v_{2})\in\mathbb{N}\cdot\Delta(\mathfrak{B}(V))$, then
there exist $v_{3}\in\Delta^{+}(\mathfrak{B}(V))$ and
$l_{1}\in\mathbb{Z}\setminus\\{0\\}$ such that
$\deg(v_{1})-k_{1}\deg(v_{2})=l_{1}\deg(v_{3})$. Keeping on the step, we
obtain $v_{i}\in\Delta(\mathfrak{B}(V))$, $k_{i}\in\mathbb{N}$ and
$l_{i}\in\mathbb{Z}\setminus\\{0\\}$ such that
$\deg(v_{i})-k_{i}\deg(v_{i+1})=l_{i}\deg(v_{i+2})\in\mathbb{N}\cdot\Delta(\mathfrak{B}(V))$
for $i=1,2,\cdots$.

(ii) $\deg(v_{i})\not=\deg(v_{j})$ when $i\not=j.$ Indeed, for any $t$, there
exist $r_{t},s_{t}\in\mathbb{R}$ such that
$\deg(v_{t})=r_{t}\deg(v_{1})+s_{t}\deg(v_{2})$ by Part (i). If
$\deg(v_{i})=\deg(v_{j})$, then
$(r_{i}-r_{j})\deg(v_{1})=\deg(v_{2})(s_{j}-s_{i})$, which contradicts [He05,
Cor. 2.5.4 ].

(iii) By Part (ii), the set $\\{v_{i}\mid i=1,2,\cdots\\}$ in Part (i) is
finite since $\Delta(\mathfrak{B}(V))$ is finite.

(iv) By Part (i). (ii) and (iii), there exists $i_{0}\in\mathbb{N}$ such that
$\deg(v_{i})-k_{i}\deg(v_{i+1})=l_{i}\deg(v_{i+2})\in\mathbb{N}\cdot\Delta(\mathfrak{B}(V))$
for $1\leq i<i_{0}$ and
$\deg(v_{i_{0}})-k\deg(v_{i_{0}+1})\notin\mathbb{N}\cdot\Delta(\mathfrak{B}(V))$
for any $k\in\mathbb{N}$. Let $\widetilde{f_{1}}:={\rm deg}(v_{i_{0}})$ and
$\widetilde{f_{2}}:={\rm deg}(v_{i_{0}+1})$. Let
$\widetilde{H}:=\mathbb{R}\\{\widetilde{f_{1}},\widetilde{f_{2}}\\}={\rm
span}\\{\widetilde{f_{1}},\widetilde{f_{2}}\\}$; By [He05, Lemma 2.7.2] and
Lemma 2.1, there exists an arithmetic root subsystem
$\Delta(\chi;f_{1},f_{2})$ such that ${\rm deg}(v)$ and ${\rm deg}(w)$ are in
$\Delta(\chi;f_{1},f_{2})$.

(v) The subsystem $\Delta(\chi;f_{1},f_{2})$ is connected when $[u]=[[v],[w]]$
is the Shirshov decomposition of $[u]\in D$. Otherwise, ${\rm deg}(v),{\rm
deg}(w),{\rm deg}(u)\in\\{f_{1},f_{2}\\}$, which is a contradiction. $\Box$

######  Proposition 2.3.

If $(\Delta,\chi,E)$ is an arithmetic root system, then for every
$\alpha\in\Delta\setminus E$, there exists a connected arithmetic root
subsystem $(\Delta\cap H,\chi_{\Gamma\times\Gamma},E_{H})$ such that
$\alpha\in\Delta\cap H$ and $\mid E_{H}\mid=2$

Proof. It follows from Lemma 2.2. $\Box$

######  Theorem 2.4.

If $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal type with $\dim
V>1$ and $\Delta(\mathfrak{B}(V))$ is an arithmetic root system, then
$p_{u,u}\neq 1$ for $\forall\ [u]\in D$ and there do not exist any
$m$-infinity elements.

