Basis of Nichols Braided Lie Algebras

TL;DR

This paper characterizes monomials in Nichols braided Lie algebras of diagonal type, establishing a connection between monomial connectivity and algebra membership, and provides bases and dimensions for finite Cartan types.

Contribution

It introduces a criterion for monomials in Nichols braided Lie algebras and computes bases and dimensions for finite Cartan type cases.

Findings

A monomial belongs to Nichols braided Lie algebra if and only if it is connected.

A basis for Nichols braided Lie algebra is constructed.

Dimensions of Nichols braided Lie algebra for finite Cartan types are determined.

Abstract

Assume that $V$ is a braided vector space with diagonal type. It is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak L(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained.

Equations

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

$$[u,v\cdot w]=p_{wu}[uv]\cdot w+v\cdot [uw].$$

$$\tau^{\prime}:={\tiny\left(\begin{array}[]{ccccccc}1&\cdots&b-1&b&\cdots&m-1&m\\ \tau_{1}\tau(1)&\cdots&\tau_{1}\tau(b-1)&\tau_{1}\tau(b+1)&\cdots&\tau_{1}\tau(m)&\tau_{1}\tau(b)\end{array}\right)},$$

$$L_{1,n}$$

$$=$$

$$\mid L_{1,n}\mid$$

$$=\sum _{j=0}^{int(\frac{n-1}{2})}(-1{)}^j{u}_j,$$

$$L_{1,n}$$

$$=$$

$$\mid L_{1,n}\mid$$

$$L_{1,n}$$

$$\mid L_{1,n}\mid$$

$$\dim \mathfrak{L}(V)=\mid L_{1,n}\mid =N^{C_{n+1}^2}-1+\sum _{j=1}^{int(\frac{n-1}{2})}(-1{)}^j\sum _{n_1=1}^{n-2}\sum _{n_2=1}^{n_1-2}\cdots \sum _{n_j=1}^{n_{j-1}-2}(N^{C_{n_j+1}^2}-1)$$

$$(N^{C_{n_{j-1}-n_j}^2}-N^{C_{n_{j-1}-n_j-1}^2})\cdots (N^{C_{n_1-n_2}^2}-N^{C_{n_1-n_2-1}^2})(N^{C_{n-n_1}^2}-N^{C_{n-n_1-1}^2}).$$

$$\dim \mathfrak{L}(V)=\mid L_{1,n}\mid =N^{n^2}-1+\sum _{j=1}^{int(\frac{n-1}{2})}(-1{)}^j\sum _{n_1=1}^{n-2}\sum _{n_2=1}^{n_1-2}\cdots \sum _{n_j=1}^{n_{j-1}-2}(N^{C_{n_j+1}^2}-1)$$

$$(N^{C_{n_{j-1}-n_j}^2}-N^{C_{n_{j-1}-n_j-1}^2})\cdots (N^{C_{n_1-n_2}^2}-N^{C_{n_1-n_2-1}^2})$$

$$(N^{(n-n_1-1{)}^2-n+n_1+1}-N^{(n-n_1-2{)}^2-n+n_1+2}),$$

$$\dim \mathfrak{L}(V)=\mid L_{1,n}\mid =(\frac{N}{2}{)}^{n^2-n}{N}^n-1+\sum _{j=1}^{int(\frac{n-1}{2})}(-1{)}^j\sum _{n_1=1}^{n-2}\sum _{n_2=1}^{n_1-2}\cdots \sum _{n_j=1}^{n_{j-1}-2}((\frac{N}{2}{)}^{C_{n_j+1}^2}-1)$$

$$((\frac{N}{2}{)}^{C_{n_{j-1}-n_j}^2}-(\frac{N}{2}{)}^{C_{n_{j-1}-n_j-1}^2})\cdots ((\frac{N}{2}{)}^{C_{n_1-n_2}^2}-(\frac{N}{2}{)}^{C_{n_1-n_2-1}^2})$$

$$((\frac{N}{2}{)}^{(n-n_1-1{)}^2-n+n_1+1}{N}^{n-n_1-1}-(\frac{N}{2}{)}^{(n-n_1-2{)}^2-n+n_1+2}{N}^{n-n_1-2}),$$

$$\dim \mathfrak{L}(V)=\mid L_{1,n}\mid =N^{n^2}-1+\sum _{j=1}^{int(\frac{n-1}{2})}(-1{)}^j\sum _{n_1=1}^{n-2}\sum _{n_2=1}^{n_1-2}\cdots \sum _{n_j=1}^{n_{j-1}-2}(N^{C_{n_j+1}^2}-1)$$

$$(N^{C_{n_{j-1}-n_j}^2}-N^{C_{n_{j-1}-n_j-1}^2})\cdots (N^{C_{n_1-n_2}^2}-N^{C_{n_1-n_2-1}^2})$$

$$(N^{(n-n_1-1{)}^2-n+n_1+1}-N^{(n-n_1-2{)}^2-n+n_1+2}),$$

$$\dim \mathfrak{L}(V)=\mid L_{1,n}\mid =(\frac{N}{2}{)}^n{N}^{n^2-n}-1+\sum _{j=1}^{int(\frac{n-1}{2})}(-1{)}^j\sum _{n_1=1}^{n-2}\sum _{n_2=1}^{n_1-2}\cdots \sum _{n_j=1}^{n_{j-1}-2}(N^{C_{n_j+1}^2}-1)$$

$$(N^{C_{n_{j-1}-n_j}^2}-N^{C_{n_{j-1}-n_j-1}^2})\cdots (N^{C_{n_1-n_2}^2}-N^{C_{n_1-n_2-1}^2})$$

$$(N^{(n-n_1-1{)}^2-n+n_1+1}(\frac{N}{2}{)}^{n-n_1-1}-N^{(n-n_1-2{)}^2-n+n_1+2}(\frac{N}{2}{)}^{n-n_1-2}),$$

$$\dim \mathfrak{L}(V)=\mid L_{1,n}\mid =N^{n^2-n}-(N-1{)}^2-\sum _{i=1}^{n-3}\{N^{C_{i+1}^2}-1+\sum _{j=1}^{int(\frac{i-1}{2})}(-1{)}^j\sum _{n_1=1}^{n-2}\sum _{n_2=1}^{n_1-2}\cdots$$

$$\cdots \sum _{n_j=1}^{n_{j-1}-2}(N^{C_{n_j+1}^2}-1)(N^{C_{n_{j-1}-n_j}^2}-N^{C_{n_{j-1}-n_j-1}^2})\cdots (N^{C_{n_1-n_2}^2}-N^{C_{n_1-n_2-1}^2})$$

$$(N^{C_{i-n_1}^2}-N^{C_{i-n_1-1}^2})\}(N^{(n-i-1{)}^2-n+i+1}-N^{(n-i-2{)}^2-n+i+2})-\{N^{C_{n-2}^2}-1$$

$$+\sum _{j=1}^{int(\frac{n-4}{2})}(-1{)}^j\sum _{n_1=1}^{n-5}\sum _{n_2=1}^{n_1-2}\cdots \sum _{n_j=1}^{n_{j-1}-2}(N^{C_{n_j+1}^2}-1)(N^{C_{n_{j-1}-n_j}^2}-N^{C_{n_{j-1}-n_j-1}^2})\cdots$$

$$\cdots (N^{C_{n_1-n_2}^2}-N^{C_{n_1-n_2-1}^2})(N^{C_{n-3-n_1}^2}-N^{C_{n-4-n_1}^2})\}(N^2-1).$$

$$\dim \mathfrak{L}(V)$$

$$\dim \mathfrak{L}(V)=N^{24}-1-(N-1)(N^3-N)-(N^3-1)(N-1),$$

$$\dim \mathfrak{L}(V)=(\frac{N}{2}{)}^{12}{N}^{12}-1-(\frac{N}{2}-1)(N^3-N)-((\frac{N}{2}{)}^3-1)(N-1),$$

Full text

2010 Mathematics Subject Classification: 16W30, 22E60, 11F23

Keywords: Braided vector space, Nichols 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, He05, He06a, He06b, WZZ15a, WZZ15b]. 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.

The theory of Lie superalgebras has been developed systematically, which includes the representation theory and classifications of simple Lie superalgebras and their varieties [Ka77] [BMZP92]. In many physical applications or in pure mathematical interest, one has to consider not only ${\bf Z}_{2}$- or ${\bf Z}$- grading but also $G$-grading of Lie algebras, where $G$ is an abelian group equipped with a skew symmetric bilinear form given by a 2-cocycle. Lie algebras in symmetric and more general categories were discussed in [Gu86] and [GRR95]. A sophisticated multilinear version of the Lie bracket was considered in [Kh99] [Pa98]. Various generalized Lie algebras have already appeared under different names, e.g. Lie color algebras, $\epsilon$ Lie algebras [Sc79], quantum and braided Lie algebras, generalized Lie algebras [BFM96], $H$-Lie algebras [BFM01] and braided $m$-Lie algebras [ZZ03].

In this paper it is shown that a monomial belongs to Nichols braided Lie algebra $\mathfrak{L}(V)$ if and only if this monomial is connected. A basis of Nichols braided Lie algebra and dimension of Nichols braided Lie algebra of finite Cartan type are obtained for a braided vector space $V$ with diagonal type.

This paper is organized as follows. In Section 1 we provide some preliminaries and set our notations. In Section 2 we obtain that a monomial belongs to $\mathfrak{L}(V)$ if and only if this monomial is connected when $V$ is a braided vector space with diagonal type. In Section 3 we obtain a basis of Nichols braided Lie algebra. In Section 4 we obtain dimension of Nichols braided Lie algebra $\mathfrak{L}(V)$ with finite Cartan type. In Section 5 we find some non-zero monomials in Nichols algebras. In Section 6 we give relations between connected components of graphes and Nichols Braided Lie Algebras.

