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\[section\] \[thm\][Proposition]{} \[thm\][Corollary]{} \[thm\][Conjecture]{} \[thm\][Example]{} \[thm\][Lemma]{} \[thm\][Remark]{} \[thm\][Definition]{} \[thm\][Hypothesis]{} addtoreset[equation]{}[section]{}

[**Regular representations of vertex operator algebras, I**]{}

[Haisheng Li[^1]\
Department of Mathematical Sciences\
Rutgers University-Camden\
Camden, NJ 08102]{}

Introduction
============

In a previous study \[Li3\], the physical super selection principle in vertex operator algebra theory seems to tell us that for a vertex operator algebra $V$, one should be able to obtain each irreducible $V$-module from the adjoint module $V$ by changing certain things. To a certain extent, what we are expecting is an analogue of the orbital construction of representations for a Lie group \[Ki\]. This was our original motivation for this study on regular representations of vertex operator algebras.

This is the first paper in a series for this study. In this paper, given a module $W$ for a vertex operator algebra $V$ and a nonzero complex number $z$ we construct a canonical (weak) $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ (a subspace of $W^{*}$ depending on $z$). We prove that for $V$-modules $W, W_{1}$ and $W_{2}$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ (\[H3\], \[HL0-3\]) exactly amounts to a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. Using Huang and Lepowsky’s one-to-one linear correspondence between the space of intertwining operators and the space of $P(z)$-intertwining maps of the same type we obtain a canonical linear isomorphism from the space ${\cal{V}}^{W'}_{W_{1}W_{2}}$ of intertwining operators of the indicated type to $\Hom _{V\otimes V}(W_{1}\otimes W_{2},{\cal{D}}_{P(z)}(W))$. In the case that $W=V$, we obtain a decomposition of Peter-Weyl type for ${\cal{D}}_{P(z)}(V)$, which are what we call the regular representations of $V$.

Let us start with the classical Peter-Weyl theory. Let $G$ be a group $G$ and $F(G,{\C})$ be the space of complex-valued functions on $G$. Then $F(G,{\C})$ is a $G\times G$-module with $$\begin{aligned}
\label{egbim}
((g_{1},g_{2})f)(g)=f(g_{1}^{-1}gg_{2})\end{aligned}$$ for $g_{1}, g_{2}, g\in G, f\in F(G,{\C})$. Furthermore, if $G$ is a topological (Lie, algebraic) group, all continuous (${\C}^{\infty}$, algebraic) functions form a $G\times G$-submodule. Now let $G$ be a compact Lie group and $C^{0}(G)$ be the space of continuous functions on $G$. A representative function on $G$ is a continuous function which generates a finite-dimensional $1\otimes G$-submodule of $C^{0}(G)$, and the regular representation of $G$ is the $G\times G$-module ${\cal{J}}(G,{\C})$ of representative functions on $G$ (cf. \[BD\]). Let $U$ be an irreducible $G$-module. For $u\in U,\; u^{*}\in U^{*}$, we can view $u^{*}\otimes u$ as a function on $G$ by $$(u^{*}\otimes u)(g)=\<u^{*},gu\>\;\;\;\mbox{ for }g\in G.$$ Then $U^{*}\otimes U$ is embedded into $C^{0}(G)$ as a $G\otimes G$-module. A theorem of Peter-Weyl type states that ${\cal{J}}(G,{\C})$ is a dense subspace of $C^{0}(G)$ and that ${\cal{J}}(G,{\C})$ is a direct sum of $U^{*}\otimes U$, where $U$ runs through a complete set of representatives of equivalence classes of finite-dimensional irreducible $G$-modules.

To start vertex operator algebra theory, let us recall from \[FLM\] and \[FHL\] (see also \[B\]) the definition of vertex operator algebra, which will be the official definition for this paper. [*A vertex operator algebra*]{} is a ${\Z}$-graded vector space $V=\coprod_{n\in {\Z}}V_{(n)}$ such that $\dim V_{(n)}<\infty$ for all $n\in {\Z}$ and $V_{(n)}=0$ for $n$ sufficiently small, equipped with a bilinear “vertex multiplication” $$\begin{aligned}
Y(\cdot,x)\cdot: & &V\otimes V\rightarrow V((x))\nonumber\\ 
& &u\otimes v\mapsto Y(u,x)v=\sum_{n\in {\Z}}u_{n}vx^{-n-1}\end{aligned}$$ such that for $m, n, k\in {\Z},\;u\in V_{(k)}$, $$\begin{aligned}
\label{egrad}
u_{n}V_{(m)}\subset V_{(m+k-n-1)},\end{aligned}$$ and such that the [*Jacobi identity*]{} holds for $u,v\in V$: $$\begin{aligned}
\label{e0jacobi}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y(u,x_{1})Y(v,x_{2})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y(v,x_{2})Y(u,x_{1})\nonumber\\
& &=x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y(Y(u,x_{0})v,x_{2}).\end{aligned}$$ It is equipped with a vector ${\bf 1}$, called the [*vacuum vector*]{}, such that for $v\in V$, $$\begin{aligned}
& &Y({\bf 1},x)v=v,\\
& &Y(v,x){\bf 1}\in V[[x]]\;\;\;\mbox{ and  }
\;\lim_{x\rightarrow 0}Y(v,x){\bf 1}=v.\end{aligned}$$ It is also equipped with a vector $\omega$, called the [*Virasoro element*]{}, such that $$\begin{aligned}
[L(m), L(n)]=(m-n)L(m+n)+\frac{m^{3}-m}{12}\delta_{m+n,0}(\mbox{rank}V)\end{aligned}$$ 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 $0, \infty$ 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 a ${\C}[x,x^{-1}]$-valued linear function on $V$ by considering $\<w',Y(v,x)w\>$ for $v\in V$. To get a ${\C}$-valued function on $V$, we use Huang and Lepowsky’s idea to evaluate the formal variable $x$ at a nonzero complex number $z$ as in \[H3\] and \[HL0-3\], so that $w'\otimes w$ can be viewed as a linear functional on $V$ by $$\begin{aligned}
(w'\otimes w)(v)=\<w',Y(v,z)w\>\;\;\;\mbox{ for }v\in V.\end{aligned}$$ Such functionals satisfy a certain nice rationality property as described below. For a linear functional $\alpha$ on $V$, we naturally extend $\alpha$ to a linear function on $V[[x,x^{-1}]]$. Then consider the formal series $\alpha(Y(u,x)v)$ with $u,v\in V$. If $\alpha=w'\otimes w$, we have $$\alpha(Y(u,x)v)=\<w',Y(Y(u,x)v,z)w\>,$$ which by associativity converges to a rational function in the domain $0<|x|<|z|$ with only two possible (finite) poles at $0$ and $-z$. Globally, from $Y$ we obtain a linear map $F$ from $W'\otimes W$ to $V^{*}$ defined by $$\begin{aligned}
\<F(w'\otimes w), v\>=\<w',Y(v,z)w\>\end{aligned}$$ for $w'\in W',\; w\in V,\; v\in V$. This map $F$ is nothing but a $Q(z)$-intertwining map as known in \[HL1\]. Linear functionals $\alpha$ with this rationality property are called $Q(z)$-linear functions and will be studied in \[Li6\].

In this paper we shall study the so-called $P(z)$-linear functionals defined below. Notice that previously we basically used the canonical intertwining operator $Y$ of type ${W\choose VW}$. It was known (\[FHL\], \[HL2\]) that there are canonical linear isomorphisms from the space of intertwining operators of type ${W\choose VW}$ to the space of intertwining operators of type ${V'\choose W'W}$. Let ${\cal{Y}}$ be an intertwining operator of type ${V'\choose W'W}$. (One can make use of the vacuum vector to get a canonical one.) Then ${\cal{Y}}(w',z)w$ is a linear functional on $V$. (To be rigorous we have to choose a branch of log function to evaluate $x$ at $z$ for ${\cal{Y}}(w',x)w$, see Section 4, or \[HL1\]. Here we neglect this issue for this introduction.) This gives another way to view $w'\otimes w$ as a linear functional on $V$. This time, for $u,v\in V$, the formal series $$\<Y(u,x){\cal{Y}}(w',z)w,v\>\;\;\;\;
\left(=\< {\cal{Y}}(w',z)w,Y^{o}(u,x)v\>\right)$$ by commutativity converges to a rational function in the domain $|x|>|z|$ with only two possible poles at $0$ and $z$, or a meromorphic function on $CP^{1}$ with only three possible poles at $0, z$ and $\infty$. Globally, evaluated intertwining operators ${\cal{Y}}(\cdot,z)\cdot$ at $x=z$ are exactly captured by the notion of $P(z)$-intertwining map (\[H3\], \[HL0-3\]). Then we consider linear functionals $\alpha$ on $V$ such that for $u,v\in V$, the formal series $\alpha(u,v,x):=\alpha(Y^{o}(u,x)v)$ converges to a rational function of $x$ with only two possible poles at $0$ and $z$, and we call such linear functionals $P(z)$-[*linear functionals*]{} on $V$ and denote by ${\cal{D}}_{P(z)}(V)$ the space of all $P(z)$-linear functionals. We define $Y^{R}(u,x)\alpha$ and $Y^{L}(u,x)\alpha$ by requiring $\< Y^{R}(u,x)\alpha,v\>$ to be the formal Laurent series expansion of the rational function $\alpha(u,v,x)$ in the domain $0<|x|<|z|$, and $\< Y^{L}(u,x)\alpha,v\>$ to be the Laurent series expansion of the rational function $\alpha(u,v,x+z)$ in the domain $0<|x|<|z|$. Our first main result is that $Y^{L}$ and $Y^{R}$ give rise to a structure of a weak $V\otimes V$-module on ${\cal{D}}_{P(z)}(V)$ in the sense that all the axioms defining the notion of a module except those involving the grading hold.

Vertex operator algebras are similar to Lie algebras. One of the similarities is that ideals are always two-sided due to the skew-symmetry. Consequently, a simple vertex operator algebra itself is an irreducible module. Most of the time, the adjoint module and other modules can be equally treated. Here, we construct a weak $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ for a general $V$-module $W$. Then, for any additional modules $W_{1}$ and $W_{2}$, we identify the space ${\cal{M}}[P(z)]^{W'}_{W_{1}W_{2}}$ of $P(z)$-intertwining maps of the indicated type with ${\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2}, {\cal{D}}_{P(z)}(W))$. If $V$ is regular, i.e., any weak $V$-module is a direct sum of irreducible (ordinary) modules, then we obtain a decomposition of ${\cal{D}}_{P(z)}(W)$ into irreducible $V\otimes V$-modules with the corresponding fusion rules as the multiplicities. In the case that $W=V$, we obtain a decomposition of ${\cal{D}}_{P(z)}(V)$ of Peter-Weyl type.

This paper is intimately related to \[H3, HL0-3\] and we use certain basic principles and techniques that have been explored herein. In the tensor product theory for a vertex operator algebra \[H3, HL0-3\], for [*two*]{} given $V$-modules $W_{1}$ and $W_{2}$, a $V$-module “$W_{1}\otimes_{P(z)} W_{2}$” was constructed such that for any $V$-module $W$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ canonically gives rise to a $V$-homomorphism from “$W_{1}\otimes _{P(z)}W_{2}$” to $W'$. In the present work, by constructing a $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ for [*one*]{} given $V$-module $W$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ becomes a homomorphism in the category of (weak) [*$V\otimes V$-modules*]{} instead of the category of [*$V$-modules*]{}.

In a recent work \[FM\], Frenkel and Malikov obtained some interesting results by using Kazhdan-Lusztig tensoring and Harish-Chandra categories. The bimodule ${\cal{D}}_{P(z)}(W)$ is analogous to the Harish-Chandra bimodule associated to a module for a Lie algebra. We hope to study ${\cal{D}}_{P(z)}(V)$ in this aspect in a future publication. During this research we have noticed that regular representations for affine Lie algebras have been studied in \[FP\] and that certain results of Peter-Weyl type on Kac-Moody groups have been obtained in \[KP\]. Presumably, \[FP\], \[KP\] (in the affine case) and the present paper are closely related. We hope to study the connections in a future publication.

In the next paper \[Li6\] we shall define and study ${\cal{D}}_{Q(z)}(W)$ for a given $V$-module $W$ and we shall study the connection between the regular representation ${\cal{D}}_{Q(z)}(V)$ and the modular invariance property of trace functions and correlation functions (\[Z\], \[DLM3\]).

This paper is organized as follows: In Section 2 we review the notion of the contragredient module and discuss certain related issues. In Section 3, we present $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$. In Section 4, we present the analogue of the Peter-Weyl theorem.

[**Acknowledgments.**]{} We thank James Lepowsky for many interesting discussions on certain subtle issues in formal calculus. In this research we greatly benefit from studying Huang and Lepowsky’s papers \[H3\], \[HL0-3\] and we adopt many of their important viewpoints into this paper. We would like also to thank IAS for its financial support and hospitality while most of this work was done during a membership in the Spring of 1997.

Contragredient modules and some related issues
==============================================

This section is preliminary. Here we review some basic notions and facts, including the notion of contragredient module and an extension of this. We shall discuss the notion of right module and prove certain results.

We shall use standard definitions and notions such as the notions of vertex operator algebra, (irreducible) module, homomorphism, for which we refer the reader to \[FHL\] and \[FLM\]. Since we here only deal with the representation theory of vertex operator algebras, we fix a vertex operator algebra $V$ throughout this paper. We shall typically use $x,y, x_{1},x_{2}, x_{3},\dots$ for mutually commuting formal variables, use letters $u,v$ for elements of $V$ and use $w, w_{(1)},w_{(2)},\dots$ for elements of $V$-modules. We use ${\Z}$ for the set of all integers, ${\N}$ for the set of all nonnegative integers and ${\C}$ for the field of complex numbers. We shall also use certain relatively new notions, which we recall next.

[*Throughout this paper, a $V$-module always stands for an ordinary $V$-module unless it is labeled as a weak or generalized $V$-module.*]{}

Vertex operator algebra $V$ is said to be [*regular*]{} \[DLM2\] if any weak $V$-module is a direct sum of irreducible (ordinary) $V$-modules. If $V$ is regular, it was proved in \[DLM2\] that $V$ has only finitely many inequivalent irreducible modules. Examples of regular vertex operator algebras were given in \[DLM2\].