Proof. Set $[u]=[[v],[w]]$ is the Shirshov decomposition of $[u]$. By Lemma
2.2, ${\rm deg}(u)$ is in a connected subsystem with rank $2$. We complete the
proof using [WZZ15b, A.2] and [He05, Th. 1.6.1]. $\Box$

##  3 The classification of Nichols Lie Algebras

In this section we give the classification of Nichols Lie algebras.

######  Lemma 3.1.

Assume that $\mathfrak{B}(V)$ is a Nichols algebra of diagonal type. Let
$a:=p_{ii}^{-1}$, $b:=p_{ij}^{-1}$, $c:=p_{ji}^{-1}$ with $i\not=j$. Then

(i) $\bar{l}_{i}^{m}[j]^{-}=0$, when $b=1,c=1$.

(ii)

$\displaystyle<y_{i}^{k},\bar{l}_{i}^{m}[j]^{-}>=\sum\limits_{l=0}^{k-1}(a^{l}-a^{m-1-l}b)<y_{i}^{k-1},\bar{l}_{i}^{m-1}[j]^{-}>+a^{k}x_{i}<y_{i}^{k},\bar{l}_{i}^{m-1}[j]^{-}>$
$\displaystyle-<y_{i}^{k},\bar{l}_{i}^{m-1}[j]^{-}>x_{i}\ \ \hbox{ for }k\leq
m.$ (5)
$\displaystyle<y_{i}^{m},\bar{l}_{i}^{m}[j]^{-}>=(1-b)^{m}(m)_{a}!x_{j}\ \
\hbox{ and }$ (6)
$\displaystyle<y_{j}y_{i}^{m},\bar{l}_{i}^{m}[j]^{-}>=(1-b)^{m}(m)_{a}!.$ (7)

(iii) Assume that $a=1$. Then $\bar{l}_{i}^{m}[j]^{-}\not=0$, when $b\not=1$
or $c\not=1$.

(iv) Assume that $a\not=1$. Then $\bar{l}_{i}^{m}[j]^{-}\not=0$, when ${\rm
ord}(a)>m$ with $c\not=1$ or $b\not=1$.

(v) $\dim(\mathfrak{L}^{-}(V))=\infty$ when there exists $i$ and $j$ with
$i\not=j$, $b\not=1$ or $c\not=1$ and ${\rm ord}(a)=1$ or $\infty$.

Proof. (i) follows from [WZZ15b, Lemma 4.1 (i)(vi)].

(ii) We show (3.1) by induction on $k$ for $k\leq m$. See

$\displaystyle<y_{i},\bar{l}_{i}^{m}[j]^{-}>$ $\displaystyle=$
$\displaystyle<y_{i},x_{i}\bar{l}_{i}^{m-1}[j]^{-}-\bar{l}_{i}^{m-1}[j]^{-}x_{i}>$
$\displaystyle=$
$\displaystyle\bar{l}_{i}^{m-1}[j]^{-}+ax_{i}<y_{i},\bar{l}_{i}^{m-1}[j]^{-}>-<y_{i},\bar{l}_{i}^{m-1}[j]^{-}>x_{i}-a^{m-1}b\bar{l}_{i}^{m-1}[j]^{-}$
$\displaystyle=$
$\displaystyle(1-a^{m-1}b)\bar{l}_{i}^{m-1}[j]^{-}+ax_{i}<y_{i},\bar{l}_{i}^{m-1}[j]^{-}>-<y_{i},\bar{l}_{i}^{m-1}[j]^{-}>x_{i}.$

