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Description

In this talk, I will discuss a natural connection of a certain $q$-Virasoro algebra with affine Lie algebras and vertex algebras. To any abelian group $S$ with a linear character, we associate an infinite-dimensional Lie algebra $D_{S}$. When $S=\Z$ with $\chi$ defined by $\chi(n)=q^{n}$ with $q$ a nonzero complex number, $D_{S}$ reduces to the $q$-Virasoro algebra $D_{q}$ which was introduced in \cite{BC}. We also introduce a 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 $\widehat{\g_{S}}$. Then we relate restricted $D_{S}$-modules of level $\ell\in \C$ with equivariant quasi modules for the vertex algebra $V_{\widehat{\g_{S}}}(\ell,0)$. 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 $B^{(1)}_{l}$. This talk is based on a joint work with Hongyan Guo, Shaobin Tan and Qing Wang.