A theory of tensor products for module categories for a vertex operator algebra. I (Q1915871)

This is the first part of the authors' series of works on a tensor theory for modules of a vertex operator algebra. In this paper, the authors give the definitions of \(P(z)\)-tensor product and \(Q(z)\)-tensor product of two modules for a vertex operator algebra, where \(z\) is a nonzero complex number and \(P(z)\) and \(Q(z)\) are two particular elements, depending on \(z\), of a certain moduli space of spheres with punctures and local coordinates. Two constructions of \(Q(z)\)-tensor product are given for a vertex operator algebra whose module category is closed under a certain operation. In the development, the authors introduce affinizations of vertex operator algebras and the \(*\)-operation. [For the reviews of Parts II--IV, cf. the following review (Zbl 0854.17033)].