Thus equation (3.1) holds when $k=1.$ Assume $k>1.$ See

$\displaystyle<y_{i}^{k},\bar{l}_{i}^{m}[j]^{-}>$ $\displaystyle=$
$\displaystyle<y_{i},\sum\limits_{l=0}^{k-2}(a^{l}-a^{m-1-l}b)<y_{i}^{k-2},\bar{l}_{i}^{m-1}[j]^{-}>$
$\displaystyle+$ $\displaystyle
a^{k-1}x_{i}<y_{i}^{k-1},\bar{l}_{i}^{m-1}[j]^{-}>-<y_{i}^{k-1},\bar{l}_{i}^{m-1}[j]^{-}>x_{i}>\
\ \hbox{ by inductive hypothesis}$ $\displaystyle=$
$\displaystyle\sum\limits_{l=0}^{k-2}(a^{l}-a^{m-1-l}b)<y_{i}^{k-1},\bar{l}_{i}^{m-1}[j]^{-}>+a^{k-1}<y_{i}^{k-1},\bar{l}_{i}^{m-1}[j]^{-}>$
$\displaystyle+$ $\displaystyle
a^{k-1}x_{i}a<y_{i}^{k},\bar{l}_{i}^{m-1}[j]^{-}>-<y_{i}^{k},\bar{l}_{i}^{m-1}[j]^{-}>x_{i}-a^{m-k}b<y_{i}^{k-1},\bar{l}_{i}^{m-1}[j]^{-}>$
$\displaystyle=$ $\displaystyle\hbox{ the right hand side of (\ref{e2.5})}.$

Consequently, (3.1 ) holds.

Now we show (6) by induction on $m$. (6) follows from [WZZ15b, Lemma 4.1(i)]
when $m=1.$

By (3.1), one obtains

$\displaystyle<y_{i}^{m},\bar{l}_{i}^{m}[j]^{-}>$ $\displaystyle=$
$\displaystyle\sum\limits_{l=0}^{m-1}(a^{l}-a^{m-1-l}b)<y_{i}^{m-1},\bar{l}_{i}^{m-1}[j]^{-}>$
$\displaystyle=$
$\displaystyle(m)_{a}(1-b)<y_{i}^{m-1},\bar{l}_{i}^{m-1}[j]^{-}>$
$\displaystyle=$ $\displaystyle\hbox{the right hand side of (\ref{e2.5.2})}.$

Therefore, ( 6 ) and (7) hold.

(iii) If $a=1$, then $<y_{i}^{m}y_{j},\bar{l}_{i}^{m}[j]^{-}>=(c-1)^{m}(m)!$
and $<y_{j}y_{i}^{m},\bar{l}_{i}^{m}[j]^{-}>=(1-b)^{m}(m)!$ by [WZZ15b, Lemma
4.1(v)] and (ii).

(iv) It follows from (ii) and [WZZ15b, Lemma 4.1 (v)].

(v) If $a=1$, it is clear by (iii). If $a\not=1$, then (v) follows from
[WZZ15b, Lemma 4.1(iii)] and (iv). $\Box$

######  Proposition 3.2.

([ZWTZ, Th.10.7(i)]) If $\mid D\mid=\infty$, then
$\dim(\mathfrak{L}^{-}(V))=\infty$.

######  Proposition 3.3.

Assume that $\mathfrak{B}(V)$ is a connected Nichols algebra of diagonal type.
Then $\dim(\mathfrak{B}(V))=\infty$ and $\dim(\mathfrak{L}^{-}(V))<\infty$ if
and only if $\dim V=1$ and ${\rm ord}(p_{11})=1$ or $\infty$.

Proof. Sufficiency is clear.

Necessity. Assume $\dim V>1.$ By [ZWTZ, Th.10.7(i)], $\Delta(\mathfrak{B}(V))$
is an arithmetic root system. Consequently, there exists $u\in D$ such that
${\rm ord}(p_{uu})=1$ or $\infty$. Considering Lemma 3.1 (v) we get a
contradiction. $\Box$

##  4 Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras

In this section we obtain an explicit necessary and sufficient condition for a
Nichols algebra and a Nichols (braided) Lie algebra to be finite dimensional

######  Proposition 4.1.

Assume that $\mathfrak{B}(V)$ is a connected Nichols algebra of diagonal type.
Then $\dim(\mathfrak{B}(V))=\infty$ and $\dim(\mathfrak{L}(V))<\infty$ if and
only if $\dim V=1$ and ${\rm ord}(p_{11})=1$.

