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

Abstract

Abstract Let sl N ˆ ( C q ) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q . In this paper, we first prove that for any complex number l, the category of restricted sl N ˆ ( C q ) -modules of level l is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ ( l , 0 ) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer l, the integrable restricted sl N ˆ ( C q ) -modules of level l are exactly the twisted modules for the quotient vertex algebra L C g ˆ ( l , 0 ) of V C g ˆ ( l , 0 ) . Finally, we classify irreducible graded twisted L C g ˆ ( l , 0 ) -modules.