$$\begin{aligned}
[L(m), L(n)]=(m-n)L(m+n)+\frac{m^{3}-m}{12}\delta_{m+n,0}(\mbox{rank}V)\end{aligned}$$ for $m,n\in {\Z}$, where $Y(\omega,x)=\sum_{n\in {\Z}}L(n)x^{-n-2}$ and $\mbox{rank}V$ is a complex number, called the [*rank*]{} of $V$, and such that for $v\in V_{(n)}, n\in {\Z}$, $$\begin{aligned}
& &Y(L(-1)v,x)={d\over dx}Y(v,x),\\
& & L(0)v=nv.\end{aligned}$$

This completes the definition. It is well known (\[FLM\], \[FHL\]) that the Jacobi identity is equivalent to the following rationality, commutativity and associativity:

(rationality) For $u,v,w\in V,\; w'\in V'$, where $V'=\coprod_{n}V_{(n)}^{*}$, the formal series $$\begin{aligned}
\label{e02}
\<w', Y(u,x_{1})Y(v,x_{2})w\>\left(=\sum_{m,n\in {\Z}}\<w',u_{m}v_{n}w\>
x_{1}^{m-1}x_{2}^{-n-1}\right)\end{aligned}$$ absolutely converges to a rational function of the form $$\begin{aligned}
\label{ef}
f(x_{1},x_{2})=\frac{g(x_{1},x_{2})}{(x_{1}-x_{2})^{k}}\end{aligned}$$ in the domain $|x_{1}|>|x_{2}|>0$, where $g\in {\C}[x_{1},x_{1}^{-1},x_{2},x_{2}^{-1}]$ and $k$ is an integer only depending on $u$ and $v$; independent of $w'$ and $w$;

(commutativity) The formal series $\<w', Y(v,x_{2})Y(u,x_{1})w\>$ absolutely converges to the same rational function $f(x_{1},x_{2})$ in the domain $|x_{2}|>|x_{1}|>0$;

(associativity) The formal series $$\begin{aligned}
\label{e03}
\<w', Y(Y(u,x_{0})v,x_{2})w\>\left(=\sum_{m,n\in {\Z}}\<w', (u_{m}v)_{n}w\>
x_{0}^{-m-1}x_{2}^{-n-1}\right)\end{aligned}$$ absolutely converges to the rational function $f(x_{0}+x_{2},x_{2})$ in the domain $|x_{2}|>|x_{0}|>0$. If we specialize $x_{2}$ to a nonzero complex number $z$, (\[e02\]) gives rise to a meromorphic function on the sphere $CP^{1}={\C}\cup \{\infty\}$ with only three possible poles at $0, \infty$ and $z$ while (\[e03\]) gives rise to a meromorphic function on the sphere $CP^{1}$ with only three possible poles at 