Preliminaries

Let $\mathfrak{B}(V)$ be the Nichols algebra generated by vector space $V$. Throughout this paper braided vector space $V$ is of diagonal type with basis $x_{1},x_{2},\cdots,x_{n}$ and $C(x_{i}\otimes x_{j})=p_{ij}x_{j}\otimes x_{i}$, $n>1$ without special announcement. 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 $\widetilde{p}_{uv}:=p_{uv}p_{vu}.$

Denote ${\rm ord}(p_{uu})$ the order of $p_{uu}$ with respect to multiplication. Let $|u|$ denote length of homogeneous element $u\in\mathfrak{B}(V).$ Let $D=:\{[u]\mid[u]\hbox{ is a hard super-letter }\}$, $\Delta^{+}(\mathfrak{B}(V)):=\{\deg(u)\mid[u]\in D\}$, $\Delta(\mathfrak{B}(V)):=\Delta^{+}(\mathfrak{B}(V))\cup\Delta^{-}(\mathfrak{B}(V))$, which is called the root system of $V.$ If $\Delta(\mathfrak{B}(V))$ is finite, then it is called an arithmetic root system. 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 $V$. The other notations are the same as in [WZZ15a].

Recall 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:

[TABLE]

[TABLE]

$\Gamma(V)$ be the generalized Dynkin diagram of $V$ omitted $p_{x_{i},x_{i}},\widetilde{p}_{x_{i},x_{j}}$ for $1\leq i,j\leq n$, which is called pure generalized Dynkin diagram of $V$.

Let $u=h_{1}h_{2}\cdots h_{m}$ be a monomial with $h_{j}=x_{i_{j}}$ for $1\leq i_{1},i_{2},\ldots,i_{m}\leq n,1\leq j\leq m$. ${\rm deg}(u)=\lambda_{1}e_{1}+\cdots+\lambda_{n}e_{n},$ where ${\rm deg}(x_{i})=e_{i}.$ Let ${\rm deg}_{x_{i}}(u):=\lambda_{i}$. Let $\mu(u):=\{x_{i_{1}},\cdots,x_{i_{m}}\}$ and $\Gamma(u)$ be a pure generalized Dynkin sub-diagram generated by $\mu(u)$. If $\Gamma(u)$ is connected and $p_{h_{i},h_{i}}\not=1$ for $1\leq i\leq m$, then $u$ is called connected (or $\mu(u)$ is called connected). If $h_{i}h_{i+1}=p_{h_{i+1}h_{i}}h_{i+1}h_{i}$, then $h_{1}h_{2}\cdots h_{m}=p_{h_{i+1}h_{i}}h_{1}h_{2}\cdots h_{i+1}h_{i}\cdots h_{m}$ and $h_{1}h_{2}\cdots h_{i+1}h_{i}\cdots h_{m}$ is called an elementary quantum equivalent of $h_{1}h_{2}\cdots h_{i+1}h_{i}\cdots h_{m}$. If $v_{1}$ is a monomial and $v_{j+1}$ is an elementary quantum equivalent of $v_{j}$ for $1\leq j\leq r-1$, then $v_{r}$ is called a quantum equivalent of $v_{1},$ written as $v_{1}\sim v_{r}.$ This is an equivalent relation since $h_{i}h_{i+1}=p_{h_{i+1}h_{i}}h_{i+1}h_{i}$ if and only if $p_{h_{i+1}h_{i}}p_{h_{i}h_{i+1}}=1.$ If there exist $x_{i}\in\mu(u)$ and $x_{j}\in\mu(v)$ such that $\widetilde{p}_{x_{i},x_{j}}\not=1$, then we say that it is connected between monomial $u$ and monomial $v$, written $u\circ v$ in short.

Throughout, $\mathbb{Z}=:\{x\mid x\hbox{ is an integer}\}.$ $\mathbb{R}=:\{x\mid x\hbox{ is a real number}\}$. $\mathbb{N}_{0}=:\{x\mid x\in\mathbb{Z},x\geq 0\}.$ $\mathbb{N}=:\{x\mid x\in\mathbb{Z},x>0\}$. $F$ denotes the base field, which is an algebraic closed field with characteristic zero. $F^{*}=F\backslash\{0\}$. $S_{n}$ denotes symmetric group, $n\in\mathbb{N}$. For any set $X$, $\mid X\mid$ is the cardinal of $X$. $int(a)$ means the biggest integer not greater than $a\in\mathbb{R}$.

2 The structure of Nichols braided Lie algebras

In this section we obtain that a monomial belongs to $\mathfrak{L}(V)$ if and only if this monomial is connected when $V$ is a braided vector space with diagonal type.

Lemma 2.1**.**

Assumed that $u,v,w$ are homogeneous elements in $\mathfrak{L}(V)$. Let $a,b,c,d,e,f$ denote $1-p_{wv}p_{vw},1-p_{uw}p_{wu},1-p_{uv}p_{vu},1-p_{uv}p_{vu}p_{uw}p_{wu},1-p_{uv}p_{vu}p_{wv}p_{vw},1-p_{wv}p_{vw}p_{uw}p_{wu}$, respectively. If $\mid\{r|r\in\{uv,uw,vw\}\hbox{ and }r\in\mathfrak{L}(V)\}\mid\geq 2$ and $\mid\{t|t\in\{a,b,c\}\hbox{ and }t\neq 0\}\mid\geq 1$, then $uvw,uwv,vwu,vuw,wuv,wvu\in\mathfrak{L}(V)$.

Proof. Without loss of generality, we let $uv,uw\in\mathfrak{L}(V)$. If $vuw\notin\mathfrak{L}(V)$, then $e=0,f=0$ by [WZZ15b, Lemma 3.2(i)] and [WZZ15a, Lemma 4.12]. If $a=0$, then $b=0,c=0,$ which is a contradiction. If $a\neq 0$, then $b\neq 0,c\neq 0$, one obtains a contradiction to [WZZ15b, Lemma 3.1]. Consequently, $vuw\in\mathfrak{L}(V)$. Similarly, we can obtain others. $\Box$

Lemma 2.2**.**

Assume $h_{i}\in\{x_{1},\cdots,x_{n}\}$ for $1\leq i\leq m.$

(i)* If it is disconnected between monomial $u$ and monomial $v$ (i.e. $\widetilde{p}_{x_{i},x_{j}}=1$ for any $x_{i}\in\mu(u)$, $x_{j}\in\mu(v)$ ), then $[u,v]=0.$*

(ii)* If $\mu(h_{1}h_{2}\cdots h_{m})$ is disconnected, then $\sigma(h_{1},h_{2},\cdots,h_{m})=0$ for any method $\sigma$ of adding bracket on $h_{1},h_{2},\cdots,h_{m}.$.*

(iii)* If $h_{1}h_{2}\cdots h_{m}\neq 0$ and $\mu(h_{1}h_{2}\cdots h_{m})$ is disconnected, then $h_{1}h_{2}\cdots h_{m}\notin\mathfrak{L}(V)$.*

Proof. (i) $u$ and $v$ are quantum commutative (i.e. $uv=p_{u,v}vu$ ) since $x_{i}x_{j}=p_{x_{i},x_{j}}x_{j}x_{i}$ for any $x_{i}\in\mu(u)$, $x_{j}\in\mu(v)$.

(ii) We show this by induction on $m$. $[h_{2},h_{1}]=[h_{1},h_{2}]=0$ for $m=2.$ For $m>2,$ $\sigma(h_{1},h_{2},\cdots,h_{m})=[\sigma_{1}(h_{\tau_{(1)}}h_{\tau_{(2)}}\cdots h_{\tau_{(t)}}),\sigma_{2}(h_{\tau_{(t+1)}}h_{\tau_{(t+2)}}\cdots h_{\tau_{(m)}})],\tau\in\mathbb{S}_{m}$. If both $h_{\tau_{(1)}}h_{\tau_{(2)}}\cdots h_{\tau_{(t)}}$ and $h_{\tau_{(t+1)}}h_{\tau_{(t+2)}}\cdots h_{\tau_{(m)}}$ are connected, then it is disconnected between $h_{\tau_{(1)}}h_{\tau_{(2)}}\cdots h_{\tau_{(t)}}$ and $h_{\tau_{(t+1)}}h_{\tau_{(t+2)}}\cdots h_{\tau_{(m)}}$. By Part (i), $\sigma(h_{1},h_{2},\cdots,h_{m})=0$. If either $h_{\tau_{(1)}}h_{\tau_{(2)}}\cdots h_{\tau_{(t)}}$ or $h_{\tau_{(t+1)}}h_{\tau_{(t+2)}}\cdots h_{\tau_{(m)}}$ is disconnected, then either $\sigma_{1}(h_{\tau_{(1)}}h_{\tau_{(2)}}\cdots h_{\tau_{(t)}})=0$ or $\sigma_{2}(h_{\tau_{(t+1)}}h_{\tau_{(t+2)}}\cdots h_{\tau_{(m)}})=0$ by inductive hypothesis.

