Axiomatic vertex algebras and the Jacobi identity
type: Item
instance of: scholarly article
title: Axiomatic vertex algebras and the Jacobi identity
zbMATH Open document ID: 0827.17029
author: Gunnar Mossberg
published in: Journal of Algebra
publication date: 1995-12-17T00:00:00Z
review text: 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.
reviewed by: Mirko Primc
Mathematics Subject Classification ID: 17B69
zbMATH DE Number: 705012
zbMATH Keywords: vertex operator algebra
zbMATH Keywords: axiomatic theory
zbMATH Keywords: bigraded space
zbMATH Keywords: generalized Jacobi identity
MaRDI profile type: Publication
full work available at URL: jabr.1994.1372
OpenAlex ID: W2014315704
DOI: 10.1006/JABR.1994.1372
Recommended article: Q3994993
Recommended article: On axiomatic approaches to vertex operator algebras and modules
Recommended article: Q4888127
Recommended article: A universal approach to vertex algebras
Recommended article: Q4395053
Recommended article: Q4839457
Recommended article: Bombay lectures on highest weight representations of infinite dimensional Lie algebras
Recommended article: On Griess algebras
Recommended article: Q4395054
Recommended article: On certain vertex algebras and their modules associated with vertex algebroids
description: scientific article; zbMATH DE number 705012