Proof. Sufficiency. It is clear. Necessity. By [WZZ15b, Th. 2.2], $\dim V=1$.
Considering $\dim(\mathfrak{B}(V))=\infty$ we have ${\rm ord}(p_{11})=1$ or
$\infty.$ By [WZZ15a, Lemma 4.3] ${\rm ord}(p_{11})=1$. $\Box$

######  Theorem 4.2.

If $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal type with $\dim
V>1$, Then $\dim(\mathfrak{B}(V))=\infty$ if and only if
$\dim(\mathfrak{L}^{-}(V))=\infty.$

Proof. Sufficiency is clear. Necessity. It follows from Proposition 3.3.
$\Box$

The fixed parameters in [He05, Table A.1, A.2], [He06a, Table B, C] are called
quantum numbers of generalied Dynkin diagrams. For example, quantum number of
Row 2 in [He05, Table A.1] is $q$; quantum numbers of Row 9 in [He05, Table
A.2] are $q$, $r$ and $s$.

######  Theorem 4.3.

Assume that $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal type
with $\dim V>1$, Then the following conditions are equivalent: (i)
$\mathfrak{B}(V)$ is finite-dimensional; (ii) $\mathfrak{L}(V)$ is finite-
dimensional; (iii) $\mathfrak{L}^{-}(V)$ is finite-dimensional. (iv)
$\Delta(\mathfrak{B}(V))$ is an arithmetic root system, ${\rm
ord}(p_{ii})\not=\infty$ and ${\rm ord}(p_{ij}p_{ji})\not=\infty$ for
$\forall\ 1\leq i\not=j\leq n$. (v) $\Delta(\mathfrak{B}(V))$ is an arithmetic
root system and the quantum numbers (i.e. The fixed parameters ) of
generalized Dynkin diagrams of $V$ are of finite order.

Proof. ${\rm(i)}\Leftrightarrow{\rm(ii)}$. It follows from [WZZ15b, Th. 2.2].

${\rm(i)}\Leftrightarrow{\rm(iii)}$. It follows from Theorem 4.3.

${\rm(iv)}\Rightarrow{\rm(i)}$. By Proposition 4.1, $p_{u,u}\not=1$ for any
$u\in D.$ We complete the proof by [He05, Th.1.4.6] or [Kh99, Th. 2] since
${\rm ord}(p_{u,u})<\infty$ for any $u\in D$.

${\rm(i)}\Rightarrow{\rm(iv)}$. Obviously, $\Delta(\mathfrak{B}(V))$ is an
arithmetic root system and ${\rm ord}(p_{ii})\not=\infty$. Assume ${\rm
ord}(p_{ij}p_{ji})=\infty$ for any $i.$ By [He05, Lemma 2.2.1], one has
$[x_{i},x_{j}]\in D$ or $[x_{j},x_{i}]\in D$. However, the heights of
$[x_{i},x_{j}]$ and $[x_{j},x_{i}]$ are infinite. Consequently we get a
contradiction.

${\rm(iv)}\Leftrightarrow{\rm(v)}$. It follows from [He05, Table A.1, A.2] and
[He06a, Table B, C]. $\Box$

######  Corollary 4.4.

Assume that $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal type
with $\dim V>1$. Then $\dim(\mathfrak{B}(V))=\infty$ (resp.
$\dim$($\mathfrak{L}(V))=\infty$$)$ $($ resp.
$\dim(\mathfrak{L}^{-}(V))=\infty$ $)$ if and only if one of following
conditions holds:

(i) $\Delta(\mathfrak{B}(V))$ is not an arithmetic root system.

(ii) $\dim V=2$, ${\rm ord}(q)=\infty$ in Row 2-6, Row 11 of [He05, Table
A.1].

(iii) $\dim V=3$, ${\rm ord}(q)=\infty$ in Row 1-8, 10 of [He05, Table A.2].

(iv) $\dim V=3$, ${\rm ord}(q)=\infty$ or ${\rm ord}(r)=\infty$ or ${\rm
ord}(s)=\infty$ in Row 9 of [He05, Table A.2].

(v) $\dim V=4$, ${\rm ord}(q)=\infty$ in Row 1-14 of [He06a, Table B].

(vi) $\dim V\geq 5$, ${\rm ord}(q)=\infty$ in Row 1-4, 7-10, 16, 20, 22 of
[He06a, Table C].