(iii) It follows from Part (ii). $\Box$

Lemma 2.3**.**

(i) If $u$ is a monomial, then there exist monomials $u_{1},u_{2},\ldots,u_{r}$ such that $u\sim u_{1}u_{2}\cdots u_{r}$ for $\{\Gamma(u_{1}),\Gamma(u_{2}),\cdots,\Gamma(u_{r})\}$ is complete set of a connected component of $\Gamma(u)$ with ${\rm deg}(u)=\sum\limits_{i=1}^{r}{\rm deg}(u_{i})$ and $\mid u_{1}\mid\geq\mid u_{2}\mid\geq\cdots\geq\mid u_{r}\mid$, (which is called a decomposition of connected components of $u$)

(ii) If a monomial $u$ is connected with $\mid u\mid>1$, then there exist two connected monomials $v$ and $w$ such that $v\circ w$ with $u\sim vw$.

Proof. (i) If $u$ is connected, it is clear. If $u$ is not connected, we show it by induction on $\mid u\mid$. It is clear when $\mid u\mid=1.$ Now assume $\mid u\mid>1$. Let $\Omega:=\{v\mid v\hbox{ is connected }\}$. Let $u=v_{1}v_{2}v_{3}$ and $v_{2}$ be in $\Omega$ such that $\mid v_{2}\mid=max\{\mid v\mid\ \ \mid v\in\Omega\}$.

If $v_{3}\in F^{*}$, then $v_{2}v_{3}\sim v_{3}v_{2}$. If $v_{2}v_{3}\neq kv_{3}v_{2}$ for $\forall\ k\in F^{*}$, then there exists $t$ such that $v_{2}\circ h_{i_{t}}$ with $v_{3}=h_{i_{1}}\cdots h_{i_{r}}$ and $v_{2}h_{i_{s}}\sim h_{i_{s}}v_{2}$ for $1\leq s<t$. Consequently, $u\sim w_{1}v_{2}h_{i_{t}}w_{3}$ and $v_{2}h_{i_{t}}$ is connected. which is a contradiction. Consequently, $v_{2}v_{3}\sim v_{3}v_{2}$.

If $v_{1}\in F^{*}$, then $v_{1}v_{2}\sim v_{2}v_{1}$. If $v_{1}v_{2}\neq kv_{2}v_{1}$ for $\forall\ k\in F^{*}$, then there exists $t$ such that $h_{i_{t}}\circ v_{2}$ with $v_{1}=h_{i_{1}}\cdots h_{i_{r}}$ and $v_{2}h_{i_{s}}\sim h_{i_{s}}v_{2}$ for $t+1<s\leq r$. Consequently, $u\sim w_{1}h_{i_{t}}v_{2}w_{3}$ and $h_{i_{t}}v_{2}$ is connected. which is a contradiction. Consequently, $v_{1}v_{2}\sim v_{2}v_{1}$.

Thus $u=v_{1}v_{2}v_{3}\sim v_{2}v_{1}v_{3}$ and $v_{1}v_{3}\sim u_{2}u_{3}\cdots u_{r}$ is a decomposition of connected components of $v_{1}v_{3}$ since $\mid v_{1}v_{3}\mid<\mid u\mid.$ Set $u_{1}:=v_{2}$.

(ii) Let $u=h_{1}h_{2}\cdots h_{m}$ and $h_{2}\cdots h_{m}\sim u_{1}u_{2}\cdots u_{r}$ be a decomposition of connected components of $h_{2}\cdots h_{m}$. Consequently, $h_{1}u_{1}\cdots u_{r-1}$ is connected since $h_{1}\circ u_{i}$ for $1\leq i\leq r$. $\Box$

Theorem 2.4**.**

Assume $h_{i}\in\{x_{1},\cdots,x_{n}\}$ and $p_{h_{1},h_{1}}\not=1$ when $h_{i}=h_{1}$ for $1\leq i\leq m$.

(i) If $h_{1}\cdots h_{b},h_{b+1}\cdots h_{m}\in\mathfrak{L}(V)$ and $h_{1}\cdots h_{b}\circ h_{b+1}\cdots h_{m}$ with $h_{1}\cdots h_{b}\not=0$ or $h_{b+1}\cdots h_{m}\not=0$, then there exist $\tau\in\mathbb{S}_{m}$ such that $h_{1}\cdots h_{m}\sim h_{\tau(1)}\cdots h_{\tau(m)}$ with

$h_{\tau(1)}\cdots h_{\tau(m-1)}\in\mathfrak{L}(V)$* and $h_{\tau(1)}\cdots h_{\tau(m-1)}\circ h_{\tau(m)}$, or with $h_{\tau(2)}\cdots h_{\tau(m)}\in\mathfrak{L}(V)$ and $h_{\tau(2)}\cdots h_{\tau(m)}\circ h_{\tau(1)}$.*

(ii) If $h_{1}h_{2}\cdots h_{m-1}\in\mathfrak{L}(V)$ and $h_{1}h_{2}\cdots h_{m-1}\circ h_{m}$ or $h_{2}h_{3}\cdots h_{m}\in\mathfrak{L}(V)$ and $h_{2}h_{3}\cdots h_{m}\circ h_{1},$ then $h_{1}h_{2}\cdots h_{m}\in\mathfrak{L}(V)$.

(iii) If $0\not=h_{1}h_{2}\cdots h_{m}\in\mathfrak{L}(V)$, then there exists $\tau\in\mathbb{S}_{m}$ such that $h_{1}\cdots h_{m}\sim h_{\tau(1)}\cdots h_{\tau(m)}$ with $0\neq h_{\tau(1)}h_{\tau(2)}\cdots h_{\tau(m-1)}\in\mathfrak{L}(V)$ and $h_{\tau(1)}h_{\tau(2)}\cdots h_{\tau(m-1)}\circ h_{\tau(m)}$, or $0\neq h_{\tau(2)}h_{\tau(3)}\cdots h_{\tau(m)}\in\mathfrak{L}(V)$ and $h_{\tau(2)}h_{\tau(3)}\cdots h_{\tau(m)}\circ h_{\tau(1)}$ .

(iv) If monomial $u$ is connected, then $u\in\mathfrak{L}(V).$

Proof. We show (i), (ii), (iii) and (iv) by induction on the length of $|h_{1}h_{2}\cdots h_{m}|=m$.

Assume $m=2$. (i) and (iv) are clear. If $h_{1}\neq h_{2}$, then (ii), (iii) follows from [WZZ15a, Lemma 4.12], [WZZ15a, Lemma 5.2], respectively. If $h_{1}=h_{2}$, then (ii) and (iii) follow from [WZZ15a, Lemma 4.3] and [He05, Lemma 1.3.3(i)].

Now $m>2$.

(i) If $b=m-1$ or $b=1$, let $\tau=id$. Now assume that $1<b<m-1$. There exist $1\leq a\leq b$ and $b+1\leq c\leq m$ such that $\widetilde{p}_{h_{a},h_{c}}\not=1$.

(1). Assumed that $h_{1}\cdots h_{b}\neq 0$, we proceed by induction over $b$.

We know there exist $\tau\in\mathbb{S}_{1,\ldots,b}$ such that $h_{1}\cdots h_{b}\sim h_{\tau(1)}\cdots h_{\tau(b)}$ with

(a) $0\neq h_{\tau(1)}\cdots h_{\tau(b-1)}\in\mathfrak{L}(V)$, $h_{\tau(1)}h_{\tau(2)}\cdots h_{\tau(b-1)}\circ h_{\tau(b)}$ or with

(b) $0\neq h_{\tau(2)}\cdots h_{\tau(b)}\in\mathfrak{L}(V)$, $h_{\tau(2)}h_{\tau(3)}\cdots h_{\tau(b)}\circ h_{\tau(1)}$ by $h_{\tau(1)}\cdots h_{\tau(b)}\in\mathfrak{L}(V)$ and induction hypotheses of (iii).

(a)1. If $h_{b+1}\cdots h_{m}\circ h_{\tau(b)}$, then $h_{\tau(b)}h_{b+1}\cdots h_{m}\in\mathfrak{L}(V)$ by induction hypotheses of (ii), and $h_{\tau(1)}\cdots h_{\tau(b-1)}\circ h_{\tau(b)}h_{b+1}\cdots h_{m}$ since $h_{\tau(1)}h_{\tau(2)}\cdots h_{\tau(b-1)}\circ h_{\tau(b)}$. It is proved since $|h_{\tau(1)}\cdots h_{\tau(b-1)}|=b-1<b$ and induction hypotheses.