It is well known (cf. \[B\], \[FFR\], \[Li1\], \[Li4\], \[MP\]) that there is a Lie algebra $g(V)$ associated to $V$. More precisely, $$\begin{aligned}
g(V)=\hat{V}/d\hat{V},\end{aligned}$$ where $$\hat{V}=V\otimes {\C}[t,t^{-1}]\;\;\mbox{and } 
d=L(-1)\otimes 1+1\otimes {d\over dt},$$ with the following Lie bracket: $$\begin{aligned}
\label{ecomm}
[u(m),v(n)]=\sum_{i\ge 0}{m\choose i}(u_{i}v)(m+n-i)\end{aligned}$$ for $u,v\in V,\; m,n\in {\Z}$, where $u(m)=u\otimes t^{m}+d\hat{V}$. Furthermore, $g(V)$ is a ${\Z}$-graded Lie algebra where $\deg u(m)=\wt u-m-1$ for homogeneous $u\in V$ and for $m\in {\Z}$. Let $W$ be a $g(V)$-module. A vector $w$ of $W$ is said to be [*restricted*]{} if for every $v\in V$, $v(m)w=0$ for $m$ sufficiently large. Denote by $W^{res}$ the space of all the restricted vectors of $W$. Since for any $u,v\in V$, $u_{i}v\ne 0$ for only finitely many nonnegative integer $i$, it follows from (\[ecomm\]) that $W^{res}$ is a submodule of $W$. A $g(V)$-module $W$ is said to be [*restricted*]{} if $W=W^{res}$. In general, it is clear that $W^{res}$ is the unique maximal restricted submodule of $W$. On the other hand, because of the truncation condition any weak $V$-module $W$ is a restricted $g(V)$-module where $v(n)$ is represented by $v_{n}$ for $v\in V, \;n\in {\Z}$.

We now recall the theory of [*contragredient module*]{} established in \[FHL\]. Let $W=\coprod_{h\in {\C}}W_{(h)}$ be a $V$-module and let $W'=\coprod_{h\in {\C}}W_{(h)}^{*}$, the [*restricted dual*]{} of $W$. For $v\in V,\; w'\in W'$, define $$\begin{aligned}
\label{ey'}
\<Y'(v,x)w',w\>
=\left\<w',Y\left(e^{xL(1)}\left(-x^{-2}\right)^{L(0)}v,x^{-1}\right)w
\right\>\end{aligned}$$ for $w\in W$. Then we have the following fundamental result due to Frenkel, Huang and Lepowsky (\[FHL\], Theorem 5.2.1 and Proposition 5.3.1).

The pair $(W',Y')$ carries the structure of a $V$-module and $(W'',Y'')=(W,Y)$.

Let $W$ be a weak $V$-module for now. For $v\in V$, set (cf. \[HL1\]) $$\begin{aligned}
\label{eyovx}
Y^{o}(v,x)=Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})\in (\End\; W)[[x,x^{-1}]].\end{aligned}$$ (Note that $e^{xL(1)}(-x^{-2})^{L(0)}v\in V[x,x^{-1}]$.) For instance, $$\begin{aligned}
& &Y^{o}({\bf 1},x)=1,\label{yo1}\\
& &Y^{o}(\omega,x)=x^{-4}Y(\omega,x^{-1})\label{eyomega}\end{aligned}$$ because $L(0){\bf 1}=L(1){\bf 1}=0$, $L(0)\omega=2\omega$ and $L(1)\omega=0$, where $\omega$ is the Virasoro element of $V$. Since $e^{xL(1)}(-x^{-2})^{L(0)}v\in V[x,x^{-1}]$ for $v\in V$ and $Y(u,x^{-1})w\in W((x^{-1}))$ for any $u\in V$, we have $$\begin{aligned}
\label{eyovxw}
Y^{o}(v,x)w\in W((x^{-1}))\;\;\;\mbox{ for }w\in W.\end{aligned}$$ It was observed in \[HL1\] that FHL’s proof in \[FHL\] for Proposition \[pfhl0\] in fact proves the following [*opposite Jacobi identity*]{}: $$\begin{aligned}
\label{ehl}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y^{o}(v,x_{2})Y^{o}(u,x_{1})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y^{o}(u,x_{1})Y^{o}(v,x_{2})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y^{o}(Y(u,x_{0})v,x_{2}).\end{aligned}$$ As it was mentioned in \[HL1\] one should think the pair $(W,Y^{o})$ as a right weak $V$-module. Since we shall discuss a little more further, let us make a formal definition here: A [*right* ]{} weak $V$-module is a vector space $W$ equipped with a linear map $Y_{W}$ from $V$ to $(\End W)[[x,x^{-1}]]$ such that for $u,v\in V$, $w\in W$, $$\begin{aligned}
& &Y_{W}(v,x)w\in W((x^{-1})),\\
& &Y_{W}({\bf 1},x)=1,\end{aligned}$$ and such that the opposite Jacobi identity (\[ehl\]) with $Y^{o}$ being replaced by $Y_{W}$ holds.

Let $U$ be a vector space together with a linear map $Y_{U}$ from the vertex operator algebra $V$ to $(\End\; U)[[x,x^{-1}]]$, e.g., $(U,Y_{U})$ is a left or right (weak) $V$-module. For $v\in V$, we define (cf. (\[eyovx\])) $$\begin{aligned}
Y_{U}^{o}(v,x)=Y_{U}(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})
\in (\End\; U)[[x,x^{-1}]].\end{aligned}$$ Then FHL’s proof in \[FHL\] for Proposition \[pfhl0\] again in fact proves the following result, which gives the equivalence between the notion of left $V$-module and the notion of right $V$-module:

Let $V$ be a vertex operator algebra and $U$ a vector space together with a linear map $Y_{U}$ from $V$ to $(\End\; U)[[x,x^{-1}]]$. Then $(U,Y_{U})$ is a left (weak) $V$-module if and only if $(U,Y^{o}_{U})$ is a right (weak) $V$-module.

Notice that $$\begin{aligned}
\Res_{x}xY^{o}(\omega,x)=\Res_{x}x^{-3}Y(\omega,x^{-1})=\Res_{x}xY(\omega,x).\end{aligned}$$ Then $\omega_{1}=L(0)$ is represented by the same operator on $U$ for $(U,Y_{U})$ and $(U,Y^{o}_{U})$. Therefore, the assertion in Proposition \[pcontragredient\] also includes the grading information.

Let $f(x)=\sum_{n\le N}a_{n}x^{n}\in U((x^{-1}))$ for a vector space $U$. Then, for any complex number $z_{0}$, $$\begin{aligned}
f(x+z_{0}):=\sum_{n\le N}a_{n}(x+z_{0})^{n}
=\sum_{n\le N}\sum_{j\ge 0}{n\choose j}a_{n}z_{0}^{j}x^{n-j}\end{aligned}$$ exists in $U((x^{-1}))$. Let $(W,Y_{W})$ be a right weak $V$-module and $z_{0}$ be a complex number. For $v\in V$, we define $Y_{W}^{(z_{0})}(v,x)\in (\End\;W)[[x,x^{-1}]]$ by $$\begin{aligned}
Y_{W}^{(z_{0})}(v,x)w=Y_{W}(v,x+z_{0})w=(Y_{W}(v,y)w)|_{y=x+z_{0}}
=e^{z_{0}{d\over dx}}(Y_{W}(v,x)w)\end{aligned}$$ for $w\in W$.

For any right weak $V$-module $(W,Y_{W})$, the pair $(W,Y_{W}^{(z_{0})})$ is also a right weak $V$-module.

Clearly, $$\begin{aligned}
Y_{W}^{(z_{0})}({\bf 1},x)=1, \;\; Y_{W}^{(z_{0})}(v,x)w\in W((x^{-1}))\end{aligned}$$ for $v\in V,\; w\in W$. Then it remains to prove the opposite Jacobi identity.

Let $u,v\in V$. Replacing $x_{1}$ and $x_{2}$ by $x_{1}+z_{0}$ and $x_{2}+z_{0}$ in the opposite Jacobi identity (\[ehl\]) with $Y^{o}$ being replaced by $Y_{W}$, respectively, we get $$\begin{aligned}
\label{ehly}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y_{W}(v,x_{2}+z_{0})Y_{W}(u,x_{1}+z_{0})\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1}+z_{0})Y_{W}(v,x_{2}+z_{0})\nonumber\\
&=&(x_{2}+z_{0})^{-1}\delta\left(\frac{x_{1}+z_{0}-x_{0}}{x_{2}+z_{0}}\right)
Y_{W}(Y(u,x_{0})v,x_{2}+z_{0}),\end{aligned}$$ noting that for $n\in {\Z}$, $$\begin{aligned}
(x_{1}+z_{0}-x_{2}-z_{0})^{n}=(x_{1}-x_{2})^{n}.\end{aligned}$$ Notice that $$\begin{aligned}
(x_{2}+z_{0})^{-1}\delta\left(\frac{x_{1}+z_{0}-x_{0}}{x_{2}+z_{0}}\right)
=e^{z_{0}({\partial\over \partial x_{1}}+{\partial\over \partial x_{2}})}
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
=x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)\end{aligned}$$ because $$\begin{aligned}
\left({\partial \over\partial x_{1}}+{\partial \over\partial x_{2}}\right)
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)=0\end{aligned}$$ (cf. \[Li2\]). Then we obtain $$\begin{aligned}
\label{ehlz}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y_{W}(v,x_{2}+z_{0})Y_{W}(u,x_{1}+z_{0})\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y_{W}(u,x_{1}+z_{0})Y_{W}(v,x_{2}+z_{0})\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y_{W}(Y(u,x_{0})v,x_{2}+z_{0}).\end{aligned}$$ (This is also a generalization of the opposite Jacobi identity for $Y_{W}$ with $z_{0}=0$.) This proves that $Y_{W}^{(z_{0})}$ satisfies the opposite Jacobi identity. $\;\;\;\;\Box$

Notice that for a $V$-module $W$, $W'$ is considerably smaller than $W^{*}$ if $W$ is infinite-dimensional — This turns out to be the case almost all of the times. Now we consider the whole space $W^{*}$. Clearly, the definition (\[ey’\]) still makes sense if one replaces $w'$ (an element of $W'$) by any element of $W^{*}$. Let $W$ be a weak $V$-module for now and let $v\in V,\;\alpha\in W^{*}$. We define $$\begin{aligned}
Y^{*}(v,x)\alpha\in W^{*}[[x,x^{-1}]]\end{aligned}$$ by $$\begin{aligned}
\label{e*1}
\<Y^{*}(v,x)\alpha, w\>=\<\alpha,Y^{o}(v,x)w\>\end{aligned}$$ for $w\in W$.

By Proposition 5.3.1 of \[FHL\] we have $$\begin{aligned}
\label{e*2}
\<\alpha,Y(v,x)w\>
=\left\<Y^{*}\left(e^{xL(1)}\left(-x^{-2}\right)^{L(0)}v,x^{-1}\right)\alpha,w
\right\>.\end{aligned}$$ Since $Y^{o}({\bf 1},x)=1$ (recall (\[yo1\])), we have $$\begin{aligned}
\label{ey*1}
Y^{*}({\bf 1},x)=1.\end{aligned}$$ For $v\in V$, we set $$\begin{aligned}
Y^{*}(v,x)=\sum_{n\in {\Z}}v^{*}_{n}x^{-n-1}.\end{aligned}$$ When $v=\omega$ (the Virasoro element), we set $$\begin{aligned}
Y^{*}(\omega,x)=\sum_{n\in {\Z}}L^{*}(n)x^{-n-2},\end{aligned}$$ i.e., $L^{*}(n)=\omega^{*}_{n+1}$ for $n\in {\Z}$. Then (by (\[eyomega\])) $$\begin{aligned}
\label{eL*n}
\< L^{*}(n)\alpha,w\>=\<\alpha, L(-n)w\>\end{aligned}$$ for $n\in {\Z},\; \alpha\in W^{*},\; w\in W$.

The opposite Jacobi identity (\[ehl\]) directly gives the following Jacobi identity in terms of matrix-coefficients: $$\begin{aligned}
\label{ejacobiy*m}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\<Y^{*}(u,x_{1})Y^{*}(v,x_{2})\alpha,w\>\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\<Y^{*}(v,x_{2})Y^{*}(u,x_{1})\alpha,w\>\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
\<Y^{*}(Y(u,x_{0})v,x_{2})\alpha,w\>\end{aligned}$$ for $u,v\in V,\; \alpha\in W^{*},\;w\in W$.

As an immediate consequence we have (cf. \[Li5\]):

For any weak $V$-module $W$, $W^{*}$ is a $g(V)$-module where $v(n)$ acts as $v_{n}^{*}$ for $v\in V, n\in {\Z}$. In particular, the following commutator formula (cf. (\[ecomm\])) holds: $$\begin{aligned}
[u_{m}^{*},v_{n}^{*}]=\sum_{i\ge 0}{m\choose i}(u_{i}v)_{m+n-i}^{*}\end{aligned}$$ for $u,v\in V,\; m,n\in {\Z}$. $\;\;\;\;\Box$

[*From now on we shall freely use the commutator formula on $W^{*}$ without explicit comments.*]{}

The following definition naturally arises from Remark \[rproof\] (cf. \[Li5\]):

As it has been noticed in \[Li5\], from (\[ehl\]) we immediately have:

The pair $(D(W),Y^{*})$ carries the structure of a weak $V$-module and $D(W)$ is the unique maximal weak $V$-module in $W^{*}$ with $Y^{*}$ being the vertex operator map.$\;\;\;\;\Box$

If $W=\coprod_{h\in {\C}}W_{(h)}$ is an ordinary $V$-module, then $W^{*}=\prod_{h\in {\C}}W_{(h)}^{*}$ is a formal completion of $W'$ and $Y^{*}$ is the natural extension of $Y'$ (cf. \[HL1\]). Let $W$ be a $V$-module and let $\alpha\in W^{*}$. Then from (\[eL\*n\]), $\alpha$ is an eigenvector of $L^{*}(0)$ with eigenvalue $h$ if and only if $\alpha\in W_{(h)}^{*}\subset W'$. Therefore, $W'$ is the largest generalized submodule of $D(W)$. Let $\cal{A}$ be the class of vertex operator algebras satisfying the condition that $L(0)$ acts semisimply on every weak module, i.e., every weak module is a generalized module. Then we immediately have (cf. \[Li5\]):

If $V$ is of class ${\cal{A}}$ and $W$ is a $V$-module, then $D(W)=W'$.$\;\;\;\;\Box$

Suppose that $V$ contains a regular vertex operator subalgebra (with the same Virasoro element) $V^{0}$. Since any weak $V$-module is a weak $V^{0}$-module, $L(0)$ (the same for both $V$ and $V^{0}$) acts semisimply on any weak $V$-module. Then $V$ is of class $\cal{A}$, so that Proposition \[pdsub\] applies to $V$.

