Vertex algebras and extended affine Lie algebras coordinated by rational quantum tori

Abstract

Let slNˆ(Cq) be the core of extended affine Lie algebra of type AN−1 coordinated by the rational quantum 2-torus Cq. In this paper, we first prove that for any complex number ℓ, the category of restricted slNˆ(Cq)-modules of level ℓ is canonically isomorphic to the category of twisted modules for the vertex algebra VCgˆ(ℓ,0) arising from a conformal Lie algebra Cg, where Cgˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer ℓ, the integrable restricted slNˆ(Cq)-modules of level ℓ are exactly the twisted modules for the quotient vertex algebra LCgˆ(ℓ,0) of VCgˆ(ℓ,0). Finally, we classify irreducible graded twisted LCgˆ(ℓ,0)-modules.