  In this paper, we give a unified construction of vertex algebras arising from
infinite-dimensional Lie algebras, including the affine Kac-Moody algebras,
Virasoro algebras, Heisenberg algebras and their higher rank analogs, orbifolds
and deformations. We define a notion of what we call quasi vertex Lie algebra
to unify these Lie algebras. Starting from any (maximal) quasi vertex Lie
algebra $\mathfrak{g}$, we construct a corresponding vertex Lie algebra
${\mathfrak{g}}_0$, and establish a canonical isomorphism between the category
of restricted $\mathfrak{g}$-modules and that of equivariant $\phi$-coordinated
quasi $V_{{\mathfrak{g}}_0}$-modules, where $V_{{\mathfrak{g}}_0}$ is the
universal enveloping vertex algebra of ${\mathfrak{g}}_0$. This unified all the
previous constructions of vertex algebras from infinite-dimensional Lie
algebras and shed light on the way to associate vertex algebras with Lie
algebras.
