$fY_{M_{1}}(v,z)=Y_{M_{2}}(v,z)f$ for all $v\in V$. We call $f$ an isomorphism if $f$ is also a linear isomorphism.

\[d1\] A (ordinary) [*$V$-module*]{} is a weak $V$-module $M$ which carries a $\C$-grading induced by the spectrum of $L(0)$. Then $M=\bigoplus_{\lambda\in \C}M_{\lambda}$ where $M_{\lambda}=\{w\in M|L(0)w=\lambda w\},$ dim$M_{\lambda}<\infty.$ Moreover, for fixed $\lambda, M_{n+{\lambda}}=0 $ for all small enough integers $n.$

The notion of module here is essentially the notion of module given in \[FLM\].

\[d2\] An [*admissible $V$-module*]{} is a weak $V$-module $M$ which carries a $\Z_{+}$ grading $M=\bigoplus_{n\in \Z_{+}}M(n)$ satisfying the following condition:$$v_{m}M(n)\subset M(n+\wt v-m-1)$$ for homogeneous $v\in
V,$ where $\Z_+$ is the set of the nonnegative integers.

The notion of admissible module here is the notion of module in \[Z\]. Using a grading shift we can always arrange the grading on $M$ so that $M(0)\ne 0.$ This shift is important in the study of algebra $A_n(V)$ below.

It is not too hard to see that any $V$-module is an admissible $V$-module. So there is a natural identification of the category of $V$-modules with a subcategory of the category of admissible $V$-modules.

\[ration\] $V$ is called rational if every admissible $V$-module is a direct sum of irreducible admissible $V$-modules.

It is proved in 