$(G,{\chi }_{\varphi })$-equivariant $\varphi$-coordinated modules for vertex algebras

Abstract

To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,{\chi }_{\varphi })$-equivariant $\varphi$-coordinated quasi modules for vertex algebras, where $G$ is a group with ${\chi }_{\varphi }$ a linear character of $G$ and $\varphi$ is an associate of the one-dimensional additive formal group. The theory of $(G,{\chi }_{\varphi })$-equivariant $\varphi$-coordinated quasi modules for nonlocal vertex algebra is established in \cite{JKLT}. 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 $(G,{\chi }_{\varphi })$-equivariant $\varphi$-coordinated quasi modules. Furthermore, for any conformal algebra $\mathcal{C}$, we construct a class of Lie algebras ${\hat{\mathcal{C}}}_{\varphi }[G]$ and prove that restricted ${\hat{\mathcal{C}}}_{\varphi }[G]$-modules are exactly $(G,{\chi }_{\varphi })$-equivariant $\varphi$-coordinated quasi modules for the universal enveloping vertex algebra of $\mathcal{C}$. As an application, we determine the $(G,{\chi }_{\varphi })$-equivariant $\varphi$-coordinated quasi modules for affine and Virasoro vertex algebras.

31 pages

References in corpus (2)

• Extended affine Lie algebras, vertex algebras, and reductive groups
• Toroidal extended affine Lie algebras and vertex algebras