The weak $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$
==================================================

In this section, given a weak $V$-module $W$ and a nonzero complex number $z$, we define a canonical subspace ${\cal{D}}_{P(z)}(W)$ of $W^{*}$, consisting of what we call $P(z)$-linear functionals on $W$. This space ${\cal{D}}_{P(z)}(W)$ contains $D(W)$ defined in Section 2 as a subspace. We prove that ${\cal{D}}_{P(z)}(W)$ has a natural weak module structure for the tensor product vertex operator algebra $V\otimes V$.

[*Throughout this section, $W$ is a weak $V$-module and $z$ is a nonzero complex number.* ]{} Now we introduce our first key notion.

All $P(z)$-linear functionals on $W$ clearly form a vector space, which we denote by ${\cal{D}}_{P(z)}(W)$.

Notice that a rational function in $x$ with only (two) possible poles at $0$ and $z$ can be considered as a meromorphic function with only three possible poles at $0, \infty$ and $z$ on the sphere $CP^{1}$. (Recall from Section 2 that $\alpha\in D(W)$ if and only if for any $v\in V,\; w\in W$, the formal series $\<\alpha,Y^{o}(v,x)w\>$ is a Laurent polynomial, i.e., a meromorphic function on the sphere $CP^{1}$ with only two possible poles at $0$ and $\infty$.)

In the following we give several equivalent definitions of $P(z)$-linear functional on $W$.

Let $\alpha\in W^{*}$. Then all the following conditions on $\alpha$ are equivalent:

\(a) $\alpha\in {\cal{D}}_{P(z)}(W)$.

\(b) For $v\in V$, there exist nonnegative integers $l$ and $k$ such that for every $w\in W$, $$\begin{aligned}
x^{l}(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>\end{aligned}$$ is a (finite) polynomial in $x$.

\(c) For $v\in V$, there exist nonnegative integers $l$ and $k$ such that $$\begin{aligned}
x^{l}(x-z)^{k}Y^{*}(v,x)\alpha\in W^{*}[[x]].\end{aligned}$$

\(d) For $v\in V$, there exists a nonnegative integer $k$ such that $$\begin{aligned}
\label{econ1}
(x-z)^{k}Y^{*}(v,x)\alpha\in W^{*}((x)).\end{aligned}$$

From the definition, $\alpha\in {\cal{D}}_{P(z)}(W)$ if and only if for $v\in V$, there exist $l, k\in {\N}$ such that for every $w\in W$, the formal series $$x^{l}(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>$$ absolutely converges in the domain $|x|>|z|$ to a rational function in $x$ without (finite) poles. Notice that a rational function without (finite) poles is simply a polynomial and any series that absolutely converges to a polynomial in the domain $|x|>|z|$ must be a polynomial itself. Then it follows that (a) and (b) are equivalent.

Conditions (c) and (d) are obviously equivalent. To finish the proof we shall show that (b) and (c) are equivalent. Clearly, (b) implies (c). Since $Y^{o}(v,x)w\in W((x^{-1}))$, from (c) we get $$\begin{aligned}
x^{l}(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>\in {\C}[[x]]\cap {\C}((x^{-1}))
={\C}[x].\end{aligned}$$ This proves that (c) implies (b).$\;\;\;\;\Box$

The following result gives the closeness of ${\cal{D}}_{P(z)}(W)$ under the actions of the component operators $v^{*}_{n}$ of each vertex operator $Y^{*}(v,x)$.

For $u,v\in V, \;\alpha\in {\cal{D}}_{P(z)}(W)$, there exists $k\in {\N}$ such that [*for all $n\in {\Z}$*]{}, $$\begin{aligned}
(x-z)^{k}Y^{*}(u,x)v^{*}_{n}\alpha\in W^{*}((x)).\end{aligned}$$ Furthermore, for $v\in V, \;n\in {\Z},\;\alpha\in {\cal{D}}_{P(z)}(W)$, $$\begin{aligned}
v^{*}_{n}\alpha\in {\cal{D}}_{P(z)}(W).\end{aligned}$$ That is, the space ${\cal{D}}_{P(z)}(W)$ is a $g(V)$-submodule of $W^{*}$.

Let $u,v\in V,\;n\in {\Z}$. From the commutator formula we have $$\begin{aligned}
\label{e3.7}
Y^{*}(u,x)v^{*}_{n}\alpha
=v^{*}_{n}Y^{*}(u,x)\alpha-\sum_{i\in {\N}}{n\choose i}x^{n-i}
Y^{*}(v_{i}u,x)\alpha.\end{aligned}$$ Since $v_{i}u=0$ for all but finitely many $i\in {\N}$, there exists $k\in {\N}$ (independent of $n$) such that $$(x-z)^{k}Y^{*}(u,x)\alpha,\;\;
(x-z)^{k}Y^{*}(v_{i}u,x)\alpha\in W^{*}((x))$$ for all $i\in {\N}$. Then using (\[e3.7\]) we have $$(x-z)^{k}Y^{*}(u,x)v^{*}_{n}\alpha\in W^{*}((x)).$$ It follows immediately from Lemma \[lequivconds\] that $v_{n}^{*}\alpha\in {\cal{D}}_{P(z)}(W)$. $\;\;\;\;\Box$

Let ${\C}(x)$ be the algebra of rational functions in $x$. Define linear maps $\iota_{x;\infty}$ and $\iota_{x;0}$ (cf. \[FLM\], \[FHL\]) from ${\C}(x)$ to ${\C}[x,x^{-1}]]$ and ${\C}((x))$, respectively, such that for $f(x)\in {\C}(x)$, $\iota_{x;\infty}f(x)$ and $\iota_{x;0}f(x)$ are the Laurent series expansions at $x=\infty$ and at $x=0$, respectively. In particular, $$\begin{aligned}
& &\iota_{x;\infty}(x-z_{0})^{r}=
\sum_{i\ge 0}{r\choose i}(-z_{0})^{i}x^{r-i};\\
& &\iota_{x;0}(x-z_{0})^{r}=
\sum_{i\ge 0}{r\choose i}(-z_{0})^{r-i}x^{i}\end{aligned}$$ for $r\in {\Z},\;z_{0}\in {\C}^{\times}$. It is clear that $\iota_{x;\infty}$ and $\iota_{x;0}$ are ${\C}[x,x^{-1}]$-linear and one-to-one, and they are algebra homomorphisms. From Definition \[dgood\], for $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W),\; w\in W$, the formal series $\<\alpha,Y^{o}(v,x)w\>$ lies in the range of $\iota_{x;\infty}$. Then $\iota_{x;\infty}^{-1}\<\alpha,Y^{o}(v,x)w\>$ is well defined. Furthermore, $\<\alpha,Y^{o}(v,x)w\>$ absolutely converges to $\iota_{x;\infty}^{-1}\<\alpha,Y^{o}(v,x)w\>$ in the domain $|x|>|z|$. Using the defined $\iota$-maps we define a new vertex operator map $Y^{R}$.

From the definition we have $$\begin{aligned}
\label{eiota-1three}
\iota_{x;0}^{-1}\<Y^{R}(v,x)\alpha,w\>=\iota_{x;\infty}^{-1}
\<\alpha,Y^{o}(v,x)w\>
=\iota_{x;\infty}^{-1}
\<Y^{*}(v,x)\alpha,w\>.\end{aligned}$$

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$ and let $l,k$ be as in Lemma \[lequivconds\] (b) or (c). Since $x^{l}\iota_{x;\infty}^{-1}\<\alpha,Y^{o}(v,x)w\>$ is analytic at $x=0$, $$\begin{aligned}
\<x^{l}Y^{R}(v,x)\alpha,w\>=\iota_{x;0}(x^{l}\iota_{x;\infty}^{-1}
\<\alpha,Y^{o}(v,x)w\>)\in {\C}[[x]]\end{aligned}$$ for every $w\in W$. Hence $$\begin{aligned}
x^{l}Y^{R}(v,x)\alpha\in W^{*}[[x]].\end{aligned}$$ This proves the following result:

For $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$, $$\begin{aligned}
Y^{R}(v,x)\alpha\in W^{*}((x)).\;\;\;\;\Box\end{aligned}$$

Let $k\in {\N}$ be as in Lemma \[lequivconds\] (b). (Note that we may take the same $k$ for (b), (c) and (d).) Noticing that $$\begin{aligned}
(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>\in {\C}[x,x^{-1}]\end{aligned}$$ and that $\iota_{x;\infty}$ and $\iota_{x;0}$ are ${\C}[x,x^{-1}]$-linear, from (\[e3.13def\]) we get $$\begin{aligned}
(x-z)^{k}\<Y^{R}(v,x)\alpha,w\>
=\iota_{x;0}\iota_{x;\infty}^{-1}(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>
=(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>\end{aligned}$$ for all $w\in W$. Therefore we have proved the following result:

For $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$, $$\begin{aligned}
\label{e3.11}
(x-z)^{k}Y^{R}(v,x)\alpha=(x-z)^{k}Y^{*}(v,x)\alpha\end{aligned}$$ whenever $(x-z)^{k}Y^{*}(v,x)\alpha\in W^{*}((x))$ for $k\in {\N}$. Furthermore $$\begin{aligned}
\label{e3.12}
(x-z)^{k}\<Y^{R}(v,x)\alpha,w\>
=(x-z)^{k}\<\alpha,Y^{o}(v,x)w\>\end{aligned}$$ for all $w\in W$. $\;\;\;\;\Box$

Set $$\begin{aligned}
Y^{R}(v,x)=\sum_{n\in {\Z}}v^{R}_{n}x^{-n-1}.\end{aligned}$$ For $v=\omega$ (the Virasoro element), following the tradition we set $$\begin{aligned}
Y^{R}(\omega,x)=\sum_{n\in {\Z}}L^{R}(n)x^{-n-2}.\end{aligned}$$

For now, $v^{R}_{n}$ are only linear maps from ${\cal{D}}_{P(z)}(W)$ to $W^{*}$. Next, we shall show that $v^{R}_{n}$ are in fact linear endomorphisms of ${\cal{D}}_{P(z)}(W)$. We shall need the following result:

Let $U$ be a vector space and let $$\begin{aligned}
f(x)=\sum_{n\in {\Z}}f_{n}x^{-n-1},\;\;g(x)=\sum_{n\in {\Z}}g_{n}x^{-n-1}
\in U[[x,x^{-1}]].\end{aligned}$$ Suppose that either $f(x)\in U((x))$ or $f(x)\in U((x^{-1}))$ and that there exist $k\in {\N}$ and $z\in {\C}^{\times}$ such that $$\begin{aligned}
\label{ebasic52}
(x-z)^{k}f(x)=(x-z)^{k}g(x).\end{aligned}$$ Then for $n\in {\Z}$, $$\begin{aligned}
f_{n}\in \mbox{\rm linear span }\{g_{m}\;|\;m\ge n\}\end{aligned}$$ if $f(x)\in U((x))$ and $$\begin{aligned}
f_{n}\in \mbox{\rm linear span }\{g_{m}\;|\;m\le n+k\}\end{aligned}$$ if $f(x)\in U((x^{-1}))$.

Assume $f(x)\in U((x))$. For $n\in {\Z}$, let $s$ be a positive integer such that $x^{n+s}f(x)\in U[[x]]$. Then $\Res_{x}x^{n+j}f(x)=0$ for $j\ge s$ and furthermore, $$\begin{aligned}
\Res_{x}x^{n+i}(x-z)^{k}f(x)=0\end{aligned}$$ for $i\ge s$. Using this and (\[ebasic52\]) we get $$\begin{aligned}
\label{e3.30f}
f_{n}&=&\Res_{x}x^{n}f(x)\nonumber\\
&=&\Res_{x}x^{n}(-z+x)^{-k}(x-z)^{k}f(x)\nonumber\\
&=&\Res_{x}\sum_{i\ge 0}{-k\choose i}(-z)^{-k-i}x^{n+i}(x-z)^{k}f(x)\nonumber\\
&=&\Res_{x}\sum_{i= 0}^{s-1}{-k\choose i}(-z)^{-k-i}x^{n+i}(x-z)^{k}f(x)
\nonumber\\
&=&\Res_{x}\sum_{i= 0}^{s-1}{-k\choose i}(-z)^{-k-i}x^{n+i}(x-z)^{k}g(x)
\nonumber\\
&\in& \mbox{linear span }\{g_{n}, g_{n+1},\cdots, g_{n+k+s-1}\}.\end{aligned}$$

If $f(x)\in U((x^{-1}))$, let $s'$ be a positive integer such that $x^{n-s'}f(x)\in x^{-2}U[[x^{-1}]]$, so that $$\begin{aligned}
\Res_{x}x^{n-k-i}(x-z)^{k}f(x)=0\end{aligned}$$ for $i\ge s'$. Then use $(x-z)^{-k}$ instead of $(-z+x)^{-k}$ in (\[e3.30f\]) to complete the proof. $\;\;\;\;\;\Box$

In view of Lemmas \[ltruncationyr\] and \[lbasic5\] we immediately have:

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$ and let $k\in {\N}$ be such that (\[e3.11\]) holds. Then for $n\in {\Z}$, $$\begin{aligned}
\label{relate}
v^{R}_{n}\alpha
\in \mbox{\rm linear span }\{v^{*}_{m}\alpha\;|\; m\ge n\}.\end{aligned}$$

As an immediate consequence of Corollary \[lbasic2\], Proposition \[pbasic1\] and Lemma \[ltruncationyr\] we have:

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$. Then $$\begin{aligned}
Y^{R}(v,x)\alpha\in \left({\cal{D}}_{P(z)}(W)\right)((x)).\end{aligned}$$

$$\begin{aligned}
\label{edef2}
Y^{R}(v,x)\alpha=(z-x)^{-k}\left[(z-x)^{k}Y^{*}(v,x)\alpha\right]
\in \left({\cal{D}}_{P(z)}(W)\right)((x)).\end{aligned}$$

By Remark \[rcontain\], $D(W)$ is contained in the intersection of all ${\cal{D}}_{P(z)}(W)$ for $z\in {\C}^{\times}$. Conversely, let $$\alpha\in {\cal{D}}_{P(z_{1})}(W)\cap {\cal{D}}_{P(z_{2})}(W),$$ where $z_{1},z_{2}$ are two distinct nonzero complex numbers. Then, for $v\in V,\; w\in W$, the formal series $\<\alpha,Y^{o}(v,x)w\>$ converges in the domain $|x|>{\rm max}\{|z_{1}|,|z_{2}|\}$ to a rational function $h(x)$ such that $$h(x)\in {\C}[x,x^{-1},(x-z_{1})^{-1}]\cap {\C}[x,x^{-1},(x-z_{2})^{-1}].$$ Consequently, $h(x)\in {\C}[x,x^{-1}]$. By Remark \[rdw\], $\alpha\in D(W)$. Thus we have proved the following simple fact:

Let $W$ be a weak $V$-module. Then $$\begin{aligned}
\cap_{z\in {\C}^{\times}}{\cal{D}}_{P(z)}(W)=D(W).\;\;\;\;\Box\end{aligned}$$

As our first key result we have:

The pair $({\cal{D}}_{P(z)}(W),Y^{R})$ carries the structure of a weak $V$-module.