Proof. It follows from Theorem 4.4, [He05, Table A.1, A.2] and [He06a, Table
B, C]. $\Box$

##  5 Appendix

The proof of Proposition 3.2

For completeness, we give the proof of Proposition 3.2.

Let $A:=\\{x_{1},x_{2},\cdots,x_{n}\\}$ be an alphabet and a basis of $V={\rm
span}(A)$; $A^{*}:=\\{u\mid u\hbox{ is a word }\\}$. Let $I$ be a graded ideal
of $T(V)$ as algebras such that $\mathfrak{B}(V)=T(V)/I$.

$u$ is called a standard word with respect to $\mathfrak{B}(V)$ if $u$ can not
be written as linear combination of strictly greater words in
$\mathfrak{B}(V)$.

Let ${\rm SW}(\mathfrak{B}(V))=:\\{u\in A^{*}\mid u\hbox{ is a standard word
with respect to }\mathfrak{B}(V)\\}$, written as SW in short;

${\rm L}:=\\{u\in A^{*}\mid u\hbox{ is a Lyndon word}\\}.$ Notice that we view
${\rm SW}(\mathfrak{B}(V))$ and ${\rm L}$ are in $\mathfrak{B}(V)$ often for
convenience.

Let ${\rm HBLLW}(\mathfrak{B}(V))=:\\{u\in{\rm L}\mid[u]\hbox{ is a hard
super-letter }\\}$,

######  Lemma 5.1.

(i) ${\rm SW}$ is a basis of $\mathfrak{B}(V).$

(ii) Any factor of a standard word is a standard word.

(iii) If $u$ is a standard word, then $u=u_{1}u_{2}\cdots u_{r}$ with
$u_{!}\geq u_{2}\geq\cdots\geq u_{r}$ and $u_{i}\in{\rm SW}\cap{\rm L}$ for
$1\leq i\leq r.$

Proof. (i) If a word $u\notin{\rm SW},$ then $u=\sum\limits_{v\in{\rm SW},\mid
v\mid=\mid u\mid,v>u}a_{v}v$. It is clear that ${\rm SW}$ is linearly
independent in $\mathfrak{B}(V).$ Consequently, ${\rm SW}$ is a basis of
$\mathfrak{B}(V).$

(ii) It is clear.

(iii) It follows from Part (ii) and [Lo83, Th. 5.1.5]. $\Box$

######  Lemma 5.2.

(i) If $l\in{\rm L},$ then $[l]^{-}=l+\sum\limits_{w>l,\mid l\mid=\mid
w\mid}a_{w}w$ in $\mathfrak{B}(V),$ where $a_{w}\in k$.

(ii) If $l\in{\rm L},$ then $[l]=a_{l}l+\sum\limits_{w>l,\mid l\mid=\mid
w\mid}a_{w}w$ in $\mathfrak{B}(V),$ where $a_{l},a_{w}\in k$ with
$a_{l}\not=0.$

(iii) ${\rm SW}\cap{\rm L}\subseteq{\rm HBLLW}.$

Proof. (ii) We show this by induction on $\mid l\mid.$ It is clear when $\mid
l\mid=1$ since $[l]=l.$ Assume that $l=uv$ is the Shirshow decomposition of
$l$. If $u^{\prime}>u$ and $v^{\prime}>v$ with $\mid u^{\prime}\mid=\mid
u\mid$ and $\mid v^{\prime}\mid=\mid v\mid$, then $u^{\prime}v^{\prime}>uv=l$
and $v^{\prime}u^{\prime}>vu>l.$

