Extensions of the Jacobi identity for generalized vertex algebras
The Jacobi identity for vertex algebras relates compositions of vertex operators \(Y(Y(u, z_0) v,z_2)\), say \(Y_2 \circ Y_0\) for short, with products of vertex operators such as \(Y(u, z_1) Y(v, z_2)\), say \(Y_1 \cdot Y_2\) for short. In [Mem. Am. Math. Soc. 507 (1993; Zbl 0798.17013)] the author has extended the Jacobi identity for vertex algebras to multi-operator identities for expressions of the form \(Y_1 \circ \cdots \circ Y_m \circ (Y_{m + 1} \cdot \dots \cdot Y_n)\). In the paper under review this result is extended further for generalized vertex algebras [cf. \textit{C. Dong} and \textit{J. Lepowsky}, Generalized vertex algebras and relative vertex operators, Prog. Math. 112, Birkhäuser, Boston (1993; Zbl 0803.17009)]. This work is motivated by vertex operator constructions of representations of affine Lie algebras and the corresponding \(Z\)-algebras [cf. \textit{C. Husu}, J. Pure Appl. Algebra 98, 163-187 (1995; Zbl 0837.17013)].
scientific article; zbMATH DE number 858638
scientific article