The truncation condition: $Y^{R}(v,x)\alpha\in {\cal{D}}_{P(z)}(W)((x))$ for $v\in V,\;\alpha\in {\cal{D}}_{P(z)}(W)$ holds (Proposition \[prd\]) and $Y^{R}({\bf 1},x)=Y^{*}({\bf 1},x)=1$. Then it remains to prove the Jacobi identity.

Let $u,v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$. By Proposition \[pbasic1\], there exists $k_{0}\in {\N}$ such that for [*all*]{} $m\in {\Z}$, $$\begin{aligned}
\label{ey*v*m}
(x_{1}-z)^{k_{0}}Y^{*}(u,x_{1})v^{*}_{m}\alpha\in W^{*}((x_{1})),\end{aligned}$$ hence (recall Lemma \[cbasic\]) $$\begin{aligned}
\label{e3.27}
(x_{1}-z)^{k_{0}}Y^{R}(u,x_{1})Y^{*}(v,x_{2})\alpha
=(x_{1}-z)^{k_{0}}Y^{*}(u,x_{1})Y^{*}(v,x_{2})\alpha.\end{aligned}$$ Let $k'\ge k_{0}$ be such that $$(x_{2}-z)^{k'}Y^{R}(v,x_{2})\alpha=
(x_{2}-z)^{k'}Y^{*}(v,x_{2})\alpha\in W^{*}((x_{2})).$$ Then by (\[e3.27\]), $$\begin{aligned}
\label{eyruv}
& &(x_{2}-z)^{k'}(x_{1}-z)^{k'}
Y^{R}(u,x_{1})Y^{R}(v,x_{2})\alpha\nonumber\\
&=&(x_{2}-z)^{k'}(x_{1}-z)^{k'}Y^{*}(u,x_{1})Y^{*}(v,x_{2})\alpha.\end{aligned}$$ Similarly, there exists $k''\in {\N}$ such that $$\begin{aligned}
\label{eyrvu}
& &(x_{1}-z)^{k''}(x_{2}-z)^{k''}Y^{R}(v,x_{2})Y^{R}(u,x_{1})\alpha
\nonumber\\
&=&(x_{1}-z)^{k''}(x_{2}-z)^{k''}Y^{*}(v,x_{2})Y^{*}(u,x_{1})\alpha.\end{aligned}$$ Set $k=k'+k''$. For any $w\in W$, using (\[eyruv\]), (\[eyrvu\]) and (\[ejacobiy\*m\]) we obtain $$\begin{aligned}
\label{e3.30}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
(x_{1}-z)^{k}(x_{2}-z)^{k}\<Y^{R}(u,x_{1})Y^{R}(v,x_{2})\alpha,w\>\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
(x_{1}-z)^{k}(x_{2}-z)^{k}\<Y^{R}(v,x_{2})Y^{R}(u,x_{1})\alpha,w\>\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
(x_{1}-z)^{k}(x_{2}-z)^{k}\<Y^{*}(u,x_{1})Y^{*}(v,x_{2})\alpha,w\>
\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
(x_{1}-z)^{k}(x_{2}-z)^{k}\<Y^{*}(v,x_{2})Y^{*}(u,x_{1})\alpha,w\>
\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
(x_{1}-z)^{k}(x_{2}-z)^{k}\<Y^{*}(Y(u,x_{0})v,x_{2})\alpha,w\>.\end{aligned}$$ Let $m$ be [*any fixed*]{} integer. Since $u_{n}v\ne 0$ for only finitely many $n\ge m$, there is a nonnegative integer $k_{1}$ ([*depending*]{} on $m$) such that for [*all*]{} $n\ge m$, $$\begin{aligned}
\label{eunv}
(x_{2}-z)^{k_{1}}Y^{*}(u_{n}v,x_{2})\alpha\in {\cal{D}}_{P(z)}(W)((x_{2})),\end{aligned}$$ hence (recall Lemma \[cbasic\]) $$\begin{aligned}
(x_{2}-z)^{k_{1}}Y^{R}(u_{n}v,x_{2})\alpha=
(x_{2}-z)^{k_{1}}Y^{*}(u_{n}v,x_{2})\alpha.\end{aligned}$$ Then $$\begin{aligned}
\label{econn1}
& &{\rm Res}_{x_{0}}x_{0}^{m}
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
(x_{2}-z)^{k_{1}}\<Y^{R}(Y(u,x_{0})v,x_{2})\alpha,w\>\nonumber\\
&=&{\rm Res}_{x_{0}}x_{0}^{m}
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
(x_{2}-z)^{k_{1}}\<Y^{*}(Y(u,x_{0})v,x_{2})\alpha,w\>.\end{aligned}$$ Choosing $k_{1}\ge k$ and then combining (\[e3.30\]) with (\[econn1\]) we obtain $$\begin{aligned}
\label{ejac}
& &{\rm Res}_{x_{0}}x_{0}^{m}
x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
(x_{1}-z)^{k_{1}}(x_{2}-z)^{k_{1}}\<Y^{R}(u,x_{1})
Y^{R}(v,x_{2})\alpha,w\>\nonumber\\
& &-{\rm Res}_{x_{0}}x_{0}^{m}
x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
(x_{1}-z)^{k_{1}}(x_{2}-z)^{k_{1}}\<Y^{R}(v,x_{2})Y^{R}(u,x_{1})\alpha,w\>
\nonumber\\
&=&{\rm Res}_{x_{0}}x_{0}^{m}
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
(x_{1}-z)^{k_{1}}(x_{2}-z)^{k_{1}}\<Y^{R}(Y(u,x_{0})v,x_{2})\alpha,w\>.\end{aligned}$$ In view of Remark \[rbim\], we may multiply both sides of (\[ejac\]) by $(-z+x_{1})^{-k_{1}}(-z+x_{2})^{-k_{1}}$ to get $$\begin{aligned}
\label{ejac2}
& &{\rm Res}_{x_{0}}x_{0}^{m}
x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\<Y^{R}(u,x_{1})Y^{R}(v,x_{2})\alpha,w\>
\nonumber\\
& &-{\rm Res}_{x_{0}}x_{0}^{m}
x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\<Y^{R}(v,x_{2})Y^{R}(u,x_{1})\alpha,w\>\nonumber\\
&=&{\rm Res}_{x_{0}}x_{0}^{m}
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
\<Y^{R}(Y(u,x_{0})v,x_{2})\alpha,w\>.\end{aligned}$$ Since there is no term depending on $m$ in (\[ejac2\]) except for $x_{0}^{m}$ and $m$ is an arbitrarily fixed integer, dropping $\Res_{x_{0}}x_{0}^{m}$ from (\[ejac2\]) we obtain the desired Jacobi identity for $Y^{R}$. This concludes the proof. $\;\;\;\;\Box$

Since $Y^{R}$ depends on $z$, it is necessary, especially in certain situations, for us to use the notation $Y^{R}_{P(z)}$ for $Y^{R}$, and we shall do so from now on.

Recall that given a left weak $V$-module $(W,Y_{W})$ and a nonzero complex number $z$, we have a right weak $V$-module $(W,(Y^{o}_{W})^{(z)})$ (Propositions \[pcontragredient\] and \[ptranslationz\]). Using this and the linear maps $\iota_{x;\infty}$ and $\iota_{x;0}$ defined earlier we now define another vertex operator map $Y_{P(z)}^{L}$; a linear map from $V$ to $(\End\; {\cal{D}}_{P(z)}(W))[[x,x^{-1}]]$.

Since $\<\alpha,Y^{o}(v,y)w\>$ absolutely converges in the domain $|y|>|z|$ to a rational function in ${\C}[y,y^{-1},(y-z)^{-1}]$, the formal series $\<\alpha,Y^{o}(v,x+z)w\>$ absolutely converges in the domain $|x|>|z|$, $|x+z|>|z|$ to a rational function in ${\C}[x,x^{-1},(x+z)^{-1}]$, where the possible poles at $y=0, z$ are transformed to $x=-z,0$. In particular, $\<\alpha,Y^{o}(v,x+z)w\>$ absolutely converges in the domain $|x|>2|z|$ (to the same rational function). Thus, in view of Remark \[rcondomain\] and Definitions \[dleftaction\] and \[dr\] we immediately have:

Let $W$ be a left weak $V$-module and $z$ a nonzero complex number. Then $$\begin{aligned}
({\cal{D}}_{P(z)}(W),Y^{L}_{P(z)})
=({\cal{D}}_{P(-z)}(W,(Y^{o}_{W})^{(z)}), Y_{P(-z)}^{R}).\;\;\;\;\Box\end{aligned}$$

In view of Remark \[rcomments\], combining Lemma \[lkey\] with Theorem \[tyR\] we immediately have:

Let $W$ be a left weak $V$-module and $z$ a nonzero complex number. Then the pair $({\cal{D}}_{P(z)}(W),Y_{P(z)}^{L})$ carries the structure of a weak $V$-module. $\;\;\;\;\Box$

Following the tradition, for the Virasoro element $\omega$ we set $$\begin{aligned}
\label{eLln}
Y_{P(z)}^{L}(\omega,x)=\sum_{n\in {\Z}}L_{P(z)}^{L}(n)x^{-n-2}.\end{aligned}$$

Recall the following delta-function properties (\[FLM\], \[FHL\], \[HL1\]): $$\begin{aligned}
x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
=x^{-1}\delta\left(\frac{x_{0}+z}{x}\right),
\;\;\;\;\;
x^{-1}\delta\left(\frac{z+x_{0}}{x}\right)
=z^{-1}\delta\left(\frac{x-x_{0}}{z}\right)\end{aligned}$$ and $$\begin{aligned}
\label{esub}
& &x_{0}^{-1}\delta\left(\frac{z-x}{x_{0}}\right)
f(x_{0},x)=x_{0}^{-1}\delta\left(\frac{z-x}{x_{0}}\right)f(z-x,x)\end{aligned}$$ for $f(x_{0},x)\in \Hom (U_{1},U_{2}[x_{0},x_{0}^{-1},x,x^{-1}])$, where $U_{1}$ and $U_{2}$ are any vector spaces. (This can be made more general (cf. \[FLM\]), but this is enough for our purpose.)

Let $f(x)\in {\C}[x,x^{-1},(x-z)^{-1}]$. Then we have (cf. \[FHL\], \[HL1\]) $$\begin{aligned}
\label{eabstractjacobi}
& &x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
\iota_{x;\infty}f(x)-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
\iota_{x;0}f(x)\nonumber\\
&=&z^{-1}\delta\left(\frac{x-x_{0}}{z}\right)(\iota_{x_{0};0}f(x_{0}+z)).\end{aligned}$$ In particular, $$\begin{aligned}
\label{e3delta}
x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
=z^{-1}\delta\left(\frac{x-x_{0}}{z}\right).\end{aligned}$$

For $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W),\;w\in W$, from the definition we have $$\begin{aligned}
\iota_{x;0}^{-1}\<Y^{L}_{P(z)}(v,x)\alpha,w\>
=\left(\iota_{y;\infty}^{-1}\<\alpha,Y^{o}(v,y)w\>\right)|_{y=x+z}.\end{aligned}$$ Recall that $$\begin{aligned}
\iota_{x;0}^{-1}\<Y^{R}_{P(z)}(v,x)\alpha,w\>
=\iota_{x;\infty}^{-1}\<\alpha,Y^{o}(v,x)w\>
=\iota_{x;\infty}^{-1}\<Y^{*}(v,x)\alpha,w\>.\end{aligned}$$ In view of this and (\[eabstractjacobi\]), we immediately have:

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$. Then $$\begin{aligned}
\label{sjacnew}
& &x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
Y^{*}(v,x)\alpha-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
Y_{P(z)}^{R}(v,x)\alpha\nonumber\\
&=&z^{-1}\delta\left(\frac{x-x_{0}}{z}\right)Y_{P(z)}^{L}(v,x_{0})\alpha.\end{aligned}$$

The following lemma (cf. Lemma \[cbasic\]) immediately follows from Proposition \[prelationnew\].

