Axiomatic vertex algebras and the Jacobi identity
A vertex operator algebra (VOA) is a \(\mathbb{Z}\)-graded vector space \(V\) equipped with a linear map \(Y\) which associates to each vector \(v\) a formal Laurent series \(Y(v, z)\) and with two distinguished vectors \textbf{1} and \(\omega\), satisfying several conditions, one of which is the Jacobi identity [cf. \textit{I. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman}, Vector operator algebras and the Monster (Pure and Applied Math. 134) (Academic Press 1988; Zbl 0674.17001)]. In the paper under review the author develops an axiomatic theory of vertex algebras in a more general setting, with a bigraded space and a generalized Jacobi identity. He introduces different properties \(({\mathcal A})-({\mathcal F})\) of the map \(Y\) and vectors \textbf{1} and \(\omega\), and studies their relationship and consequences. The main result is that the (generalized) commutativity property for \(Y\) implies the (generalized) Jacobi identity. In the second part of the paper the author presents a nontrivial example of the (generalized) VOA structure: the vertex operator algebra \(V_A\) constructed from a free abelian group \(A\) of finite rank, a generalization of \(V_L\) in the reference cited above.
scientific article; zbMATH DE number 705012