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

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

Introduction
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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 