Associating vertex algebras with the unitary Lie algebra

In this paper, we associate vertex algebras and their two different kinds of module categories with the unitary Lie algebra uˆN(CΓ˜) for N≥2 being a positive integer and Γ˜={qn|n∈Z}, where the nonzero complex number q is not a root of unity. It is proved that for any complex number ℓ, the category of restricted uˆN(CΓ˜)-modules of level ℓ is canonically isomorphic to the category of quasi modules for certain vertex algebra. And we also prove that the category of restricted uˆN(CΓ˜)-modules of level ℓ is isomorphic to the category of Γ-equivariant ϕ-coordinated quasi modules for the same vertex algebra, where Γ is an automorphism group of this vertex algebra.