A new construction of vertex algebras and quasi-modules for vertex algebras. (Q2490798)

Many examples of vertex operator algebras have been constructed by using infinite dimensional Lie algebras such as affine Lie algebras and Virasoro algebras \textit{I. B. Frenkel} and \textit{Y. Zhu} [Duke Math. J. 66, 123--168 (1992; Zbl 0848.17032)]. The construction is based on the notion of locality of vertex operators on highest weight modules \textit{H.-S. Li} [J. Pure Appl. Algebra 109, 143--195 (1996; Zbl 0854.17035)]. In the paper under review the author studies quasilocality of vertex operators, which is a generalization of the ordinary locality. A notion of quasimodules for vertex algebras is also introduced. The author defines a new \(n\)-th product \(a(x)_n b(x)\) for mutually quasilocal vertex operators \(a(x)\) and \(b(x)\) on a vector space \(W\). It is shown that any quasilocal set of vertex operators on \(W\) generates a vertex algebra with respect to the \(n\)-th product and \(W\) is a quasimodule for the vertex algebra. The motivation of the work is to establish a connection between a wider class of infinite dimensional Lie algebras and vertex algebras. Two families of examples of quasimodules for vertex algebras are presented. One is related to modules for twisted affine Lie algebras and the other is related to modules for quantum torus Lie algebras.