Let $v\in V,\; \alpha\in {\cal{D}}_{P(z)}(W)$. Then $$\begin{aligned}
(z+x_{0})^{l}Y_{P(z)}^{L}(v,x_{0})\alpha&=&
\Res_{x}x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)x^{l}Y^{*}(v,x)\alpha\\
&=&(z+x_{0})^{l}Y^{*}(v,x_{0}+z)\alpha\end{aligned}$$ if $x^{l}Y_{P(z)}^{R}(v,x)\alpha\in W^{*}[[x]]$ for $l\in {\N}$.$\;\;\;\;\Box$

For $u,v\in V$, $$\begin{aligned}
\label{elrrl}
Y_{P(z)}^{L}(u,x_{1})Y_{P(z)}^{R}(v,x_{2})
=Y_{P(z)}^{R}(v,x_{2})Y_{P(z)}^{L}(u,x_{1}),\end{aligned}$$ acting on ${\cal{D}}_{P(z)}(W)$.

Let $\alpha\in {\cal{D}}_{P(z)}(W)$ and let $k\in {\N}$ be such that $$\begin{aligned}
(y-z)^{k}Y_{P(z)}^{R}(v,y)\alpha=(y-z)^{k}Y^{*}(v,y)\alpha.\end{aligned}$$ By Proposition \[pbasic1\] and Lemma \[cbasic\], we may choose $k$ so large that $$(y-z)^{k}Y_{P(z)}^{R}(v,y)Y^{*}(u,x)\alpha=(y-z)^{k}Y^{*}(v,y)Y^{*}(u,x)\alpha$$ also holds. Furthermore, since both $Y_{P(z)}^{R}$ and $Y^{*}$ satisfy the weak commutativity, which follows from the commutator formula, we may choose $k$ such that the following also hold: $$\begin{aligned}
& &(x-y)^{k}Y^{*}(u,x)Y^{*}(v,y)=(x-y)^{k}Y^{*}(v,y)Y^{*}(u,x),\\
& &(x-y)^{k}Y_{P(z)}^{R}(u,x)Y_{P(z)}^{R}(v,y)
=(x-y)^{k}Y_{P(z)}^{R}(v,y)Y_{P(z)}^{R}(u,x).\end{aligned}$$ Then using (\[sjacnew\]) and all the above identities we obtain $$\begin{aligned}
\label{elrcom1}
& &z^{-1}\delta\left(\frac{x-x_{0}}{z}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{L}(u,x_{0})Y_{P(z)}^{R}(v,y)\alpha\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y^{*}(u,x)Y_{P(z)}^{R}(v,y)\alpha\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{R}(u,x)Y_{P(z)}^{R}(v,y)\alpha
\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y^{*}(u,x)Y^{*}(v,y)\alpha
\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{R}(u,x)Y_{P(z)}^{R}(v,y)\alpha\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y^{*}(v,y)Y^{*}(u,x)\alpha
\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{R}(v,y)Y_{P(z)}^{R}(u,x)\alpha
\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{x-z}{x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{R}(v,y)Y^{*}(u,x)\alpha
\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{z-x}{-x_{0}}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{R}(v,y)Y_{P(z)}^{R}(u,x)\alpha
\nonumber\\
&=&z^{-1}\delta\left(\frac{x-x_{0}}{z}\right)
(y-z)^{k}(x-y)^{k}Y_{P(z)}^{R}(v,y)Y_{P(z)}^{L}(u,x_{0})\alpha.\end{aligned}$$ Taking $\Res_{x}$ of (\[elrcom1\]) and then using (\[esub\]) we get $$\begin{aligned}
\label{elrcom2}
& &(y-z)^{k}(z+x_{0}-y)^{k}Y_{P(z)}^{L}(u,x_{0})Y_{P(z)}^{R}(v,y)\alpha\nonumber\\
&=&(y-z)^{k}(z+x_{0}-y)^{k}Y_{P(z)}^{R}(v,y)Y_{P(z)}^{L}(u,x_{0})\alpha.\end{aligned}$$ In view of Remark \[rbim\] we may multiply both sides of (\[elrcom2\]) by $(-z+y)^{-k}(z+x_{0}-y)^{-k}$, to get $$Y_{P(z)}^{L}(u,x_{0})Y_{P(z)}^{R}(v,y)\alpha
=Y_{P(z)}^{R}(v,y)Y_{P(z)}^{L}(u,x_{0})\alpha.$$ This proves (\[elrrl\]), $\;\;\;\;\Box$

Next we shall combine the two weak $V$-module structures $Y_{P(z)}^{L}$ and $Y_{P(z)}^{R}$ on ${\cal{D}}_{P(z)}(W)$ into a weak $V\otimes V$-module structure. Recall (\[FHL\], Proposition 3.7.1) that $V\otimes V$ has a natural vertex operator algebra structure, where the vertex operator map $Y$ is defined by $$\begin{aligned}
Y(u\otimes v,x)=Y(u,x)\otimes Y(v,x)\;\;\;\mbox{for }u,v\in V. \end{aligned}$$ It was proved (\[FHL\], Propositions 4.6.1) that for any $V$-modules $W_{1}$ and $W_{2}$, $W_{1}\otimes W_{2}$ has a natural $V\otimes V$-module structure. Let $Y_{P(z)}(\cdot,x)$ be the (unique) linear map from $V\otimes V$ to $(\End \;{\cal{D}}_{P(z)}(W))[[x,x^{-1}]]$ such that $$\begin{aligned}
Y_{P(z)}(u\otimes v,x)=Y_{P(z)}^{L}(u,x)Y_{P(z)}^{R}(v,x)
\;\;\;\mbox{ for }u,v\in V.\end{aligned}$$ Now we present our second main theorem of the paper.

The pair $({\cal{D}}_{P(z)}(W),Y_{P(z)})$ carries the structure of a weak $V\otimes V$-module.

The proof of Proposition 4.6.1 of \[FHL\] in fact proves the following result: If we have two commuting left weak $V$-module structures $Y_{1}$ and $Y_{2}$ on a vector space $M$ in the sense that $$\begin{aligned}
Y_{1}(u,x_{1})Y_{2}(v,x_{2})=Y_{2}(v,x_{2})Y_{1}(u,x_{1})\end{aligned}$$ for $u,v\in V$, then $Y_{1}\otimes Y_{2}$ gives rise to a left weak $V\otimes V$-module structure on $M$. Then it immediately follows from Theorem \[tyR\] and Propositions \[plrc\] and \[pyLnew\].$\;\;\;\;\;\Box$

Decomposition of ${\cal{D}}_{P(z)}(W)$ into irreducible $V\otimes V$-modules
============================================================================

In this section, for $V$-modules $W_{1},W_{2}$ and $W$, we shall identify a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ in the sense of \[H3\] and \[HL0-3\] with a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. By using Huang and Lepowsky’s one-to-one linear correspondences \[HL3\] between the space of intertwining operators and the space of $P(z)$-intertwining maps of the same type we obtain canonical linear isomorphisms from the space of intertwining operators of type ${W'\choose W_{1}W_{2}}$ to the space of $V\otimes V$-homomorphisms from $W_{1}\otimes W_{2}$ to ${\cal{D}}_{P(z)}(W)$. When $V$ is regular, we obtain a decomposition of ${\cal{D}}_{P(z)}(W)$ into irreducible $V\otimes V$-modules. In the case that $W=V$, we obtain an analogue of Peter-Weyl theorem.

A [*generalized*]{} $V$-module \[HL1\] is a weak $V$ module on which $L(0)$ acts semisimply. That is, a generalized $V$-module satisfies all the axioms in the notion of a module except the two grading restrictions on homogeneous spaces. A [*lower truncated generalized*]{} $V$-module \[H2\] is a generalized $V$-module that also satisfies the lower truncation condition, one of the two grading restrictions on homogeneous spaces. That is, the only difference between a module and a generalized module is that the homogeneous subspaces of a generalized module could be infinite-dimensional.

Following \[FHL\] and \[HL1\], for a vector space $U$, we set $$\begin{aligned}
U\{x\}=\left\{A(x)=\sum_{h\in {\C}}a_{h}x^{h}\;|\; a_{h}\in U
\;\;\;\mbox{ for }h\in {\C}\right\}.\end{aligned}$$

[*Throughout this section, $W, W_{1}, W_{2}$ and $ W_{3}$ are generalized $V$-modules.*]{} Now we recall the definition of an intertwining operator from \[FHL\] and \[HL1\]:

The following result can be found in \[HL2\] (cf. \[FHL\]):

Let $W_{1},W_{2}$ and $W_{3}$ be (ordinary) $V$-modules. We have the following (canonical) linear isomorphism relations: $$\begin{aligned}
{\cal{V}}^{W_{3}}_{W_{1}W_{2}}\cong {\cal{V}}^{W_{3}}_{W_{2}W_{1}},\;\;\;\;\;
{\cal{V}}^{W_{3}}_{W_{1}W_{2}}\cong {\cal{V}}^{W_{2}'}_{W_{1}W_{3}'}.\end{aligned}$$

Notice that in Sections 2 and 3, only integral powers of $z$ were involved. In this section, intertwining operators will play an important role, so that in general complex powers of $z$ will be involved. Let ${\cal{Y}}(\cdot,x)$ be an intertwining operator of type ${W'\choose W_{1}W_{2}}$. Since for $w_{(1)}\in W_{1},\; w_{(2)}\in W_{2}$, ${\cal{Y}}(w_{(1)},x)w_{(2)}$ involves complex powers of $x$, to consider the evaluation “${\cal{Y}}(w_{(1)},z)w_{(2)}$” we need to choose a branch of the log function as in \[HL1\]. Following \[HL1\] we choose log $z$ so that $$\begin{aligned}
\log z =\log |z|+i \arg z\;\;\;\mbox{ with }\;\;0\le \arg z< 2\pi,\end{aligned}$$ and arbitrary values of the log function will be denoted by $$\begin{aligned}
l_{p}(z)=\log z+2p\pi i\end{aligned}$$ for $p\in {\Z}$.

For a $V$-module $W=\coprod_{h\in {\C}}W_{(h)}$, following \[HL1\] we define the formal completion $$\begin{aligned}
\overline{W}=\prod_{h\in {\C}}W_{(h)}.\end{aligned}$$ Each $f\in \overline{W}$ can be viewed as a formal sum $\sum_{h\in {\C}}f_{h}$, which is a well defined element of $(W')^{*}$. In view of this we have $$\begin{aligned}
\overline{W}=(W')^{*},\;\;\;\overline{W'}=W^{*}.\end{aligned}$$ Furthermore, as noticed in \[HL1\] the action of a vertex operator $Y(v,x)$ on $W$ can be naturally extended to $\overline{W}$. Then $Y^{*}(v,x)$ acting on $W^{*}$ $(=\overline{W'})$ is the natural extension of $Y'(v,x)$ on $W'$. Because of this, we shall also use $Y$ for $Y^{*}$ (as it was done in \[HL1\]).

Let $U_{1}$ and $U_{2}$ be vector spaces and let $$A(x)=\sum_{h\in {\C}}a_{h}x^{h}\in (\Hom (U_{1},U_{2}))\{x\}$$ be such that for every $u_{1}\in U_{1}$, $a_{h}u_{1}=0$ for all but finitely many $h$, so that $A(x)u_{1}$ is a finite sum. Then we define (cf. \[HL1\]) $$\begin{aligned}
\label{eaxz}
A(e^{l_{p}(z)})=A(x)|_{x=e^{l_{p}(z)}}:
=\sum_{h\in {\C}}e^{hl_{p}(z)}a_{h},\end{aligned}$$ which is a well defined element of $\Hom (U_{1},U_{2})$. In the case that $U_{2}={\C}$ and $U_{1}=W$, a $V$-module, we have $A(y)\in W'\{y\}$ and $$\begin{aligned}
(f(y)A(y))|_{y=e^{l_{p}(z)}}=f(z)A(e^{l_{p}(z)})\end{aligned}$$ for $f(y)\in {\C}[y,y^{-1}]$. If ${\cal{Y}}$ is an intertwining operator of type ${W'\choose W_{1}W_{2}}$, then (cf. \[HL1\]) $$\begin{aligned}
{\cal{Y}}(w_{(1)},e^{l_{p}(z)})w_{(2)}
={\cal{Y}}(w_{(1)},x)w_{(2)}|_{x=e^{l_{p}(z)}}\in W^{*}\end{aligned}$$ for $w_{(1)}\in W_{1},\; w_{(2)}\in W_{2}$.

We recall the following notion of $P(z)$-intertwining map from \[HL1\]:

All $P(z)$-intertwining maps of such type clearly form a vector space, denoted by ${\cal{M}}[P(z)]_{W_{1}W_{2}}^{W'}$.

For $p\in {\Z}$ and for any intertwining operator ${\cal{Y}}$ of type ${W'\choose W_{1}W_{2}}$, we define a linear map (cf. \[HL3\]) $$\begin{aligned}
F_{{\cal{Y}},p}^{P(z)}: 
& &W_{1}\otimes W_{2}\rightarrow \overline{W'}\;\;(=W^{*})
\nonumber\\
& &\sum_{j=1}^{n}w^{1j}\otimes w^{2j}\mapsto 
\sum_{j=1}^{n}{\cal{Y}}(w_{(1j)},e^{l_{p}(z)})w_{(2j)}\end{aligned}$$ for $w_{(1j)}\in W_{1},\;w_{(2j)}\in W_{2}$. It is clear that $F_{{\cal{Y}},p}^{P(z)}$ is an intertwining map of the same type. Furthermore, we have (\[HL3\], Proposition 12.2):

Let $W_{1}, W_{2}$ and $W$ be lower truncated generalized $V$-modules, let $z$ be a nonzero complex number and let $p\in {\Z}$. Then the correspondence ${\cal{Y}}\mapsto F_{{\cal{Y}},p}^{P(z)}$ is a linear isomorphism from the space ${\cal{V}}_{W_{1}W_{2}}^{W'}$ of intertwining operators of type ${W'\choose W_{1}W_{2}}$ onto the space ${\cal{M}}[P(z)]_{W_{1}W_{2}}^{W'}$ of $P(z)$-intertwining maps of type ${W'\choose W_{1}W_{2}}$.

Now we present our main result in this section.

