q-Virasoro algebra and affine Kac-Moody Lie algebras

In this paper, we introduce an infinite-dimensional Lie algebra D_S for any abelian group S. If S is the additive group of integers, D_S reduces to the q-Virasoro algebra D_q 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 g_S with S as an automorphism group and we prove that D_S is isomorphic to the S-covariant algebra of the affine Lie algebra g_Sˆ. We then relate restricted D_S-modules of level ℓ∈C to equivariant quasi modules for the vertex algebra V_gSˆ(ℓ,0) associated to g_Sˆ with level ℓ. Furthermore, we establish an intrinsic connection between the q-Virasoro algebra D_q and affine Kac-Moody Lie algebras. More specifically, we show that if S is a finite abelian group of order 2l+1, D_S is isomorphic to the affine Kac-Moody algebra of type B_l ⁽¹⁾.