Q-Virasoro algebra and affine Kac-Moody Lie algebras

Abstract

In this paper, we introduce an infinite-dimensional Lie algebra DS for any abelian group S. If S is the additive group of integers, DS reduces to the q-Virasoro algebra Dq introduced by Belov and Chaltikian in the study of lattice conformal theories. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra gS with S as an automorphism group and we prove that DS is isomorphic to the S-covariant algebra of the affine Lie algebra gSˆ. We then relate restricted DS-modules of level ℓ∈C to equivariant quasi modules for the vertex algebra VgSˆ(ℓ,0) associated to gSˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q-Virasoro algebra Dq and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2l+1, DS is isomorphic to the affine Kac-Moody algebra of type Bl(1).