modules (cf. \[FLM\], \[DLM2\] and \[Z\]).

A [*weak $V$-module*]{} $M$ is a vector space equipped with a linear map $$\begin{array}{lll}
V&\to &(\End\,M)\{z\}\\
v&\mapsto&\displaystyle{ Y_M(v,z)=\sum_{n\in\Q}v_nz^{-n-1}\ \ \ (v_n\in
\End\,M)}
\end{array}$$ which satisfies the following properties for all $u\in V$, $v\in V,$ $w\in M$, $$\begin{aligned}
& &u_lw=0\ \ \                  
\mbox{for}\ \ \ l>>0\label{vlw0}\\
& &Y_M({\mathbf 1},z)=1;\label{vacuum}\end{aligned}$$ $$\label{jacobi}
\begin{array}{c}
\displaystyle{z^{-1}_0\delta\left(\frac{z_1-z_2}{z_0}\right)
Y_M(u,z_1)Y_M(v,z_2)-z^{-1}_0\delta\left(\frac{z_2-z_1}{-z_0}\right)
Y_M(v,z_2)Y_M(u,z_1)}\\
\displaystyle{=z_2^{-1}
\delta\left(\frac{z_1-z_0}{z_2}\right)
Y_M(Y(u,z_0)v,z_2)},
\end{array}$$ where $\delta(z)=\sum_{n\in\Z}z^n$ and all binomial expressions (here and below) are to be expanded in nonnegative integral powers of the second variable. Elementary properties of the $\delta$-function can be found in \[FLM\] and \[FHL\]

(\[jacobi\]) is called the [*Jacobi identity*]{}. One can prove that the Jacobi identity is equivalent to the following associativity formula (see \[FLM\] and \[FHL\]) $$\begin{aligned}
\label{ea}
(z_{0}+z_{2})^{k}Y_{M}(u,z_{0}+z_{2})Y_{M}(v,z_{2})w
=(z_{2}+z_{0})^{k}Y_M(Y(u,z_0)v,z_2)w\end{aligned}$$ and commutator relation $$\begin{aligned}
& &\ \ \ \  [Y_{M}(u,z_{1}),Y_{M}(v,z_{2})]\nonumber\\
& &=\Res_{z_{0}}z_2^{-1}
\delta\left(\frac{z_1-z_0}{z_2}\right)Y_M(Y(u,z_0)v,z_2)\label{ec}\end{aligned}$$ where $w\in M$ and $k$ is a nonnegative integer such that $z^{k}Y_{M}(u,z)w$ involves only nonnegative integral powers of $z.$

We may also deduce from (\[vlw0\])-(\[jacobi\]) the usual Virasoro algebra axioms (see \[DLM2\]). Namely, if $Y_M(\om,z)=\sum_{n\in\Z}L(n)z^{-n-2}$ then $$\label{g3.8}
[L(m),L(n)]=(m-n)L(m+n)+\frac{1}{12}(m^3-m)\delta_{m+n,0}(\mbox{rank}\,V)$$ and $$\label{2.5}
\frac{d}{dz}Y_M(v,z)=Y_M(L(-1)v,z).$$

Suppose that $(M_{i},Y_{i})$ are two weak $V$-modules, $i$=1,2. A homomorphism from $M_{1}$ to $M_{2}$ is a linear map $f$:$M_{1}\rightarrow M_{2}$ which satisfied 