A new construction of vertex algebras and quasi modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasi module for vertex algebras is introduced and studied. More specifically, a notion of quasi local subset(space) of Hom (W,W ((x))) for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasi local subspace there exists a natural vertex algebra structure and that any quasi local subset of Hom (W,W ((x))) generates a vertex algebra. Furthermore, a notion of quasi module for a vertex algebra is introduced and it is proved that W is a quasi module for each of the vertex algebras generated by quasi local subsets of Hom (W,W ((x))). A notion of Γ-vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C× of nonzero complex numbers. It is proved that any maximal quasi local subspace of Hom (W,W ((x))) is naturally a Γ-vertex algebra and that any quasi local subset of Hom (W,W ((x))) generates a Γvertex algebra. It is also proved that a Γ-vertex algebra exactly amounts to a vertex algebra equipped with a Γ-module structure which satisfies a certain compatibility condition. Finally, three families of examples are given, involving twisted affine Lie algebras, certain quantum Heisenberg algebras and certain quantum torus Lie algebras.