(G,χ)-equivariant ϕ-coordinated quasi modules for vertex algebras

To give a unified treatment on the association of Lie algebras and vertex algebras, we study ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for vertex algebras, where G is a group with χ ϕ a linear character of G and ϕ is an associate of the one-dimensional additive formal group. The theory of ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for nonlocal vertex algebra is established in [10] . In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules. Furthermore, for any conformal algebra C , we construct a class of Lie algebras C ˆ ϕ [ G ] and prove that restricted C ˆ ϕ [ G ] -modules are exactly ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for the universal enveloping vertex algebra of C . As an application, we determine the ( G , χ ϕ ) -equivariant ϕ-coordinated quasi modules for affine and Virasoro vertex algebras.