Let $W_{1}, W_{2}$ and $W$ be generalized $V$-modules and let $F$ be a linear map from $W_{1}\otimes W_{2}$ to $\overline{W'}$ $(=W^{*})$. Then $F$ is a $P(z)$-intertwining map of type ${W' \choose W_{1}W_{2}}$ if and only if $$\begin{aligned}
F(W_{1}\otimes W_{2})\subset {\cal{D}}_{P(z)}(W)\end{aligned}$$ and $F$ is a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$ (a subspace of $W^{*}$). Equivalently, $$\begin{aligned}
{\cal{M}}[P(z)]^{W'}_{W_{1}W_{2}}={\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2},
{\cal{D}}_{P(z)}(W)).\end{aligned}$$ In particular, $$\begin{aligned}
\dim {\cal{M}}[P(z)]^{W'}_{W_{1}W_{2}}
=\dim {\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2},
{\cal{D}}_{P(z)}(W)).\end{aligned}$$

Suppose that $F$ is a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. Let $v\in V, \;w_{(1)}\in W_{1},\;w_{(2)}\in W_{2}$. Since $F(w_{(1)}\otimes w_{(2)})\in {\cal{D}}_{P(z)}(W)$, using Proposition \[prelationnew\] and the fact that $F$ is a $V\otimes V$-homomorphism we get $$\begin{aligned}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-z}{x_{0}}\right)
Y(v,x_{1})F(w_{(1)}\otimes w_{(2)})\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{z-x_{1}}{-x_{0}}\right)
Y_{P(z)}^{R}(v,x_{1})F(w_{(1)}\otimes w_{(2)})+
z^{-1}\delta\left(\frac{x_{1}-x_{0}}{z}\right)
Y_{P(z)}^{L}(v,x_{0})F(w_{(1)}\otimes w_{(2)})\nonumber\\
&=&x_{0}^{-1}\delta\left(\frac{z-x_{1}}{-x_{0}}\right)
F(w_{(1)}\otimes Y(v,x_{1})w_{(2)})+
z^{-1}\delta\left(\frac{x_{1}-x_{0}}{z}\right)
F(Y(v,x_{0})w_{(1)}\otimes w_{(2)}).\end{aligned}$$ Then $F$ is a $P(z)$-intertwining map.

Conversely, suppose that $F$ is a $P(z)$-intertwining map. For any $v\in V, \;w_{(1)}\in W_{1}, \;w_{(2)}\in W_{2}$, let $k\in {\N}$ such that $x_{0}^{k}Y(v,x_{0})w_{(1)}\in W_{1}[[x_{0}]]$. Then by taking $\Res_{x_{0}}x_{0}^{k}$ from (\[ejacobimap\]) we obtain $$\begin{aligned}
\label{ercomm}
(x_{1}-z)^{k}Y(v,x_{1})F(w_{(1)}\otimes w_{(2)})
=(x_{1}-z)^{k}F(w_{(1)}\otimes Y(v,x_{1})w_{(2)}).\end{aligned}$$ Since $Y(v,x_{1})w_{(2)}\in W_{2}((x_{1}))$, the right-hand side of (\[ercomm\]) lies in $W^{*}((x_{1}))$, hence $$\begin{aligned}
\label{e4.25}
(x_{1}-z)^{k}Y(v,x_{1})F(w_{(1)}\otimes w_{(2)})\in W^{*}((x_{1})).\end{aligned}$$ Therefore $F(w_{(1)}\otimes w_{(2)})\in {\cal{D}}_{P(z)}(W)$ by Lemma \[lequivconds\]. Furthermore, by Lemma \[cbasic\] we have $$\begin{aligned}
(x_{1}-z)^{k}Y_{P(z)}^{R}(v,x_{1})F(w_{(1)}\otimes w_{(2)})
=(x_{1}-z)^{k}Y(v,x_{1})F(w_{(1)}\otimes w_{(2)}).\end{aligned}$$ Combining this with (\[ercomm\]) we get $$\begin{aligned}
\label{etem}
(x_{1}-z)^{k}Y_{P(z)}^{R}(v,x_{1})F(w_{(1)}\otimes w_{(2)})
=(x_{1}-z)^{k}F(w_{(1)}\otimes Y(v,x_{1})w_{(2)}).\end{aligned}$$ Because $$Y_{P(z)}^{R}(v,x_{1})F(w_{(1)}\otimes w_{(2)}),\;\;\;\; 
F(w_{(1)}\otimes Y(v,x_{1})w_{(2)})\in {\cal{D}}_{P(z)}(W)((x_{1})),$$ we can multiply (\[etem\]) by $(-z+x_{1})^{-k}$ to obtain $$\begin{aligned}
\label{erhom}
Y_{P(z)}^{R}(v,x_{1})F(w_{(1)}\otimes w_{(2)})
=F(w_{(1)}\otimes Y(v,x_{1})w_{(2)}).\end{aligned}$$

Let $l\in {\N}$ be such that $x_{1}^{l}Y(v,x_{1})w_{(2)}\in W_{2}[[x_{1}]]$. Then by taking $\Res_{x_{1}}x_{1}^{l}$ from (\[ejacobimap\]) we obtain $$\begin{aligned}
\label{e4.24}
\Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x_{1}-z}{x_{0}}\right)
x_{1}^{l}Y(v, x_{1})F(w_{(1)}\otimes w_{(2)})=
(x_{0}+z)^{l}F(Y(v,x_{0})w_{(1)}\otimes w_{(2)}).\end{aligned}$$ Similarly, let $l'\in {\N}$ be such that $$x_{1}^{l'}Y^{R}_{P(z)}(v,x_{1})F(w_{(1)}\otimes w_{(2)})\in 
({\cal{D}}_{P(z)}(W))[[x_{1}]].$$ By taking $\Res_{x_{1}}x_{1}^{l'}$ from (\[sjacnew\]) with $\alpha=F(w_{(1)}\otimes w_{(2)})$ we obtain $$\begin{aligned}
\label{e4.252}
\Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x_{1}-z}{x_{0}}\right)
x_{1}^{l'}Y(v, x_{1})F(w_{(1)}\otimes w_{(2)})=
(x_{0}+z)^{l'}Y_{P(z)}^{L}(v,x_{0})F(w_{(1)}\otimes w_{(2)}).\end{aligned}$$ Combing (\[e4.24\]) with (\[e4.252\]) we get $$\begin{aligned}
\label{eteml}
(x_{0}+z)^{l+l'}Y_{P(z)}^{L}(v,x_{0})F(w_{(1)}\otimes w_{(2)})
=(x_{0}+z)^{l+l'}F(Y(v,x_{0})w_{(1)}\otimes w_{(2)}).\end{aligned}$$ Again, because $$Y_{P(z)}^{L}(v,x_{0})F(w_{(1)}\otimes w_{(2)}),\;\;\;\; 
F(Y(v,x_{0})w_{(1)}\otimes w_{(2)})\in ({\cal{D}}_{P(z)}(W))((x_{0})),$$ we can multiply (\[eteml\]) by $(z+x_{0})^{-l-l'}$ to obtain $$\begin{aligned}
\label{elhom}
Y_{P(z)}^{L}(v,x_{0})F(w_{(1)}\otimes w_{(2)})
=F(Y(v,x_{0})w_{(1)}\otimes w_{(2)}).\end{aligned}$$ It follows from (\[erhom\]) and (\[elhom\]) that $F$ is a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. $\;\;\;\;\Box$

From Theorem \[tsub\], the image of any $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ is contained in ${\cal{D}}_{P(z)}(W)$. Let ${\cal{Y}}$ be an intertwining operator of type ${W'\choose W_{1}W_{2}}$. It follows from Proposition \[phl12.2\] and Theorem \[tsub\] that $$\begin{aligned}
{\cal{Y}}(w_{(1)},e^{l_{p}(z)})w_{(2)}\in {\cal{D}}_{P(z)}(W)\end{aligned}$$ for any $w_{(1)}\in W_{1},\;w_{(2)}\in W_{2}, \;p\in {\Z}$. Furthermore, the linear map $F_{{\cal{Y}},p}^{P(z)}$ is a $V\otimes V$-homomorphism. In view of this, for $p\in {\Z}$ we obtain a linear map $$\begin{aligned}
F_{p}^{P(z)}:& & {\cal{V}}^{W'}_{W_{1}W_{2}}\rightarrow 
{\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2}, {\cal{D}}_{P(z)}(W))\nonumber\\
& &{\cal{Y}}\mapsto F_{{\cal{Y}},p}^{P(z)}.\end{aligned}$$ Combining Proposition \[phl12.2\] with Theorem \[tsub\] we immediately have:

Let $W_{1},W_{2}, W$ be lower truncated generalized $V$-modules and let $p\in {\Z}$. Then $F_{p}^{P(z)}$ is a linear isomorphism from ${\cal{V}}^{W'}_{W_{1}W_{2}}$ onto ${\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2},{\cal{D}}_{P(z)}(W))$. In particular, $$\begin{aligned}
\dim {\cal{V}}^{W'}_{W_{1}W_{2}}
=\dim {\rm Hom}_{V\otimes V}(W_{1}\otimes W_{2},
{\cal{D}}_{P(z)}(W)).\end{aligned}$$

Recall that ${\cal{D}}_{P(z)}(W)$ is a weak $V\otimes V$-module in general. We introduce the following notion in terms of ordinary $V\otimes V$-submodules of ${\cal{D}}_{P(z)}(W)$:

Note that on an ordinary $V\otimes V$-submodule of ${\cal{D}}_{P(z)}(W)$, $L_{P(z)}^{L}(0)$ semisimply acts if and only if $L_{P(z)}^{R}(0)$ semisimply acts. With this definition, any $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ is a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into $R_{P(z)}(W)$. Because of the two grading restrictions in the definition of the notion of a module, $R_{P(z)}(W)$ is a generalized $V\otimes V$-module in general.

For the rest of this section, [*let $S$ be a (non-canonical) complete set of representatives of equivalence classes of irreducible $V$-modules*]{}. From \[FHL\], if $W_{1}$ and $W_{2}$ are irreducible $V$-modules, $W_{1}\otimes W_{2}$ is an irreducible $V\otimes V$-module and furthermore, any irreducible $V\otimes V$-module is isomorphic to such a module. Then $W_{1}\otimes W_{2}$ for $W_{1},W_{2}\in S$ form a complete set of representatives of equivalence classes of irreducible $V\otimes V$-modules. Set $$\begin{aligned}
\label{ehvsw}
H_{V,S}^{P(z)}(W)
=\coprod_{W_{1},W_{2}\in S}{\cal{M}}[P(z)]_{W_{1}W_{2}}^{W'}
\otimes (W_{1}\otimes W_{2}),\end{aligned}$$ which is viewed as a generalized $V\otimes V$-module with the action on the second parts $(W_{1}\otimes W_{2})$. With Theorem \[tsub\] we obtain a $V\otimes V$-homomorphism $\Psi_{W}$ from $H_{V,S}^{P(z)}(W)$ into $R_{P(z)}(W)$, defined by $$\begin{aligned}
\label{epi}
\Psi_{W} (F\otimes (w_{(1)}\otimes w_{(2)}))=F(w_{(1)}\otimes w_{(2)})\end{aligned}$$ for $F\in {\cal{M}}[P(z)]_{W_{1}W_{2}}^{W'},\;w_{(1)}\in W_{1},
\; w_{(2)}\in W_{2}$ with $W_{1},W_{2}\in S$.

As we shall need, we discuss certain properties of modules for a tensor product vertex operator algebra. For vertex operator algebras $V_{1}$ and $V_{2}$, we naturally consider $V_{1}$ and $V_{2}$ as subalgebras of $V_{1}\otimes V_{2}$ by identifying $V_{1}$ with $V_{1}\otimes {\C}{\bf 1}$ and $V_{2}$ with ${\C}{\bf 1}\otimes V_{2}$, respectively. In this way, any weak $V_{1}\otimes V_{2}$-module is automatically a weak $V_{1}$-module and a weak $V_{2}$-module and the actions of $V_{1}$ and $V_{2}$ commute. (See \[FHL\], Section 4.7 for more details.) We denote by $L^{1}(n)$ and $L^{2}(n)$ for $n\in {\Z}$ the corresponding Virasoro operators. Then $L^{1}(n)+L^{2}(n)$ for $n\in {\Z}$ are the Virasoro operators for $V_{1}\otimes V_{2}$.

Let $V_{1}$ and $V_{2}$ be vertex operator algebras, let $W$ be a weak $V_{1}\otimes V_{2}$-module, and let $W_{2}$ be a finitely generated weak $V_{2}$-module, e.g., $W_{2}$ is an irreducible $V_{2}$-module. Then ${\rm Hom}_{V_{2}}(W_{2},W)$ is a weak $V_{1}$-module with $$\begin{aligned}
(Y(v_{(1)},x)f)(w_{(2)})=Y(v_{(1)},x)f(w_{(2)})\end{aligned}$$ for $v_{(1)}\in V_{1},\; f\in \Hom_{V_{2}}(W_{2},W),\;w_{(2)}\in W_{2}$. Furthermore, if $W_{2}$ is a $V_{1}$-module and $W$ is a $V_{1}\otimes V_{2}$-module on which $L^{1}(0)$ or $L^{2}(0)$ semisimply acts, then ${\rm Hom}_{V_{2}}(W_{2},W)$ is a $V_{1}$-module.

Since the actions of $V_{1}$ and $V_{2}$ on $W$ commute, we easily see that $$\begin{aligned}
Y(v_{(1)},x)f\in (\Hom_{V_{2}}(W_{2},W))[[x,x^{-1}]]\end{aligned}$$ for $v_{(1)}\in V_{1},\; f\in \Hom_{V_{2}}(W_{1},W)$. Clearly, the vacuum property holds and the Jacobi identity will also hold provided that the truncation condition holds.