(a)2. If all $j_{1}\in\{b+1,\ldots,m\}$ such that $\tilde{p}_{h_{\tau(b)},h_{j_{1}}}=1$, then $h_{\tau(b)}h_{b+1}\cdots h_{m}\sim h_{b+1}\cdots h_{m}h_{\tau(b)}$. On the other hand, we know $h_{\tau(1)}\cdots h_{\tau(b-1)}\circ h_{b+1}\cdots h_{m}$ since $h_{1}\cdots h_{b}\circ h_{b+1}\cdots h_{m}$, then $h_{\tau_{1}\tau(1)}\cdots h_{\tau_{1}\tau(b-1)}h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}\in\mathfrak{L}(V)$, $h_{\tau(1)}\cdots h_{\tau(b-1)}h_{b+1}\cdots h_{m}\sim h_{\tau_{1}\tau(1)}\cdots h_{\tau_{1}\tau(b-1)}h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}$ for some $\tau_{1}\in\mathbb{S}_{\tau(1),\ldots,\tau(b-1),b+1,\ldots,m}$ by the induction hypotheses of (i) and (ii). We know $h_{\tau_{1}\tau(1)}\cdots h_{\tau_{1}\tau(b-1)}h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}\circ h_{\tau(b)}$ since

$h_{\tau(1)}h_{\tau(2)}\cdots h_{\tau(b-1)}\circ h_{\tau(b)}$. We obtain $h_{1}\cdots h_{m}\sim h_{\tau(1)}\cdots h_{\tau(b)}h_{b+1}\cdots h_{m}$

$\sim h_{\tau(1)}\cdots h_{\tau(b-1)}h_{b+1}\cdots h_{m}h_{\tau(b)}\sim h_{\tau_{1}\tau(1)}\cdots h_{\tau_{1}\tau(b-1)}h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}h_{\tau(b)}$

$=h_{\tau_{1}\tau(1)}\cdots h_{\tau_{1}\tau(b-1)}$ $h_{\tau_{1}\tau(b+1)}\cdots h_{\tau_{1}\tau(m)}h_{\tau_{1}\tau(b)}$ by $h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}$ $=h_{\tau_{1}\tau(b+1)}\cdots h_{\tau_{1}\tau(m)}$ since $\tau\in\mathbb{S}_{1,2,\ldots,b}$, and $h_{\tau_{1}\tau(b)}=h_{\tau(b)}$ since $\tau_{1}\in\mathbb{S}_{\tau(1),\tau(2),\ldots,\tau(b-1),b+1,\ldots,m}$. Set

[TABLE]

then we obtain $h_{\tau^{\prime}(1)}\cdots h_{\tau^{\prime}(m-1)}\in\mathfrak{L}(V),h_{\tau^{\prime}(1)}\cdots h_{\tau^{\prime}(m-1)}\circ h_{\tau^{\prime}(m)}$ and

$h_{1}\cdots h_{m}\sim h_{\tau^{\prime}(1)}\cdots h_{\tau^{\prime}(m)}$ for some $\tau^{\prime}\in\mathbb{S}_{1,\ldots,m}.$

(b)1. If $h_{\tau(2)}\cdots h_{\tau(b)}\circ h_{b+1}\cdots h_{m}$, then $h_{\tau_{1}\tau(2)}\cdots h_{\tau_{1}\tau(b)}h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}\in\mathfrak{L}(V)$ and $h_{\tau(2)}\cdots h_{\tau(b)}h_{b+1}\cdots h_{m}\sim h_{\tau_{1}\tau(2)}\cdots h_{\tau_{1}\tau(b)}$  $h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}$ for some $\tau_{1}\in\mathbb{S}_{\tau(2),\ldots,\tau(b),b+1,\ldots,m}$ by the induction hypotheses of (i) and (ii). We know $h_{\tau_{1}\tau(2)}\cdots h_{\tau_{1}\tau(b)}h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}\circ h_{\tau(1)}$ since $h_{\tau(2)}h_{\tau(3)}\cdots h_{\tau(b)}\circ h_{\tau(1)}$. We obtain $h_{1}\cdots h_{m}\sim h_{\tau(1)}\cdots h_{\tau(b)}h_{b+1}\cdots h_{m}\sim h_{\tau(1)}h_{\tau_{1}\tau(2)}\cdots h_{\tau_{1}\tau(b)}h_{\tau_{1}(b+1)}$ $\cdots h_{\tau_{1}(m)}=h_{\tau_{1}\tau(1)}h_{\tau_{1}\tau(2)}\cdots h_{\tau_{1}\tau(b)}$ $h_{\tau_{1}\tau(b+1)}\cdots h_{\tau_{1}\tau(m)}$ by

$h_{\tau_{1}(b+1)}\cdots h_{\tau_{1}(m)}=h_{\tau_{1}\tau(b+1)}\cdots h_{\tau_{1}\tau(m)}$ since $\tau\in\mathbb{S}_{1,\ldots,b}$ and $h_{\tau_{1}\tau(1)}=h_{\tau(1)}$ since $\tau_{1}\in\mathbb{S}_{\tau(2),\ldots,\tau(b),b+1,\ldots,m}$. Set $\tau^{\prime}=\tau_{1}\tau$, then we obtain $h_{\tau^{\prime}(2)}\cdots h_{\tau^{\prime}(m)}\in\mathfrak{L}(V),h_{\tau^{\prime}(2)}\cdots h_{\tau^{\prime}(m)}\circ h_{\tau^{\prime}(1)}$ and $h_{1}\cdots h_{m}\sim h_{\tau^{\prime}(1)}\cdots h_{\tau^{\prime}(m)}$ for some $\tau^{\prime}\in\mathbb{S}_{1,\ldots,m}$.

(b)2. If all $j_{1}\in\{2,\ldots,b\},l_{2}\in\{b+1,\ldots,m\}$ such that $\tilde{p}_{h_{\tau(j_{1})},h_{j_{2}}}=1$, then $h_{\tau(2)}\cdots h_{\tau(b)}$ $h_{b+1}\cdots h_{m}\sim h_{b+1}\cdots h_{m}h_{\tau(2)}\cdots h_{\tau(b)}$. On the other hand, we know $h_{\tau(1)}\circ h_{b+1}\cdots h_{m}$ since $h_{1}\cdots h_{b}\circ h_{b+1}\cdots h_{m}$, then $h_{\tau(1)}h_{b+1}\cdots h_{m}\in\mathfrak{L}(V)$ by induction hypotheses of (ii). We know $h_{1}\cdots h_{m}\sim h_{\tau(1)}\cdots h_{\tau(b)}h_{b+1}\cdots h_{m}\sim h_{\tau(1)}h_{b+1}\cdots h_{m}h_{\tau(2)}\cdots h_{\tau(b)}$, and $h_{\tau(1)}h_{b+1}\cdots h_{m}\circ h_{\tau(2)}\cdots h_{\tau(b)}$ since $h_{\tau(1)}\circ h_{\tau(2)}\cdots h_{\tau(b)}$. It is proved since $|h_{\tau(2)}\cdots h_{\tau(b)}|=b-1<b$ and induction hypotheses.

(2). If $h_{1}\cdots h_{b}=0$, then $h_{b+1}\cdots h_{m}\neq 0$ and $h_{b+1}\cdots h_{m}$ is connected by Lemma 2.2(iii). Consequently, $0=h_{1}\cdots h_{m-1}\in\mathfrak{L}(V)$ and $h_{1}\cdots h_{m-1}\circ h_{m}.$

(ii) Assume that $h_{1}h_{2}\cdots h_{m}\notin\mathfrak{L}(V)$. Obviously, $h_{1}h_{2}\cdots h_{m-1}\not=0$ and $h_{2}h_{3}\cdots h_{m}\not=0.$ Set $i_{0}:=1$ and $j_{0}:=m-1$ when $h_{1}h_{2}\cdots h_{m-1}\in\mathfrak{L}(V)$; $i_{0}:=2$ and $j_{0}:=m$ when $h_{2}h_{3}\cdots h_{m}\in\mathfrak{L}(V)$; $N_{0}:=\{i_{0},i_{0}+1,\cdots,j_{0}\};$ $A_{0}:=\{1,\ldots,m\}-N_{0}$,

Now we prove the following $\mathbf{Assertion(k)}$ by induction on $k$, $1\leq k\leq m-2.$

$\mathbf{Assertion(k):}$ There exist $1\leq i_{k}\leq j_{k}\leq m$, $\tau_{k}\in\mathbb{S}_{N_{k-1}}$ such that the following conditions hold: $(C_{1}).$ $0\neq h_{\tau^{k}(i_{k})}\cdots h_{\tau^{k}(j_{k})}\in\mathfrak{L}(V);$ $(C_{2}).$ $\tilde{p}_{h_{r},h_{t}}=1$ for $\forall\ r\neq t\in A_{k}$; $(C_{3}).$ $h_{\tau^{k}(i_{k})}\cdots h_{\tau^{k}(j_{k})}\circ h_{r}$ for $\forall\ r\in A_{k}$; $(C_{4}).$ $\tilde{p}_{h_{\tau^{k}(i_{k})}\cdots h_{\tau^{k}(j_{k})},h_{r}}=1$ for $\forall\ r\in A_{k}$; $(C_{5}).$ $h_{\tau^{k}(i_{k-1})}h_{\tau^{k}(i_{k-1}+1)}\cdots h_{\tau^{k}(j_{k-1})}\sim h_{\tau^{k-1}(i_{k-1})}h_{\tau^{k-1}(i_{k-1}+1)}\cdots h_{\tau^{k-1}(j_{k-1})}$ and $h_{\tau^{k}(1)}\cdots h_{\tau^{k}(m)}\sim h_{1}h_{2}\cdots h_{m}$; $(C_{6}).$ $i_{k-1}\leq i_{k}\leq j_{k}\leq j_{k-1}$ with $j_{k-1}-i_{k-1}=j_{k}-i_{k}+1$, where $N_{k}:=\{\tau^{k}({i_{k}}),\tau^{k}({i_{k}+1}),\cdots,\tau^{k}({j_{k}})\}$ and $A_{k}:=\{1,\ldots,m\}-N_{k}$, $\tau^{k}:=\tau_{k}\tau_{k-1}\cdots\tau_{1}$.

Step 1. For $k=1,$ now we construct $i_{1}$ and $j_{1}$ as follows.

