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

Abstract

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal 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 quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of Hom ( W , W ( ( x ) ) ) generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal 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 quasilocal subspace of Hom ( W , W ( ( x ) ) ) is naturally a Γ -vertex algebra and that any quasilocal subset of Hom ( W , W ( ( x ) ) ) generates a Γ -vertex 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, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.