Let $P$ be a finite-dimensional subspace of $W_{2}$, which generates $W_{2}$ as a weak $V_{2}$-module. Let $v_{(1)}\in V_{1},\; f\in {\Hom}_{V_{2}}(W_{2},W)$. Since $\dim\; P<\infty$ and $$(Y(v_{(1)},x)f)(p)=Y(v_{(1)},x)f(p)\in W((x))\;\;\;\mbox{ for every }
p\in P,$$ there exists $r\in {\N}$ such that $$\begin{aligned}
x^{r}(Y(v_{(1)},x)f)(p)=x^{r}Y(v_{(1)},x)f(p)\in W[[x]]\;\;\;\;
\mbox{ for all }p\in P.\end{aligned}$$ Furthermore, for $v_{(2)}\in V_{2}$ we have $$\begin{aligned}
\label{e4.37}
x_{1}^{r}(Y(v_{(1)},x_{1})f)(Y(v_{(2)},x_{2})p)
&=&Y(v_{(2)},x_{2})x_{1}^{r}Y(v_{(1)},x_{1})f(p)\nonumber\\
&\in& W[[x_{1},x_{2},x_{2}^{-1}]]\;\;\;\;\mbox{ for all }p\in P.\end{aligned}$$ Since $P$ generates $W_{2}$ as a $V_{2}$-module, by repeatedly using (\[e4.37\]) we get $$\begin{aligned}
x^{r}(Y(v_{(1)},x)f)(w_{(2)})\in W[[x]]\;\;\;\mbox{ for all }w_{(2)}\in W_{2}.\end{aligned}$$ Thus $x^{r}Y(v_{(1)},x)f\in \left({\Hom}_{V_{2}}(W_{2},W)\right)[[x]]$. This proves the truncation condition, so ${\Hom}_{V_{2}}(W_{2},W)$ is a weak $V_{1}$-module.

Now, assume that $W_{2}$ is a $V_{2}$-module and $W$ is a $V_{1}\otimes V_{2}$-module on which $L^{1}(0)$ or $L^{2}(0)$ semisimply acts. Since $L^{1}(0)+L^{2}(0)$ semisimply acts on $W$, both $L^{1}(0)$ and $L^{2}(0)$ semisimply act on $W$. Then $$\begin{aligned}
W=\coprod_{\alpha\in {\C}}W[\alpha],\end{aligned}$$ where $$\begin{aligned}
W[\alpha]=\{ w\in W\;|\; L^{1}(0)w=\alpha w\}.\end{aligned}$$ Clearly, each $W[\alpha]$ is a weak $V_{2}$-submodule of $W$. On the other hand, we may take the generating space $P$ of $W_{2}$ to be a graded subspace with $$\begin{aligned}
P=P_{(h_{1})}\oplus \cdots \oplus P_{(h_{m})},\end{aligned}$$ where $h_{1},\dots,h_{m}$ are finitely many distinct complex numbers.

Let $f\in {\Hom}_{V_{2}}(W_{2},W)$. Since $\dim f(P)<\infty$, there are finitely many (distinct) complex numbers $\alpha_{1},\dots,\alpha_{n}$ such that $$\begin{aligned}
f(P)\subset W[\alpha_{1}]\oplus \cdots\oplus W[\alpha_{n}].\end{aligned}$$ Since $P$ generates $W_{2}$ under the action of $V_{2}$, $$\begin{aligned}
f(W_{2})\subset W[\alpha_{1}]\oplus \cdots\oplus W[\alpha_{n}].\end{aligned}$$ Let $f_{i}$ be the projection of $f$ onto $W[\alpha_{i}]$ for $1\le i\le n$. Clearly, each $f_{i}$ is a $V_{2}$-homomorphism, hence $f_{i}\in \Hom_{V_{2}}(W_{2},W)$. For $1\le i\le n$, we have $$\begin{aligned}
L(0)f_{i}=\alpha_{i}f_{i}\end{aligned}$$ because $$(L(0)f_{i})(w_{(2)})=L^{1}(0)f_{i}(w_{(2)})=\alpha_{i}f_{i}(w_{(2)})$$ for $w_{(2)}\in W_{2}$. This proves that $L(0)$ semisimply acts on $\Hom_{V_{2}}(W_{2},W)$. If $f\in (\Hom_{V_{2}}(W_{2},W))_{(\alpha)}$ and $p\in P_{(h)}$, then $$(L^{1}(0)+L^{2}(0))f(p)=(L(0)f)(p)+f(L(0)p)=(\alpha+h)f(p).$$ That is, $f(p)\in W_{(\alpha+h)}$. Since $P$ generates $W_{2}$ under the action of $V_{2}$, through the restriction map we may identify $(\Hom_{V_{2}}(W_{2},W))_{(\alpha)}$ as a subspace of $\sum_{j=1}^{m} \Hom(P_{(h_{j})},W_{(\alpha+h_{j})})$. Then it follows from the finite-dimensionality of $P$ and the two grading restrictions on $W$ that the two grading restrictions on $\Hom_{V_{2}}(W_{2},W)$ also hold. Therefore, $\Hom_{V_{2}}(W_{2},W)$ is a $V_{1}$-module.$\;\;\;\;\Box$

We also have the following simple fact:

Let $V$ be a vertex operator algebra, $W_{j}$ $(j\in J)$ be weak $V$-modules and $W$ be a finitely generated weak $V$-module. Then as a vector space, $$\begin{aligned}
\Hom_{V}\left(W,\coprod_{j\in J}W_{j}\right)
=\coprod_{j\in J}\Hom_{V}(W,W_{j}).\end{aligned}$$

If $J$ is a finite index set, it follows easily from the proof of the corresponding classical result. In general, we shall need the assumption on $W$. It is clear that $$\begin{aligned}
\label{econverse}
\coprod_{j\in J}\Hom_{V}(W,W_{j})\subset 
\Hom_{V}\left(W,\coprod_{j\in J}W_{j}\right).\end{aligned}$$ For the converse, let $P$ be a finite-dimensional generating space of $W$. Then for every $f\in \Hom_{V}(W,\coprod_{j\in J}W_{j})$, there are finitely many $j_{1},\dots,j_{n}$ such that $$\begin{aligned}
f(P)\subset W_{j_{1}}\oplus \cdots \oplus W_{j_{n}}.\end{aligned}$$ Since $P$ generates $W$, we have $$\begin{aligned}
f(W)\subset W_{j_{1}}\oplus \cdots \oplus W_{j_{n}}.\end{aligned}$$ Then $$f\in \Hom_{V}(W,W_{j_{1}})\oplus \cdots \oplus \Hom_{V}(W,W_{j_{n}}).$$ This proves the converse of (\[econverse\]), hence completes the proof. $\;\;\;\;\Box$

Because modules for vertex operator algebras have finite-dimensional homogeneous subspaces and we work on ${\C}$, Schur’s Lemma for irreducible modules holds (cf. \[FHL\], Remark 4.7.1). Then using the proof of the corresponding classical result (see for example \[BD\], Chapter II, Proposition 1.14, where Lemma \[lhomdirect\] is used) we have:

Let $V_{1}$ and $V_{2}$ be vertex operator algebras and $W$ be a weak $V_{1}\otimes V_{2}$-module such that $W$ is a direct sum of irreducible (ordinary) $V_{2}$-modules. Let $S_{2}=\{ M_{i}\;|\;i\in I\}$ be a complete set of representatives of equivalence classes of irreducible $V_{2}$-modules. For each $i\in I$, define a linear map $$\begin{aligned}
\Psi_{i} : & &{\rm Hom}_{V_{2}}(M_{i},W)\otimes M_{i}\rightarrow W
\nonumber\\
& & (f,w)\mapsto f(w).\end{aligned}$$ Then the natural linear map $$\begin{aligned}
\Psi =(\Psi_{i}): 
\coprod_{i\in I}{\rm Hom}_{V_{2}}(M_{i},W)\otimes M_{i}\rightarrow W\end{aligned}$$ is a $V_{1}\otimes V_{2}$-isomorphism.

If every $V\otimes V$-module is completely reducible, then $R_{P(z)}(W)$, being a sum of $V\otimes V$-modules, is completely reducible, Then it follows from Theorem \[tsub\] and Proposition \[lfact\] with $V_{1}={\C}$ and $V_{2}=V\otimes V$ that for any $V$-module $W$, $\Psi_{W}$, defined previously, is a $V\otimes V$-isomorphism onto $R_{P(z)}(W)$.

The following is a version of a result of \[DMZ\] about the rationality of the tensor product of rational vertex operator algebras.

Let $V_{1}$ and $V_{2}$ be vertex operator algebras such that every module is completely reducible. Then any $V_{1}\otimes V_{2}$-module $W$, on which $L^{1}(0)$ or $L^{2}(0)$ semisimply acts, is completely reducible.

Since $L^{1}(0)$ and $L^{2}(0)$ commute, each vector of $W$ is a sum of common eigenvectors for $L^{1}(0)$ and $L^{2}(0)$. Let $w\in W$ be a common eigenvector for $L^{1}(0)$ and $L^{2}(0)$ with $$\begin{aligned}
L^{1}(0)w=h_{1}w,\;\;L^{2}(0)w=h_{2}w.\end{aligned}$$ Then $w\in W_{(h_{1}+h_{2})}$, the $(h_{1}+h_{2})$-eigenspace of $L^{1}(0)+L^{2}(0)$. Then under the action of $V\otimes {\C}$ (or ${\C}\otimes V$), $w$ generates an ordinary $V$-module, which is completely reducible from the assumption. Thus $W$ is a completely reducible $V$-module under the action of $V\otimes {\C}$ (or ${\C}\otimes V$). Let $\{W_{i}\;|\; i\in I\}$ be a complete set of non-isomorphic irreducible $V$-submodules of $W$ under the action of $V\otimes {\C}$. In view of Proposition \[lfact\] we have $$W=\oplus_{i\in I} \Hom _{V}(W_{i},W)\otimes W_{i}.$$ Furthermore, in view of Proposition \[phom\] $\Hom_{V}(W_{i},W)$ is a $V$-module under the action of ${\C}\otimes V$, which is completely reducible. Proposition 4.7.2 of \[FHL\] states that the tensor product of irreducible modules for factors is an irreducible module for the tensor product vertex operator algebra. Then it follows that $W$ is a completely reducible $V\otimes V$-module. $\;\;\;\;\Box$

If $V$ is rational in the sense of \[HL1\], then $H_{V,S}^{P(z)}(W))$ is an (ordinary) $V\otimes V$-module. Furthermore, it follows from Lemma \[ltensorrationality\] that $R_{P(z)}(W)$ is completely reducible. Then using Proposition \[lfact\] and Theorem \[tsub\] we immediately have:

Let $V$ be a vertex operator algebra such that every $V$-module is completely reducible and let $S$ be a (non-canonical) complete set of representatives of equivalence classes of irreducible $V$-modules. Then for any $V$-module $W$, $\Psi_{W}$ is a $V\otimes V$-isomorphism from $H_{V,S}^{P(z)}(W)$ onto $R_{P(z)}(W)$. In particular, if $V$ is rational in the sense of \[HL1\], the above assertion holds and $R_{P(z)}(W)$ is an (ordinary) $V\otimes V$-module. $\;\;\;\;\Box$

Let us consider a regular vertex operator algebra $V$, i.e., any weak $V$-module is a direct sum of irreducible (ordinary) $V$-modules. It was proved in \[DLM2\] that $V\otimes V$ is also regular. (Proposition \[lfact\] with $V_{1}=V_{2}=V$ together with Proposition 4.7.2 of \[FHL\] gives another slightly different proof.) Consequently, ${\cal{D}}_{P(z)}(W)=R_{P(z)}(W)$ is a direct sum of irreducible (ordinary) $V\otimes V$-modules. It follows from Theorem \[tw\] that $\Psi_{W}$ is a $V\otimes V$-isomorphism onto ${\cal{D}}_{P(z)}(W)$. Furthermore, it was proved (\[Li5\], Theorem 3.13) that fusion rules for triples of irreducible modules are finite. Thus, ${\cal{D}}_{P(z)}(W)$ is an (ordinary) $V\otimes V$-module. To summarize we have:

Let $V$ be a regular vertex operator algebra and let $S$ be a (non-canonical) complete set of representatives of equivalence classes of irreducible $V$-modules. Then for any $V$-module $W$, $$\begin{aligned}
{\cal{D}}_{P(z)}(W)=R_{P(z)}(W),\end{aligned}$$ ${\cal{D}}_{P(z)}(W)$ is an (ordinary) $V\otimes V$-module, and the linear map $$\begin{aligned}
\Psi_{W}:  \coprod_{W_{1},W_{2}\in S}
{\cal{M}}[P(z)]^{W'}_{W_{1}W_{2}}\otimes (W_{1}\otimes W_{2})
\rightarrow {\cal{D}}_{P(z)}(W)\end{aligned}$$ is a $V\otimes V$-isomorphism.$\;\;\;\;\Box$

Next, we consider the special case that $W=V$. It was known (cf. \[Li0\], Remark 2.9) that ${\cal{V}}^{W_{2}'}_{VW_{1}}\cong {\rm Hom}_{V}(W_{1},W_{2}')$. Then using Schur’s lemma, we find that for irreducible $V$-modules $W_{1}$ and $W_{2}$, $\dim {\cal{V}}^{W_{2}'}_{VW_{1}}=1$ if $W_{1}\cong W_{2}'$, and $0$ otherwise. In view of Proposition \[phl2\] we have $\dim {\cal{V}}^{V'}_{W_{1}W_{1}'}=1$.

