q-Virasoro algebra and affine Kac-Moody Lie algebras

Abstract

We establish a natural connection of the $q$-Virasoro algebra $D_q$ introduced by Belov and Chaltikian with affine Kac-Moody Lie algebras. More specifically, for each abelian group $S$ together with a one-to-one linear character $\chi$, we define an infinite-dimensional Lie algebra $D_S$ which reduces to $D_q$ when $S=\mathbb{Z}$. Guided by the theory of equivariant quasi modules for vertex algebras, we introduce another Lie algebra ${\mathfrak{g}}_S$ with $S$ as an automorphism group and we prove that $D_S$ is isomorphic to the $S$-covariant algebra of the affine Lie algebra $\widehat{{\mathfrak{g}}_S}$. We then relate restricted $D_S$-modules of level $\ell \in \mathbb{C}$ to equivariant quasi modules for the vertex algebra $V_{\widehat{{\mathfrak{g}}_S}}(\ell ,0)$ associated to $\widehat{{\mathfrak{g}}_S}$ with level $\ell$. Furthermore, 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 $B_l^{(1)}$.

20 pages