q-Virasoro algebra and affine Kac-Moody Lie algebras

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 $B^{(1)}_{l}$.