We know $0\not=h_{i_{0}}\cdots h_{j_{0}}\in\mathfrak{L}(V)$, then $\tilde{p}_{h_{i_{0}}\cdots h_{j_{0}},h_{r}}=1$ for $r\in A_{1}$ by [WZZ15a, Lemma 4.12] and there exist $\tau_{1}\in\mathbb{S}_{N_{0}}$ such that $h_{i_{0}}\cdots h_{j_{0}}\sim h_{\tau_{1}(i_{0})}\cdots h_{\tau_{1}(j_{0})}$ with (a) $0\neq h_{\tau_{1}(i_{0})}\cdots h_{\tau_{1}(j_{0}-1)}\in\mathfrak{L}(V),$ $i_{1}:=i_{0}$ and $j_{1}:=j_{0}-1$ or with (b) $0\neq h_{\tau_{1}(i_{0}+1)}\cdots h_{\tau_{1}(j_{0})}\in\mathfrak{L}(V),$ $i_{1}:=i_{0}+1$ and $j_{1}:=j_{0}$ by induction hypotheses of (iii). Obviously, $0\neq h_{\tau^{1}(i_{1})}\cdots h_{\tau^{1}(j_{1})}\in\mathfrak{L}(V)$ and $h_{\tau_{1}(r)}=h_{r}$ for $r\in A_{0}$.

Obviously, $(C_{1})$, $(C_{5})$ and $(C_{6})$ hold.

$\tilde{p}_{h_{t},h_{r}}=1$ for any $t\not=r\in A_{1},$ i.e. $(C_{2})$ holds. Indeed, if $\tilde{p}_{h_{t},h_{r}}\neq 1$ for $t\in A_{1}-A_{0},$ $r\in A_{0}$, then $\tilde{p}_{h_{i_{1}}\cdots h_{j_{1}},h_{r}}\neq 1$ since $\tilde{p}_{h_{i_{0}}\cdots h_{j_{0}},h_{r}}=1$. We obtain $h_{i_{1}}\cdots h_{j_{1}}h_{t}h_{r},$ $h_{t}h_{i_{1}}\cdots h_{j_{1}}h_{r},$ $h_{r}h_{i_{1}}\cdots h_{j_{1}}h_{t}$ $\in\mathfrak{L}(V)$ by Lemma 2.1. However, $h_{1}\cdots h_{m}$ is a quantum equivalent with one among $h_{i_{1}}\cdots h_{j_{1}}h_{t}h_{r},$ $h_{t}h_{i_{1}}\cdots h_{j_{1}}h_{r},$ $h_{r}h_{i_{1}}\cdots h_{j_{1}}h_{t}$, which is a contradiction to $h_{1}\cdots h_{m}\notin\mathfrak{L}(V)$.

$(C_{3})$ and $(C_{4})$ follow from $(C_{2})$.

Step 2. Assumed that $\mathbf{Assertion(k)}$ hold, we prove that $\mathbf{Assertion(k+1)}$ holds, $k\leq m-3$. Let $i:=i_{k}$ and $j:=j_{k}$ in this proof for convenience.

We know $0\neq h_{\tau^{k}(i)}\cdots h_{\tau^{k}(j)}\in\mathfrak{L}(V)$, then there exists $\tau_{k+1}\in\mathbb{S}_{N_{k}}$ such that

$h_{\tau^{k}(i)}\cdots h_{\tau^{k}(j)}\sim h_{\tau^{k+1}(i)}\cdots h_{\tau^{k+1}(j)}$ with ($1^{\circ}$) $0\neq h_{\tau^{k+1}(i)}\cdots h_{\tau^{k+1}(j-1)}\in\mathfrak{L}(V),$ $i_{k+1}:=\tau^{k+1}(i)$ and $j_{k+1}:=\tau^{k+1}(j-1)$, or with ($2^{\circ}$) $0\neq h_{\tau^{k+1}(i+1)}\cdots h_{\tau^{k+1}(j)}\in\mathfrak{L}(V),$ $i_{k+1}:=\tau^{k+1}(i+1)$ and $j_{k+1}:=\tau^{k+1}(j)$ by induction hypotheses of (iii). We obtain $h_{1}\cdots h_{m}\sim h_{\tau^{k+1}(1)}\cdots h_{\tau^{k+1}(m)}$. For convenience, $h_{s}^{\prime}:=h_{\tau^{k+1}(s)}$ for all $s\in\{1,\ldots,m\}$. We know $\tau_{k+1}(t)=t$ for all $t\in A_{k}$ by the definition of $\tau^{k+1}$.

Obviously, $(C_{1})$, $(C_{5})$ and $(C_{6})$ hold.

($1^{\circ}$) If there exists $r\in A_{k+1}$ such that $\tilde{p}_{h_{\alpha}^{\prime},h_{r}}=1$ for all $\alpha\in\{i,\ldots,j-1\}$, then $\tilde{p}_{h_{j}^{\prime},h_{r}}\neq 1$ by $\mathbf{Assertion(k)}$, we know $\tilde{p}_{h_{i}^{\prime}\cdots h_{j}^{\prime},h_{r}}\neq 1$, it is a contradiction to $\mathbf{Assertion(k)}$. Thus for all $r\in A_{k+1}$, there exists $\alpha\in\{i,\ldots,j-1\}$ such that $\tilde{p}_{h_{\alpha}^{\prime},h_{r}}\neq 1$ , we obtain

$h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i}^{\prime}\cdots h_{j-1}^{\prime}\in\mathfrak{L}(V)$ by induction hypotheses.

Set $\{g_{1},\cdots,g_{\beta},l_{1},\cdots,l_{\eta}\}=\{\tau^{k+1}(j+1),\ldots,\tau^{k+1}(m-1)\}$ with $g_{1}<\cdots<g_{\beta},l_{1}<\cdots<l_{\eta}$ such that $\tilde{p}_{h_{j}^{\prime},h_{g_{\lambda}}}\not=1$ and $\tilde{p}_{h_{j}^{\prime},h_{l_{\xi}}}=1$ for all $\lambda\in\{1,\ldots,\beta\}$, all  $\xi\in\{1,\ldots,\eta\}$. Then $h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{r}}^{\prime}\in\mathfrak{L}(V)$ for all $r\in\{1,\ldots,\eta\}$ and $h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{t}}^{\prime}\in\mathfrak{L}(V)$ for all $t\in\{1,\ldots,\beta\}$ by induction hypotheses. If $\tilde{p}_{h_{j}^{\prime},h_{m}^{\prime}}\not=1$, then $\tilde{p}_{h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime},h_{m}^{\prime}}\not=1$ and $\tilde{p}_{h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}}^{\prime},h_{m}^{\prime}}\not=1$ by $\mathbf{Assertion(k)}$. Then $h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{m}^{\prime}\in\mathfrak{L}(V)$ by Lemma 2.1, $h_{1}\cdots h_{m}\sim h_{1}^{\prime}\cdots h_{m}^{\prime}\sim h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{m}^{\prime}$. It is a contradiction. We obtain $\tilde{p}_{h_{j}^{\prime},h_{m}^{\prime}}=1$. We obtain $h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{m}^{\prime}\in\mathfrak{L}(V)$ by induction hypotheses. If $\beta\geq 1$, then $\tilde{p}_{h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta-1}}^{\prime},h_{g_{\beta}}^{\prime}}\neq 1$ and $\tilde{p}_{h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{m}^{\prime},h_{g_{\beta}}^{\prime}}$ $\neq 1$ by $\mathbf{Assertion(k)}$. then $h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{m}^{\prime}h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}-1}^{\prime}h_{g_{\beta}}^{\prime}\in\mathfrak{L}(V)$ by Lemma 2.1. $h_{1}h_{2}\cdots h_{m}\sim h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{m}^{\prime}h_{j}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}-1}^{\prime}h_{g_{\beta}}^{\prime}$. It is contradiction. Then $\beta=0$. $\eta=m-j-1$, i.e. $\tilde{p}_{h_{j}^{\prime},h_{r}^{\prime}}=1$ for all $r\in\{j+1,\ldots,m\}$. So $\tilde{p}_{h_{i}^{\prime}\cdots h_{j-1}^{\prime},h_{r}^{\prime}}=1$ for all $r\in\{j+1,\ldots,m\}$ by $\mathbf{Assertion(k)}$. Assumed that there exists $\theta\in\{1,\ldots,i-1\}$ such that $\tilde{p}_{h_{\theta}^{\prime},h_{j}^{\prime}}\neq 1$. If $\tilde{p}_{h_{j}^{\prime},h_{1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{i-1}^{\prime}h_{i}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}}=1$, then $\tilde{p}_{h_{j}^{\prime},h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}}\neq 1$, $h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ is clear by induction hypotheses. then $h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}h_{j}^{\prime}$

$\in\mathfrak{L}(V)$ by [WZZ15a, Lemma 4.12]. $h_{1}h_{2}\cdots h_{m}\sim h_{1}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}h_{j}^{\prime}$. It is contradiction. If $\tilde{p}_{h_{j}^{\prime},h_{1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{i-1}^{\prime}h_{i}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}}\neq 1$, $h_{1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{i-1}^{\prime}h_{i}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ is clear by induction hypotheses. Then $h_{\theta}^{\prime}h_{1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{i-1}^{\prime}h_{i}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}h_{j}^{\prime}$

$\in\mathfrak{L}(V)$ by Lemma 2.1. $h_{1}h_{2}\cdots h_{m}\sim h_{\theta}^{\prime}h_{1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{i-1}^{\prime}h_{i}^{\prime}\cdots h_{j-1}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}h_{j}^{\prime}$. It is contradiction. Then $\tilde{p}_{h_{\theta}^{\prime},h_{j}^{\prime}}=1$ for all $\theta\in\{1,\ldots,i-1\}$, which implies that $(C_{2})$ holds.

$(C_{3})$ and $(C_{4})$ follows from $(C_{2})$.