Let $(W,Y_{W})$ be a $V$-module. Then $Y_{W}$ is an intertwining operator of type ${W\choose VW}$. As usual, we use $Y$ for $Y_{W}$. From \[FHL\], we have an intertwining operator of type ${W\choose WV}$, defined by $$\begin{aligned}
Y(w,x)v=e^{xL(-1)}Y(v,-x)w\end{aligned}$$ for $v\in V,\; w\in W$. For convenience, we refer the canonical intertwining operators $Y$ of types ${W\choose VW}$ and ${W\choose WV}$ as the [*standard*]{} intertwining operators. Furthermore, from \[HL2\] (see also \[FHL\]), we have an intertwining operator ${\cal{Y}}$ of type ${V'\choose WW'}$, defined by $$\begin{aligned}
\label{eww'v'}
\<{\cal{Y}}(w,x)w',v\>
&=&\<w',Y(e^{xL(1)}e^{\pi i L(0)}x^{-2L(0)}w,x^{-1})v\>\nonumber\\
&=&\<w',e^{x^{-1}L(-1)}Y(v,-x^{-1})e^{xL(1)}e^{\pi i L(0)}x^{-2L(0)}w\>
\nonumber\\
&=&\<e^{x^{-1}L(1)}w',Y(v,-x^{-1})e^{xL(1)}e^{\pi i L(0)}x^{-2L(0)}w\>\end{aligned}$$ for $w\in W,\;w'\in W',\; v\in V$. Then for an irreducible $V$-module $W$ we have $$\begin{aligned}
{\cal{V}}_{WW'}^{V'}={\C}{\cal{Y}}.\end{aligned}$$ Then in view of Theorem \[tw\] we immediately have:

Let $V$ be a vertex operator algebra such that every $V$-module is completely reducible and let $S$ be as before. Define a linear map $$\begin{aligned}
\Phi^{P(z)}_{V}:  \coprod_{W\in S}W\otimes W' \rightarrow R_{P(z)}(V)
\subset {\cal{D}}_{P(z)}(V)\end{aligned}$$ by $$\begin{aligned}
\<\Phi^{P(z)}_{V}(w\otimes w'),v\>
=\<e^{z^{-1}L(1)}w', Y(v,-z^{-1})e^{zL(1)}e^{\pi i L(0)}e^{-2L(0)\log z}w\>\end{aligned}$$ (cf. (\[eww’v’\])) for $w\in W,\; w'\in W',\; v\in V$. Then $\Phi^{P(z)}_{V}$ is a $V\otimes V$-isomorphism onto $R_{P(z)}(V)$. In particular, the above assertion holds if $V$ is regular, or rational in the sense of \[HL1\].$\;\;\;\;\Box$

From Theorem \[tw\], for any nonzero complex numbers $z_{1}$ and $z_{2}$, $R_{P(z_{1})}(W)$ and $R_{P(z_{2})}(W)$ are isomorphic generalized $V\otimes V$-modules if every $V$-module is completely reducible. In the following we shall give a canonical $V\otimes V$-isomorphism between ${\cal{D}}_{P(z_{1})}(W)$ and ${\cal{D}}_{P(z_{2})}(W)$ without the complete reducibility assumption on $V$-modules.

We recall the following conjugation formula from \[FHL\] (Lemma 5.2.3):

Let $(M,Y_{M})$ be a generalized $V$-module. Then $$\begin{aligned}
\label{L(0)con}
x_{0}^{L(0)}Y_{M}(v,x)x_{0}^{-L(0)}=Y_{M}(x_{0}^{L(0)}v,xx_{0})\end{aligned}$$ for $v\in V$, where $x$ and $x_{0}$ are independent commuting formal variables. Furthermore, when replacing $x_{0}$ by a nonzero complex number $z$, we have $$\begin{aligned}
\label{eL(0)conz}
e^{L(0)\log z}Y_{M}(v,x)e^{-L(0)\log z}=Y_{M}(z^{L(0)}v,zx).\end{aligned}$$

Notice that for a nonzero complex number $z$ and for $p\in {\Z}$, the linear endomorphism $e^{l_{p}(z)L'(0)}$ of $W'$ is naturally extended to the endomorphism $e^{l_{p}(z)L^{*}(0)}$ of $W^{*} (=\overline{W'})$. From (\[eL(0)conz\]) we have $$\begin{aligned}
\label{zL(0)con}
e^{l_{p}(z)L^{*}(0)}Y^{*}(v,x)e^{-l_{p}(z)L^{*}(0)}=Y^{*}(z_{0}^{L(0)}v,zx)\end{aligned}$$ on $W^{*}$ for $v\in V$. Furthermore, we have (cf. Lemma \[lfhl523\]):

Let $z, z_{1}$ be nonzero complex numbers and let $W$ be a $V$-module. Then the linear automorphism $e^{l_{p}(z)L^{*}(0)}$ of $W^{*}$ maps ${\cal{D}}_{P(z_{1})}(W)$ onto ${\cal{D}}_{P(zz_{1})}(W)$ such that $$\begin{aligned}
& &e^{l_{p}(z)L^{*}(0)}Y^{R}_{P(z_{1})}(v,x)\alpha
=Y^{R}_{P(zz_{1})}(z^{L(0)}v,zx)e^{l_{p}(z)L^{*}(0)}\alpha\label{eisom1}\\
& &e^{l_{p}(z)L^{*}(0)}Y^{L}_{P(z_{1})}(v,x)\alpha
=Y^{L}_{P(zz_{1})}(z^{L(0)}v,zx)e^{l_{p}(z)L^{*}(0)}\alpha\label{eisom2}\end{aligned}$$ for $v\in V, \;\alpha\in {\cal{D}}_{P(z_{1})}(W)$.

For $v\in V,\;\alpha\in {\cal{D}}_{P(z_{1})}(W)$, let $k\in {\N}$ be such that $$\begin{aligned}
\label{exz1}
(x-z_{1})^{k}Y^{*}(v,x)\alpha\in W^{*}((x)),\end{aligned}$$ hence (by Lemma \[cbasic\]), $$\begin{aligned}
\label{e4.40}
(x-z_{1})^{k}Y^{R}_{P(z_{1})}(v,x)\alpha=(x-z_{1})^{k}Y^{*}(v,x)\alpha.\end{aligned}$$ Without losing much generality we may assume that $v$ is homogeneous. Using (\[zL(0)con\]) and (\[exz1\]) we get $$\begin{aligned}
& &(x-zz_{1})^{k}Y^{*}(v,x)e^{l_{p}(z)L^{*}(0)}\alpha\nonumber\\
&=&(x-zz_{1})^{k}e^{l_{p}(z)L^{*}(0)}Y^{*}(z^{-L(0)}v,z^{-1}x)\alpha
\nonumber\\
&=&z^{k-\wt v}(z^{-1}x-z_{1})^{k}e^{l_{p}(z)L^{*}(0)}Y^{*}(v,z^{-1}x)\alpha
\nonumber\\
&\in &W^{*}((x)).\end{aligned}$$ By Lemma \[lequivconds\], $e^{l_{p}(z)L^{*}(0)}\alpha\in {\cal{D}}_{P(zz_{1})}(W)$. Furthermore, by Lemma \[cbasic\], $$\begin{aligned}
\label{e4.42}
(x-zz_{1})^{k}Y^{R}_{P(zz_{1})}(v,x)e^{l_{p}(z)L^{*}(0)}\alpha
=(x-zz_{1})^{k}Y^{*}(v,x)e^{l_{p}(z)L^{*}(0)}\alpha.\end{aligned}$$ Using (\[e4.40\]), (\[zL(0)con\]) and (\[e4.42\]) we get $$\begin{aligned}
& &z^{k}(x-z_{1})^{k}e^{l_{p}(z)L^{*}(0)}Y^{R}_{P(z_{1})}(v,x)\alpha
\nonumber\\
&=&z^{k}(x-z_{1})^{k}e^{l_{p}(z)L^{*}(0)}Y^{*}(v,x)\alpha\nonumber\\
&=&(zx-zz_{1})^{k}Y^{*}(z^{L(0)}v,zx)e^{l_{p}(z)L^{*}(0)}\alpha\nonumber\\
&=&(zx-zz_{1})^{k}Y^{R}_{P(zz_{1})}(z^{L(0)}v,zx)e^{l_{p}(z)L^{*}(0)}\alpha.\end{aligned}$$ Then multiplying by $z^{-k}(-z_{1}+x)^{-k}$ we obtain (\[eisom1\]).

Since $\alpha\in {\cal{D}}_{P(z_{1})}(W)$ and $e^{l_{p}(z)L^{*}(0)}\alpha\in {\cal{D}}_{P(zz_{1})}(W)$, by Lemma \[lleft1\] there exists $l\in {\N}$ such that $$\begin{aligned}
& &(x_{0}+z_{1})^{l}Y^{L}_{P(z_{1})}(v,x_{0})\alpha
=\Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x_{1}-z_{1}}{x_{0}}\right)
(x_{0}+z_{1})^{l}Y^{*}(v,x_{1})\alpha,\label{e4.46}\\
& &(x_{0}+zz_{1})^{l}Y^{L}_{P(zz_{1})}(v,x_{0})e^{l_{p}(z)L^{*}(0)}
\alpha\nonumber\\
&=&\Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x_{1}-zz_{1}}{x_{0}}\right)
(x_{0}+zz_{1})^{l}Y^{*}(v,x_{1})e^{l_{p}(z)L^{*}(0)}\alpha.\label{e4.47}\end{aligned}$$ Using (\[e4.46\]), (\[zL(0)con\]) and (\[e4.47\]) we get $$\begin{aligned}
& &z^{l}(x_{0}+z_{1})^{l}e^{l_{p}(z)L^{*}(0)}Y^{L}_{P(z_{1})}(v,x_{0})\alpha
\nonumber\\
&=&\Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x_{1}-z_{1}}{x_{0}}\right)
z^{l}(x_{0}+z_{1})^{l}e^{l_{p}(z)L^{*}(0)}Y^{*}(v,x_{1})\alpha
\nonumber\\
&=&\Res_{x_{1}}x_{0}^{-1}\delta\left(\frac{x_{1}-z_{1}}{x_{0}}\right)
z^{l}(x_{0}+z_{1})^{l}Y^{*}(z^{L(0)}v,zx_{1})e^{l_{p}(z)L^{*}(0)}\alpha
\nonumber\\
&=&\Res_{x_{1}}z^{-1}x_{0}^{-1}
\delta\left(\frac{z^{-1}x_{1}-z_{1}}{x_{0}}\right)
z^{l}(x_{0}+z_{1})^{l}Y^{*}(z^{L(0)}v,x_{1})e^{l_{p}(z)L^{*}(0)}\alpha
\nonumber\\
&=&\Res_{x_{1}}(zx_{0})^{-1}
\delta\left(\frac{x_{1}-zz_{1}}{zx_{0}}\right)
z^{l}(x_{0}+z_{1})^{l}Y^{*}(z^{L(0)}v,x_{1})e^{l_{p}(z)L^{*}(0)}\alpha
\nonumber\\
&=&z^{l}(x_{0}+z_{1})^{l}
Y^{L}_{P(zz_{1})}(z^{L(0)}v,zx_{0})e^{l_{p}(z)L^{*}(0)}\alpha.\end{aligned}$$ Multiplying by $z^{-l}(z_{1}+x_{0})^{-l}$ we obtain (\[eisom2\]). $\;\;\;\;\Box$

By setting $v=\omega$ (the Virasoro element) in (\[eisom1\]) and (\[eisom2\]) we obtain $$\begin{aligned}
e^{l_{p}(z)L^{*}(0)}L^{R}_{P(z_{1})}(0)
&=&L^{R}_{P(zz_{1})}(0)e^{l_{p}(z)L^{*}(0)},\\
e^{l_{p}(z)L^{*}(0)}L^{L}_{P(z_{1})}(0)
&=&L^{L}_{P(zz_{1})}(0)e^{l_{p}(z)L^{*}(0)}.\end{aligned}$$ Consequently, $e^{l_{p}(z)L^{*}(0)}$ preserves the weight subspaces. By (\[eisom1\]) and (\[eisom2\]) again, $e^{l_{p}(z)L^{*}(0)}$ maps an ordinary $V\otimes V$-submodule of ${\cal{D}}_{P(z_{1})}(W)$ to an ordinary $V\otimes V$-submodule of ${\cal{D}}_{P(z_{2})}(W)$. Therefore we have:

The linear map $e^{l_{p}(z)L^{*}(0)}$ maps $R_{P(z_{1})}(W)$ onto $R_{P(zz_{1})}(W)$. $\;\;\;\;\Box$

To achieve our goal we also need the following fact (cf. Remark \[rchangemodule\]):

Let $(M,Y_{M})$ be a generalized $V$-module (for now). Define $$Y_{M}^{z}(v,x)=Y_{M}(z^{L(0)}v,zx)$$ for $v\in V$. Then $(M,Y_{M}^{z}(\cdot,x))$ is also a generalized $V$-module and $e^{l_{p}(z)L(0)}$ is a $V$-isomorphism from $(M,Y_{M}(\cdot,x))$ onto $(M,Y_{M}^{z}(\cdot,x))$.

Using the conjugation formula (\[L(0)con\]) for $Y_{M}$, we get $$e^{L(0)\log (z)}Y_{M}(v,x)e^{-L(0)\log (z)}
=Y_{M}(z^{L(0)}v,zx)=Y_{M}^{z}(v,x).$$ Then $(M,Y_{M}^{z}(\cdot,x))$ carries the structure of a generalized $V$-module, which is the transported structure from $(M,Y_{M}(\cdot,x))$ through the linear isomorphism $e^{l_{p}(z)L(0)}$, and furthermore, $e^{l_{p}(z)L(0)}$ is a $V$-isomorphism from $(M,Y_{M})$ onto $(M,Y_{M}^{z})$.$\;\;\;\;\Box$

Now we are ready to prove our last result:

Let $W$ be a $V$-module, $z, z_{1}$ be nonzero complex numbers and let $p\in {\Z}$. Then the linear map $$\begin{aligned}
\label{eisopz}
\sigma_{(p,z,z_{1})}:=
e^{-l_{p}(z)L^{L}_{P(zz_{1})}(0)}e^{-l_{p}(z)L^{R}_{P(zz_{1})}(0)}
e^{l_{p}(z)L^{*}(0)}\end{aligned}$$ is a $V\otimes V$-isomorphism from $R_{P(z_{1})}(W)$ onto $R_{P(zz_{1})}(W)$.

[**Proof.**]{} By definition, $$\begin{aligned}
Y_{P(z')}(u\otimes v,x)=Y^{L}_{P(z')}(u,x)Y^{R}_{P(z')}(v,x)\end{aligned}$$ for $u,v\in V$ and for any nonzero complex number $z'$. Using Corollary \[crz12\], Proposition \[pisom\] and Lemma \[lfact1\] we get $$\begin{aligned}
\sigma_{(p,z,z_{1})}Y_{P(z_{1})}(u\otimes v,x)
&=&\sigma_{(p,z,z_{1})}Y_{P(z_{1})}^{L}(u, x)Y_{P(z_{1})}^{R}(v,x)
\nonumber\\
&=&Y_{P(zz_{1})}^{L}(u,x)Y_{P(zz_{1})}^{R}(v,x)\sigma_{(p,z,z_{1})}
\nonumber\\
&=&Y_{P(zz_{1})}(u\otimes v,x)\sigma_{(p,z,z_{1})}.
\;\;\;\;\Box\end{aligned}$$

[FLM1]{}

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[^1]: Supported by NSF grant DMS-9616630