$\displaystyle[l]$ $\displaystyle=$ $\displaystyle[v][u]-p_{v,u}[u][v]$
$\displaystyle=$ $\displaystyle(a_{v}^{\prime}v+\sum\limits_{v^{\prime}>v,\mid
v^{\prime}\mid=\mid
v\mid}a_{v^{\prime}}^{\prime}v^{\prime})(a_{u}^{\prime}u+\sum\limits_{u^{\prime}>u,\mid
u^{\prime}\mid=\mid u\mid}a_{u^{\prime}}^{\prime}u^{\prime})$ $\displaystyle-
p_{vu}(a_{u}^{\prime}u+\sum\limits_{u^{\prime}>u,\mid u^{\prime}\mid=\mid
u\mid}a_{u^{\prime}}^{\prime}u^{\prime})(a_{v}^{\prime}v+\sum\limits_{v^{\prime}>v,\mid
v^{\prime}\mid=\mid v\mid}a_{v^{\prime}}^{\prime}v^{\prime})\ \ (\hbox{by
inductive hypothesis})$ $\displaystyle=$ $\displaystyle
a_{l}l+\sum\limits_{w>l,\mid l\mid=\mid w\mid}a_{w}w.$

(i) The proof is similar to the proof of (ii).

(iii) If $l\notin{\rm HBLLW}$ with $l\in{\rm L}$, then

$\displaystyle[l]$ $\displaystyle=$
$\displaystyle\sum\limits_{r=1}^{m}\sum\limits_{u_{i}>l,1\leq i\leq r,\mid
u_{1}u_{2}\cdots u_{r}\mid=\mid
l\mid}a_{u}^{\prime}[u_{1}][u_{2}]\cdots[u_{r}]$ $\displaystyle=$
$\displaystyle\sum\limits_{r=1}^{m}\sum\limits_{u_{i}>l,1\leq i\leq r,\mid
u_{1}u_{2}\cdots u_{r}\mid=\mid
l\mid}a_{u}^{\prime}\prod_{i=1}^{r}(a_{u_{i}}^{\prime\prime}u_{i}+\sum\limits_{w_{ij}>u_{i},\mid
w_{ij}\mid=\mid u_{i}\mid}a^{\prime\prime}_{w_{ij}}w_{ij})$ $\displaystyle=$
$\displaystyle\sum\limits_{u>l,\mid u\mid=\mid l\mid}a_{u}u\ \ \ \hbox{ and }$
$\displaystyle[l]=a_{l}^{\prime\prime\prime}l+\sum\limits_{w>l,\mid l\mid=\mid
w\mid}a_{w}^{\prime\prime\prime}w\ \ (\hbox{ by }{\rm(ii)}).$

Consequently, $l=\sum\limits_{w>l,\mid l\mid=\mid w\mid}a_{w}w$ and
$l\notin{\rm SW}\cap{\rm L}.$.

$\Box$

######  Theorem 5.3.

If $\mathfrak{B}(V)$ is a graded braided Hopf algebras with diagonal braiding,
then ${\rm SW}(\mathfrak{B}(V))\cap{\rm L}={\rm HBLLW}(\mathfrak{B}(V)).$
Furthermore, if $\Delta(\mathfrak{B}(V))$ is not an arithmetic root system,
then $\dim\mathfrak{L}^{-}(\mathfrak{B}(V))=\infty.$

Proof. ${\rm SW}\cap{\rm L}\subseteq{\rm HBLLW}$ by Lemma 5.2 (iii). If there
exists $u\in{\rm HBLLW}$ and $u\notin{\rm SW}\cap{\rm L},$ then
$u=\sum\limits_{v_{1}\geq v_{2}\geq\cdots\geq v_{r},v_{i}\in{\rm SW}\cap{\rm
L},\mid v_{1}v_{2}\cdots v_{r}\mid=\mid u\mid,v_{1}v_{2}\cdots
v_{r}>u}a_{v}v_{1}v_{2}\cdots v_{r}$ by Lemma 5.1 (iii). Consequently,
$[u]=\sum\limits_{v_{1}\geq v_{2}\geq\cdots\geq v_{r},v_{i}\in{\rm SW}\cap{\rm
L},\mid v_{1}v_{2}\cdots v_{r}\mid=\mid u\mid,v_{1}v_{2}\cdots
v_{r}>u}a_{v}[v_{1}][v_{2}]\cdots[v_{r}],$ which contradicts to
$[u],[v_{1}][v_{2}]\cdots[v_{r}]\in P.$ $\Box$

## Acknowledgement

Y.Z.Z. was partly supported by the Australian Research Council through
Discovery Project DP140101492.

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