($2^{\circ}$) If there exists $r\in A_{k+1}$ such that $\tilde{p}_{h_{\alpha}^{\prime},h_{r}}=1$ for all $\alpha\in\{i+1,\ldots,j\}$, then $\tilde{p}_{h_{i}^{\prime},h_{r}}\neq 1$ by $\mathbf{Assertion(k)}$, we know $\tilde{p}_{h_{i}^{\prime}\cdots h_{j}^{\prime},h_{r}}\neq 1$, it is a contradiction to $\mathbf{Assertion(k)}$. Thus for all $r\in A_{k+1}$, there exists $\alpha\in\{i+1,\ldots,j\}$ such that $\tilde{p}_{h_{\alpha}^{\prime},h_{r}}\neq 1$ , we obtain

$h_{i+1}^{\prime}\cdots h_{j}^{\prime}h_{j+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ by induction hypotheses.

Set $\{g_{1},\cdots,g_{\beta},l_{1},\cdots,l_{\eta}\}=\{\tau^{k+1}(2),\ldots,\tau^{k+1}(i-1)\}$ with $g_{1}<\cdots<g_{\beta},l_{1}<\cdots<l_{\eta}$ such that $\tilde{p}_{h_{i}^{\prime},h_{g_{\lambda}}}\not=1$ and $\tilde{p}_{h_{i}^{\prime},h_{l_{\xi}}}=1$ for all $\lambda\in\{1,\ldots,\beta\}$, all $\xi\in\{1,\ldots,\eta\}$. Then $h_{l_{1}}^{\prime}\cdots h_{l_{r}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ for all $r\in\{1,\ldots,\eta\}$ and $h_{g_{1}}^{\prime}\cdots h_{g_{t}}^{\prime}h_{i}^{\prime}\in\mathfrak{L}(V)$ for all $t\in\{1,\ldots,\beta\}$ by induction hypotheses. If $\tilde{p}_{h_{i}^{\prime},h_{1}^{\prime}}\not=1$, then $\tilde{p}_{h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime},h_{1}^{\prime}}\not=1$ and $\tilde{p}_{h_{g_{1}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{i}^{\prime},h_{1}^{\prime}}\not=1$ by $\mathbf{Assertion(k)}$. Then $h_{1}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{i}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ by Lemma 2.1, $h_{1}\cdots h_{m}\sim h_{1}^{\prime}\cdots h_{m}^{\prime}\sim h_{1}^{\prime}h_{g_{1}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{i}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}$. It is a contradiction. We obtain $\tilde{p}_{h_{i}^{\prime},h_{1}^{\prime}}=1$. We obtain $h_{1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ by induction hypotheses. If $\beta\geq 1$, then $\tilde{p}_{h_{g_{2}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{i}^{\prime},h_{g_{1}}^{\prime}}\neq 1$ and $\tilde{p}_{h_{1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime},h_{g_{1}}^{\prime}}$ $\neq 1$ by $\mathbf{Assertion(k)}$. then $h_{g_{1}}^{\prime}h_{g_{2}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{i}^{\prime}h_{1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ by Lemma 2.1. $h_{1}h_{2}\cdots h_{m}\sim h_{g_{1}}^{\prime}h_{g_{2}}^{\prime}\cdots h_{g_{\beta}}^{\prime}h_{i}^{\prime}h_{1}^{\prime}h_{l_{1}}^{\prime}\cdots h_{l_{\eta}}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}$. It is contradiction. Then $\beta=0$. $\eta=i-2$, i.e. $\tilde{p}_{h_{i}^{\prime},h_{r}^{\prime}}=1$ for all $r\in\{1,\ldots,i-1\}$. So $\tilde{p}_{h_{i+1}^{\prime}\cdots h_{j}^{\prime},h_{r}^{\prime}}=1$ for all $r\in\{1,\ldots,i-1\}$ by $\mathbf{Assertion(k)}$. Assumed that there exists $\theta\in\{j+1,\ldots,m\}$ such that $\tilde{p}_{h_{\theta}^{\prime},h_{i}^{\prime}}\neq 1$. If $\tilde{p}_{h_{i}^{\prime},h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{j}^{\prime}h_{j+1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{m}^{\prime}}=1$, then $\tilde{p}_{h_{i}^{\prime},h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}}\neq 1$, $h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ is clear by induction hypotheses. then $h_{i}^{\prime}h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ by [WZZ15a, Lemma 4.12]. $h_{1}h_{2}\cdots h_{m}\sim h_{i}^{\prime}h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{m}^{\prime}$. It is contradiction. If

$\tilde{p}_{h_{i}^{\prime},h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{j}^{\prime}h_{j+1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{m}^{\prime}}\neq 1$, $h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{j}^{\prime}h_{j+1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{m}^{\prime}\in\mathfrak{L}(V)$ is clear by induction hypotheses. Then $h_{i}^{\prime}h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{j}^{\prime}h_{j+1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{m}^{\prime}h_{\theta}^{\prime}$

$\in\mathfrak{L}(V)$ by Lemma 2.1. $h_{1}h_{2}\cdots h_{m}\sim h_{i}^{\prime}h_{1}^{\prime}\cdots h_{i-1}^{\prime}h_{i+1}^{\prime}\cdots h_{j}^{\prime}h_{j+1}^{\prime}\cdots h_{\theta-1}^{\prime}h_{\theta+1}^{\prime}\cdots h_{m}^{\prime}h_{\theta}^{\prime}$. It is contradiction. Then $\tilde{p}_{h_{\theta}^{\prime},h_{i}^{\prime}}=1$ for all $\theta\in\{j+1,\ldots,m\}$, which implies that $(C_{2})$ holds.

$(C_{3})$ and $(C_{4})$ follows from $(C_{2})$.

Step 3. In $\mathbf{Assertion(m-2)}$, It is a contradiction by $(C_{3})$ and $(C_{4})$.

(iii) By Lemma 2.2, $u$ is connected. which implies that there exist two connected monomials $v$ and $w$ such that $u\sim vw$. By inductive assumption, $v$ and $w$ belong to $\mathfrak{L}(V).$ Consequently, (iii) holds by (i).

(iv) By Lemma 2.3, $u\sim vw$ such that $v$ and $w$ are connected, as well as, $v\circ w.$ By inductive assumption, $v,w\in\mathfrak{L}(V).$ It follows from (i) and (ii). $\Box$

Corollary 2.5**.**

If $h_{1}h_{2}\cdots h_{m}\neq 0$, then $h_{1}h_{2}\cdots h_{m}\in\mathfrak{L}(V)$ if and only if $\mu(h_{1}h_{2}\cdots h_{m})$ is connected.

Proof. It follows from Lemma 2.2 and Theorem 2.4 (iv). $\Box$

3 A basis of Nichols braided Lie algebras

In this section we obtain a basis of Nichols braided Lie algebra.

Lemma 3.1**.**

Assume that $V$ is a braided vector space of diagonal type. If $u\neq 0$ is homogeneous elements in $\mathfrak{B}(V)$, Then $\mu(u)$ is connected (i.e. every monomial of $u$ is connected ) if and only if $u\in\mathfrak{L}(V)$.

Proof. The necessity follows from Corollary 2.5. The sufficiency. If $u$ is not connected and $u=\sum\limits_{i=1}^{r}k_{i}\sigma_{i}(u_{i})$ with $k_{i}\in F^{*}$, where $u_{i}$ is a non-zero disconnected monomial and $\sigma_{i}$ is a method of bracket on letters of $u_{i}$ for $1\leq i\leq r$. By Lemma 2.2, $\sigma_{i}(u_{i})=0$ for $1\leq i\leq r$ and $u=0$, which is a contradiction. $\Box$

Lemma 3.2**.**

If $[u]\in D$, then $u$ is connected and $u\not=0$.

Proof. By [Kh99, Cor. 1], $u\not=0.$ Obviously $[u]\not=0$ and $[u]\in\mathfrak{L}(V)$. Considering Lemma 3.1 we complete the proof. $\Box$

Theorem 3.3**.**

If $\mathfrak{B}(V)$ is Nichols algebra of diagonal type with $\dim V\geq 2$, then the set $\{[u_{1}]^{k_{1}}[u_{2}]^{k_{2}}\cdots[u_{s}]^{k_{s}}\ \mid\ [u_{i}]\in D,\mid D\mid=s;0\leq k_{i}<h_{u_{i}};1\leq i\leq s;u_{s}<u_{s-1}<\cdots<u_{1},\mu([u_{1}]^{k_{1}}[u_{2}]^{k_{2}}\cdots[u_{s}]^{k_{s}})\hbox{ is connect },\sum\limits_{i=1}^{s}k_{i}>0\}$ is a basis of $\mathfrak{L}(V)$.

Proof. It follows from [He05, Th. 1.4.6], and Corollary 2.5 and Lemma 3.1. $\Box$

4 The dimension of $\mathfrak{L}(V)$

In this section we obtain dimension of Nichols braided Lie algebra $\mathfrak{L}(V)$ with finite Cartan type.

Let $V_{i_{1},\cdots,i_{r}}$ denote the braided vector subspace generated by $\{x_{i_{1}},x_{i_{2}},\cdots,x_{i_{r}}\}$ of $V$; $D_{i_{1},\cdots,i_{r}}$ denote $\{[u]\mid[u]\hbox{ is a hard super-letter of }\mathfrak{B}(V_{i_{1},i_{2},\cdots,i_{r}})\}$;

$L_{i_{1},\cdots,i_{r}}:=\{[u_{1}]^{k_{1}}[u_{2}]^{k_{2}}\cdots[u_{s}]^{k_{s}}\ \mid\ [u_{j}]\in D_{i_{1},\cdots,i_{r}};0\leq k_{j}<{\rm ord}(p_{u_{j},u_{j}});1\leq j\leq s;\mid D_{i_{1},\cdots,i_{r}}\mid=s;u_{s}<u_{s-1}<\cdots<u_{1},\mu([u_{1}]^{k_{1}}[u_{2}]^{k_{2}}\cdots[u_{s}]^{k_{s}})\hbox{ is connected },\sum\limits_{j=1}^{s}k_{j}>0\}$.

