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 $\text{Hom}(W,W((x)))$ 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 $\Hom (W,W((x)))$\mathrm{.}Anotionof$\Gamma$-vertex algebra is also introduced and studied, where $\mathrm{\Gamma }$ is a subgroup of the multiplicative group ${\text{C}}^{\times }$ of nonzero complex numbers. It is proved that any maximal quasi local subspace of $\text{Hom}(W,W((x)))$ is naturally a $\mathrm{\Gamma }$-vertex algebra and that any quasi local subset of $\text{Hom}(W,W((x)))$ generates a $\mathrm{\Gamma }$-vertex algebra. It is also proved that a $\mathrm{\Gamma }$-vertex algebra exactly amounts to a vertex algebra equipped with a $\mathrm{\Gamma }$-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.