An algebra $(\mathcal{E},\cdot)$ is called a Leibniz algebra if

$$x\cdot(y\cdot z)=(x\cdot y)\cdot z+y\cdot(x\cdot z)$$

(2.2)

for all $x,y,z\in L$ .

. (ML Diagram) An IL diagram $\{D,m\}$ is called a Multiplicity-Labeled Diagram (ML Diagram) if $m(c)=1,2~{}(c\in C(D))$ . In figures, we draw a chord $c$ with $m(c)=1$ by a thin line and a chord $c$ with $m(c)=2$ by a thin line with a letter ”2” as follows:

$\displaystyle\begin{picture}(6.0,0.0)(4.0,-4.0)\special{pn 8}\special{pa 400 400}\special{pa 1000 400}\special{fp}\end{picture}~{}~{}\hbox{m(c)=1},\hskip 42.679134pt\begin{picture}(6.0,1.65)(4.0,-4.0)\special{pn 8}\special{pa 400 400}\special{pa 1000 400}\special{fp}\put(7.0,-3.4){\makebox(0.0,0.0){\scriptsize$2$}}\end{picture}~{}~{}\hbox{m(c)=2}.$

We give two examples of ML diagrams,

$\displaystyle\begin{picture}(3.4,2.4)(1.5,-3.6)\special{pn 16}\special{ar 320 320 120 120 0.0000000 6.2831853}\special{pn 4}\special{pa 320 200}\special{pa 320 440}\special{fp}\special{pn 4}\special{pa 209 275}\special{pa 431 275}\special{fp}\put(4.7,-2.76){\makebox(0.0,0.0){\tiny 2}}\special{pn 4}\special{pa 209 365}\special{pa 431 365}\special{fp}\end{picture},\hskip 42.679134pt\begin{picture}(6.9,2.4)(1.5,-3.6)\special{pn 16}\special{ar 320 320 120 120 0.0000000 6.2831853}\special{pn 16}\special{ar 680 320 120 120 0.0000000 6.2831853}\special{pn 4}\special{pa 401 230}\special{pa 600 230}\special{fp}\special{pn 4}\special{pa 399 410}\special{pa 602 410}\special{fp}\special{pn 4}\special{pa 200 320}\special{pa 440 320}\special{fp}\put(3.2,-2.9){\makebox(0.0,0.0){\tiny 2}}\end{picture}.$

. (Dotted Diagram) An IL diagram $\{D,\kappa\}$ is called a Dotted diagram if $\kappa(c)=0,1~{}(c\in C(D))$ . A chord $c$ is called a normal chord if $\kappa(c)=1$ and a dotted chord if $\kappa(c)=0$ . In figures, we draw a normal chord $(\kappa(c)=1)$ by a thin line and a dotted chord $(\kappa(c)=0)$ by a dotted line as follows:

$\displaystyle\begin{picture}(6.0,0.0)(4.0,-4.0)\special{pn 8}\special{pa 400 400}\special{pa 1000 400}\special{fp}\end{picture}~{}~{}\hbox{normal chord}~{}(\kappa(c)=1),\hskip 42.679134pt\begin{picture}(6.0,2.0)(4.0,-4.5)\special{pn 8}\special{pa 400 400}\special{pa 1000 400}\special{dt 0.045}\special{pa 1000 400}\special{pa 999 400}\special{dt 0.045}\end{picture}~{}~{}\hbox{dotted chord}~{}(\kappa(c)=0).$

We give two exmples of dotted diagrams,

$\displaystyle\begin{picture}(6.9,2.4)(1.5,-3.6)\special{pn 16}\special{ar 320 320 120 120 0.0000000 6.2831853}\special{pn 16}\special{ar 680 320 120 120 0.0000000 6.2831853}\special{pn 8}\special{pa 425 260}\special{pa 576 260}\special{dt 0.035}\special{pa 576 260}\special{pa 575 260}\special{dt 0.035}\special{pn 4}\special{pa 425 380}\special{pa 576 380}\special{fp}\special{pn 8}\special{pa 200 320}\special{pa 440 320}\special{dt 0.035}\special{pa 440 320}\special{pa 439 320}\special{dt 0.035}\end{picture},\hskip 42.679134pt\begin{picture}(3.4,2.4)(1.5,-3.6)\special{pn 16}\special{ar 320 320 120 120 0.0000000 6.2831853}\special{pn 8}\special{pa 320 200}\special{pa 320 440}\special{dt 0.04}\special{pa 320 440}\special{pa 320 439}\special{dt 0.04}\special{pn 4}\special{pa 209 275}\special{pa 431 275}\special{fp}\special{pn 8}\special{pa 209 365}\special{pa 429 365}\special{dt 0.035}\special{pa 429 365}\special{pa 428 365}\special{dt 0.035}\end{picture}.$

Given two partitions $\alpha$ and $\beta$ , denote by $\alpha\wedge\beta$ the meet of $\alpha$ and $\beta$ , that is, the minimal partition consisting of whole blocks of both $\alpha$ and $\beta$ . We say that $\alpha$ and $\beta$ are transversal and write $\alpha\perp\beta$ if

$$\ell(\alpha)+\ell(\beta)-\ell(\alpha\wedge\beta)=n\,.$$

Define the weight of a monomial $\xi^{\mu}$ by $\operatorname{wt}(\xi^{\mu})=\operatorname{wt}(\mu)=|\mu|+\ell(\mu)$ or, in other words,

$$\textup{weight}=\textup{degree}+\#\textup{ of variables}\,.$$

Let $(G,+)$ be a commutative monoid. A commutation factor on $G$ with values in $K$ is a mapping $\epsilon:G\times G\rightarrow K$ such that :

	$\displaystyle\epsilon(g,h)\epsilon(h,g)=1$		(1)
	$\displaystyle\epsilon(g+h,k)=\epsilon(g,k)\epsilon(h,k)$		(2)

Let $V$ be a $G$ -graded $K$ -space, and $[.,.]$ be a bilinear map from $V\times V$ to $V$ , such that $\forall_{h}$ $x,y,z\in V$ :

$$[x,y]=-\epsilon(\bar{y},\bar{x})[y,x]$$

(5)

$$\epsilon(\bar{z},\bar{x})[x,[y,z]]+\epsilon(\bar{y},\bar{z})[z,[x,y]]+\epsilon(\bar{x},\bar{y})[y,[z,x]]=0$$

(6)

$V$ is called an $\epsilon$ -Lie algebra.