$B_{i_{1},\cdots,i_{r}}:=\{[u_{1}]^{k_{1}}[u_{2}]^{k_{2}}\cdots[u_{s}]^{k_{s}}\ \mid\ [u_{j}]\in D_{i_{1},\cdots,i_{r}};0\leq k_{j}<{\rm ord}(p_{u_{j},u_{j}});1\leq j\leq s;\mid D_{i_{1},\cdots,i_{r}}\mid=s,\sum\limits_{j=1}^{s}k_{j}>0\}$; Let $V_{s,t}$ denote $V_{s,s+1,\cdots,t}$ in short; similarly we have $B_{s,t}$ and $L_{s,t}$. Let $B_{i,j}:=\emptyset$ and $L_{i,j}:=\emptyset$ when $i>j.$

Lemma 4.1**.**

Assumed that $\mathfrak{B}(V)$ is connected Nichols algebra of diagonal type with $\dim V>2$ and $\Delta(\mathfrak{B}(V))$ is an arithmetic root system. The following hold.

(i)* If pure generalized Dynkin diagram is*

\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots123n-2n-1n

then

[TABLE]

[TABLE]

[TABLE]

where $u_{0}=\mid B_{1,n}\mid$ and $u_{j}=\sum\limits_{n_{1}=1}^{n-2}\sum\limits_{n_{2}=1}^{n_{1}-2}\cdots\sum\limits_{n_{j}=1}^{n_{j-1}-2}\mid B_{1,n_{j}}\mid(\mid B_{n_{j}+2,n_{j-1}}\mid-\mid B_{n_{j}+3,n_{j-1}}\mid)\cdots(\mid B_{n_{2}+2,n_{1}}\mid-\mid B_{n_{2}+3,n_{1}}\mid)(\mid B_{n_{1}+2,n}\mid-\mid B_{n_{1}+3,n}\mid)$ for $j>0$.

(ii)* If pure generalized Dynkin diagram is

\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots12n - 3n - 2n - 1n

then

[TABLE]

[TABLE]

where $\mid L_{1,i}\mid$ is obtained by the formula (7) when $1\leq i\leq n-3$.

(iii)* If pure generalized Dynkin diagram is

\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots123n-3n-1nn-2$\bullet$

then

[TABLE]

[TABLE]

where $\mid L_{1,i}\mid$ is obtained by the formula (7) when $1\leq i\leq n-2$.

Proof. (i) We only determine which element in $B_{1,n}$ is connected. It is clear that the left hand of (3) $\subseteq$ the right hand of (3). If $u\in B_{1,n}-L_{1,n}$, let $i_{u}:={\rm min}\{j\mid x_{j}\notin\mu(u)\hbox{ and there exists }x_{i}\in\mu(u)\hbox{ such that }1\leq i<j\leq n\}$. By Lemma 3.2, there exist $v\in L_{1,i_{u}-1}$ and $w\in B_{i_{u}+1,n}$ such that $u=vw$. Consequently, the right hand of (3) $\subseteq$ the left hand of (3). therefore (3) holds.

[TABLE]

Similarly, we can show (ii) and (iii). $\Box$

By [Hu78], $\mid D(A_{n})\mid=C_{n+1}^{2},$ $\mid D(B_{n})\mid=n^{2}=\mid D(C_{n})\mid,$ $\mid D(D_{n})\mid=n^{2}-n,\mid D(E_{6})\mid=36$, $\mid D(E_{7})\mid=63$, $\mid D(E_{8})\mid=120$, $\mid D(F_{4})\mid=24$, $\mid D(G_{2})\mid=6.$ By Lemma 4.1 and [WZZ15a, Lemma 6.4], we have the following results.

Example 4.2**.**

*Let ${\rm ord}(q):=N$.

*(i) For $A_{n}$, $n\geq 1$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots123n-1nq$$q$$q$$q$$q$$q^{-1}$$q^{-1}$$q^{-1}$$,q\in F^{*}/\{1\}.

then $\dim\mathfrak{L}(V)=\sum\limits_{j=0}^{int(\frac{n-1}{2})}(-1)^{j}u_{j}$, where $u_{0}=\mid B_{1,n}\mid$ and $u_{j}=\sum\limits_{n_{1}=1}^{n-2}\sum\limits_{n_{2}=1}^{n_{1}-2}\cdots\sum\limits_{n_{j}=1}^{n_{j-1}-2}\mid B_{1,n_{j}}\mid(\mid B_{n_{j}+2,n_{j-1}}\mid-\mid B_{n_{j}+3,n_{j-1}}\mid)\cdots(\mid B_{n_{2}+2,n_{1}}\mid-\mid B_{n_{2}+3,n_{1}}\mid)(\mid B_{n_{1}+2,n}\mid-\mid B_{n_{1}+3,n}\mid)$ for $j>0$ and $\mid B_{i,k}\mid=N^{C_{k-i+2}^{2}}-1$ for $1\leq i\leq k\leq n$. Furthermore,*

[TABLE]

*(ii) For $B_{n}$, $n\geq 2$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots123n-2n-1nq^{2}$$q^{2}$$q^{2}$$q^{2}$$q^{2}$$q$$q^{-2}$$q^{-2}$$q^{-2}$$q^{-2}$$,q\in F^{*}/\{1,-1\}.

then $\dim\mathfrak{L}(V)=\sum\limits_{j=0}^{int(\frac{n-1}{2})}(-1)^{j}u_{j}$, where $u_{0}=\mid B_{1,n}\mid$ and $u_{j}=\sum\limits_{n_{1}=1}^{n-2}\sum\limits_{n_{2}=1}^{n_{1}-2}\cdots\sum\limits_{n_{j}=1}^{n_{j-1}-2}\mid B_{1,n_{j}}\mid(\mid B_{n_{j}+2,n_{j-1}}\mid-\mid B_{n_{j}+3,n_{j-1}}\mid)\cdots(\mid B_{n_{2}+2,n_{1}}\mid-\mid B_{n_{2}+3,n_{1}}\mid)(\mid B_{n_{1}+2,n}\mid-\mid B_{n_{1}+3,n}\mid)$ for $j>0$; $\mid B_{i,n}\mid=N^{(n-i+1)^{2}}-1$ for $1\leq i<n$ and $\mid B_{i,k}\mid=N^{C_{k-i+2}^{2}}-1$ for $1\leq i\leq k<n$, when N is odd; $\mid B_{i,n}\mid=(\frac{N}{2})^{(n-i+1)^{2}-n+i-1}N^{n-i+1}-1$ for $1\leq i<n$ and $\mid B_{i,k}\mid=(\frac{N}{2})^{C_{k-i+2}^{2}}-1$ for $1\leq i\leq k<n$, when N is even. Furthermore,*

[TABLE]

when N is odd;

[TABLE]

*when N is even.

(iii) For $C_{n}$, $n>2$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots123n-2n-1nq$$q$$q$$q$$q$$q^{2}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-2}$$,q\in F^{*}/\{1,-1\}.

then $\dim\mathfrak{L}(V)=\sum\limits_{j=0}^{int(\frac{n-1}{2})}(-1)^{j}u_{j}$, where $u_{0}=\mid B_{1,n}\mid$ and $u_{j}=\sum\limits_{n_{1}=1}^{n-2}\sum\limits_{n_{2}=1}^{n_{1}-2}\cdots\sum\limits_{n_{j}=1}^{n_{j-1}-2}\mid B_{1,n_{j}}\mid(\mid B_{n_{j}+2,n_{j-1}}\mid-\mid B_{n_{j}+3,n_{j-1}}\mid)\cdots(\mid B_{n_{2}+2,n_{1}}\mid-\mid B_{n_{2}+3,n_{1}}\mid)(\mid B_{n_{1}+2,n}\mid-\mid B_{n_{1}+3,n}\mid)$ for $j>0$; $\mid B_{i,n}\mid=N^{(n-i+1)^{2}}-1$ for $1\leq i<n$ and $\mid B_{i,k}\mid=N^{C_{k-i+2}^{2}}-1$ for $1\leq i\leq k<n$, when N is odd; $\mid B_{i,n}\mid=N^{(n-i+1)^{2}-n+i-1}(\frac{N}{2})^{n-i+1}-1$ for $1\leq i<n$ and $\mid B_{i,k}\mid=N^{C_{k-i+2}^{2}}-1$ for $1\leq i\leq k<n$, when N is even. Furthermore,*

[TABLE]

when N is odd;

[TABLE]

*when N is even.

*(iv) For $D_{n}$, $n>3$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\cdots\cdots\cdots\cdots12n - 3n - 2n - 1nq$$q$$q$$q$$q$$q$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$,q\in F^{*}/\{1\}.

then $\dim\mathfrak{L}(V)=\mid B_{1,n}\mid-\mid B_{n-1,n-1}\mid\mid B_{n,n}\mid-\sum\limits_{i=1}^{n-3}\mid L_{1,i}\mid(\mid B_{i+2,n}\mid-\mid B_{i+3,n}\mid)-\mid L_{1,n-3}\mid\mid B_{n-1,n}\mid$, where $\mid B_{k,n}\mid=N^{({n-k+1})^{2}-n+k-1}-1$ for $n-k+1>2$, $\mid B_{n-1,n}\mid=N^{3}-1$, $\mid B_{n,n}\mid=\mid B_{n-1,n-1}\mid=N-1$; $\mid L_{1,t}\mid$ is obtained by the formula (4.2) when $1\leq t\leq n-3$. Furthermore,*

[TABLE]

*(v) For $E_{6}$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet123456q$$q$$q$$q$$q$$q$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$,q\in F^{*}/\{1\}.

then*

[TABLE]

*where $\mid L_{1,t}\mid$ is obtained by the formula (4.2) when $1\leq t\leq 4$.

