$0, \infty$ and $-z$. The rationality, commutativity and associativity is the basis for Huang’s geometric interpretation of a vertex operator algebra \[H1\].

For a vertex operator algebra $V$, we have the following fundamental results established in \[FHL\]. It was proved that $V\otimes V$ has a natural vertex operator algebra structure and that for any $V$-modules $W_{1}$ and $W_{2}$, $W_{1}\otimes W_{2}$ is a natural $V\otimes V$-module; Let $W=\coprod_{h\in {\C}}W_{(h)}$ be a $V$-module and let $W'=\coprod_{h\in {\C}}W_{(h)}^{*}$, the restricted dual. For $v\in V$, define $Y'(v,x)$ to be an element of $({\rm End}\;W')[[x,x^{-1}]]$ by $$\begin{aligned}
\<Y'(v,x)w',w\>=\<w',Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})w\>\end{aligned}$$ for $w'\in W',\; w\in W$. Then $(W', Y')$ carries the structure of a $V$-module; For $v\in V$, set $$\begin{aligned}
\label{eyo}
Y^{o}(v,x)=Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1}).\end{aligned}$$ It was proved (\[HL1\], \[FHL\]) that $Y^{0}$ satisfies the opposite Jacobi identity (\[ehl\]), so that the pair $(W,Y^{o})$ carries the structure of a right $V$-module. These facts should be compared with those mentioned in the second paragraph for a group.

Now we consider what the space of a regular representation should be. The classical analogue suggests that we consider “functions on $V$ of the form $w'\otimes w$” for $w'\in W',\; w\in W$, where $W$ is an irreducible $V$-module. By analogy we should view $w'\otimes w$ as 