*(vi) For $E_{7}$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet1234567q$$q$$q$$q$$q$$q$$q$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$,q\in F^{*}/\{1\}.

then*

[TABLE]

*where $\mid L_{1,t}\mid$ is obtained by the formula (4.2) when $1\leq t\leq 5$.

*(vii) For $E_{8}$, \bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet12345678q$$q$$q$$q$$q$$q$$q$$q$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$q^{-1}$$,q\in F^{*}/\{1\}.

then*

[TABLE]

*where $\mid L_{1,t}\mid$ is obtained by the formula (4.2) when $1\leq t\leq 6$.

*(viii) For $F_{4}$. \bullet$$\bullet$$\bullet$$\bullet1234q^{2}$$q^{2}$$q$$q$$q^{-2}$$q^{-2}$$q^{-1}$$,q\in F^{*}/\{1,-1\}.

then $\dim\mathfrak{L}(V)=\sum\limits_{j=0}^{1}(-1)^{j}u_{j}$, where $u_{0}=\mid B_{1,4}\mid$ and $u_{1}=\sum\limits_{n_{1}=1}^{2}(\mid B_{1,n_{1}}\mid)(\mid B_{n_{1}+2,4}\mid-\mid B_{n_{1}+3,4}\mid)$; $\mid B_{1,4}\mid=N^{24}-1$, $\mid B_{1,1}\mid=N-1$, $\mid B_{1,2}\mid=N^{C_{3}^{2}}-1=N^{3}-1$, $\mid B_{3,4}\mid=N^{C_{3}^{2}}-1=N^{3}-1$ and $\mid B_{4,4}\mid=N-1$ when $N$ is odd; $\mid B_{1,4}\mid=N^{12}(\frac{N}{2})^{12}-1$, $\mid B_{1,1}\mid=\frac{N}{2}-1$, $\mid B_{1,2}\mid=(\frac{N}{2})^{C_{3}^{2}}-1=(\frac{N}{2})^{3}-1$, $\mid B_{3,4}\mid=N^{C_{3}^{2}}-1=N^{3}-1$ and $\mid B_{4,4}\mid=N-1$ when $N$ is even. Furthermore*

[TABLE]

when $N$ is odd;

[TABLE]

when $N$ is even.

*(ix) For $G_{2}$, \bullet$$\bullet12q$$q^{3}$$q^{-3}$$,q\in F^{*}/\{1,-1\},q^{3}\not=1.

then $\dim\mathfrak{L}(V)=N^{6}-1$ when $3\nmid N$; $\dim\mathfrak{L}(V)=(\frac{N}{3})^{3}N^{3}-1$ when $3\mid N$.

5 Non-zero monomials in Nichols algebras

In this section we find some non-zero monomials in Nichols algebras.

Let $(s)_{q}:=1+q+q^{2}+\cdots+q^{s-1}$ and $(s)_{q}!:=(1)_{q}(2)_{q}\cdots(s)_{q}.$

Lemma 5.1**.**

Assume $h_{i}\in\{x_{1},\cdots,x_{n}\}$ for $1\leq i\leq m$ and $u_{j}$ is 1 or a monomial with $x_{k}\notin\mu(u_{j})$ for $1\leq j\leq l+1$. Let $q:=p_{x_{k},x_{k}}^{-1}$ for convenience.

(i)**

[TABLE]

(ii)**

[TABLE]

(iii)* If ${\rm ord}(p_{h_{i},h_{i}})>\mid{\rm deg}_{h_{i}}(h_{1}\cdots h_{m})\mid$ for all $i\in\{1,\ldots,m\}$, then $h_{1}\cdots h_{m}\neq 0$.*

Proof. (i) It can be obtained by induction on $l.$

(ii) We show this by induction on $l$. If $l=1$, $<y_{k},u_{1}x_{k}u_{2}>=p_{k,u_{1}}^{-1}u_{1}u_{2}$. Assume $l>1.$ See that

[TABLE]

(iii) We show this by induction on $t:=\mid\mu(h_{1}h_{2}\cdots h_{m})\mid$. If $t=1$, we obtain $h_{1}\cdots h_{m}\neq 0$ by [He05, Lemma 1.3.3 (i)]. Assume $t>1$ and $h_{1}h_{2}\cdots h_{m}=u_{1}x_{k}u_{2}x_{k}\cdots u_{l}x_{k}u_{l+1}$ with $\mid\mu(u_{1}u_{2}\cdots u_{l+1})\mid$ $=t-1$. Thus $u_{1}u_{2}\cdots u_{l+1}\neq 0$ by induction hypotheses, $(1+p_{kk}^{-1})(1+p_{kk}^{-1}+p_{kk}^{-2})\cdots(1+p_{kk}^{-1}+\cdots+p_{kk}^{-l+1})\neq 0$ since ${\rm ord}(p_{x_{k},x_{k}})>\mid{\rm deg}_{x_{k}}(h_{1}\cdots h_{m})\mid=l$. Hence $u_{1}x_{k}u_{2}x_{k}\cdots u_{l}x_{k}u_{l+1}\neq 0$, completing the proof. $\Box$

Corollary 5.2**.**

If $u$ and $v$ are monomials in $\mathfrak{B}(V)$, $\mu(u)\cap\mu(v)=\emptyset$, $p_{ij}p_{ji}=1$ for any $x_{i}\in\mu(u)$, $x_{j}\in\mu(v)$, then the following conditions are equivalent: (i) $u\neq 0,v\neq 0$. (ii) $uv\neq 0$. (iii) $uv\notin\mathfrak{L}(V)$.

Proof. We know (ii) $\Longleftrightarrow$ (iii) by Lemma 2.2. (ii)$\Longrightarrow$ (i) is clear. (ii) $\Longleftarrow$ (i): Assume $\mid u\mid=k$, then $\exists y_{i_{1}},\ldots,y_{i_{k}}\in d(u)$ such that $<y_{i_{1}}\cdots y_{i_{k}},u>\in F^{*}$ and $<y_{i_{1}}\cdots y_{i_{k}},uv>$ $=<y_{i_{1}}\cdots y_{i_{k}},u>v\neq 0$. $\Box$

Lemma 5.3**.**

If $u$ and $v$ are two homogeneous elements with $\mu(u)\cap\mu(v)=\emptyset$, then $uv=0$ implies $u=0$ or $v=0$.

Proof. Without lost general, there exists $i_{0}$ such that $\mu(v)\subseteq\{1,2,\cdots i_{0}\}$ and $\mu(u)\subseteq\{i_{0}+1,i_{0}+2,\cdots n\}$. Assume $u\not=0$ and $v\not=0$ with $u=\sum\limits_{i=1}^{s}k_{i}u_{i}$ and $v=\sum\limits_{j=1}^{t}k_{i}^{\prime}v_{i}$, where $k_{i}\not=0,$ $k_{j}^{\prime}\not=0,$ $u_{i}\in B_{1,i_{0}}$, $v_{j}\in B_{i_{0}+1,n}$ for $1\leq i\leq i_{0},$ $1+i_{0}\leq j\leq n.$ Consequently, $uv=\sum\limits_{i=1}^{s}\sum\limits_{j=1}^{t}k_{i}k_{j}^{\prime}u_{i}v_{j}\not=0$ with $u_{i}v_{j}\in B_{1,n}$ and $k_{i}k_{j}\not=0$ for $1\leq i\leq s,$ $1\leq j\leq t.$ $\Box$

6 Relations between graphes and Nichols Braided Lie Algebras

In this section we give relations between connected components of graphes and Nichols Braided Lie Algebras.

Obviously, every pure generalized Dynkin diagram is a graph in terms of graph theory (see [Ha69]). Conversely, Assume that $\Gamma$ is a graph in terms of graph theory with vertex $\{1,2,\cdots,n\}$. We define a matrix $(q_{ij})_{n\times n}$ as follows: $q_{ij}q_{ji}\neq 1$ if and only if there exists an edge between $i$ and $j.$ Let $V$ be the braided vector space with braiding matrix $(q_{ij})_{n\times n}$. $\Gamma$ is a pure generalized Dynkin diagram of $V.$

Corollary 6.1**.**

(i) $\Gamma(u)$ is a connected component of $\Gamma(V)$ with a non-zero monomial $u$ if and only if $\mu(u)$ is a maximal element in $\{\mu(v)\mid v\hbox{ is a non-zero monomial in }\mathfrak{L}(V)\}$ under order $\subseteq$.

(ii) The following conditions are equivalent.

(a). $\Gamma(V)$ is connected

(b). $x_{n}x_{n-1}\cdots x_{1}\in\mathfrak{L}(V)$.

(c). $x_{1}x_{2}\cdots x_{n}\in\mathfrak{L}(V)$

(d). there exists a non-zero monomial $u\in\mathfrak{L}(V)$ with $\mid\mu(u)\mid=\dim V$.

Proof. (i) It follows from Corollary 2.5.

(ii) Considering Lemma 5.3 we have $x_{n}x_{n-1}\cdots x_{1}\not=0$ and $x_{1}x_{2}\cdots x_{n}\not=0$. Using Corollary 2.5 we complete the proof. $\Box$