Introduction to vertex operator algebras and their representations
type: Item
instance of: scholarly article
title: Introduction to vertex operator algebras and their representations
zbMATH Open document ID: 1055.17001
author: J. Lepowsky
author: Haisheng Li
published in: Progress in Mathematics
publication date: 2004-03-25T00:00:00Z
review text: After \textit{R. E. Borcherds}' seminal contribution [``Vertex algebras, Kac-Moody algebras, and the Monster'', Proc. Natl. Acad. Sci. USA 84, 3068--3071 (1986; Zbl 0613.17012)] several books and reviews have been written about vertex algebras, vertex operator algebras and related structures. In the retrospective the most important and by far the most exciting aspects of this theory were obtained by \textit{I. Frenkel, J. Lepowsky} and \textit{A. Meurman}, in [Vertex operator algebras and the Monster (Pure and Applied Mathematics 134) (Boston: Academic Press) (1988; Zbl 0674.17001)], where vertex operator algebra structure on \(V^\natural\) (the Moonshine Module) was used in an essential way for a construction of the largest sporadic finite simple group, the Monster! Even though considered a major breakthrough the book needed a proper algebraic and axiomatic framework. That was partially achieved by \textit{I. Frenkel, Y.-Z. Huang} and \textit{J. Lepowsky} in [On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc. 494 (1993; Zbl 0789.17022)], where among other things, the authors introduced the notion of module for vertex operator algebras. Another circle of ideas, also based on Frenkel, Lepowsky and Meurman's construction, led to a generalization of the notion of vertex operator algebra [cf. \textit{C. Dong} and \textit{J. Lepowsky}, Generalized vertex algebras and relative vertex operators (Prog. Math. 112) (Boston: Birkhäuser) (1993; Zbl 0803.17009)]. A few years later Kac [cf. \textit{V. Kac}, Vertex algebras for beginners (University Lecture Series 10) (Providence: AMS) (1996; Zbl 0861.17017), 2nd ed. (1998; Zbl 0924.17023)] offered yet another approach to vertex algebras which largely overlaps with Frenkel, Huang and Lepowsky's book and eventually led to the theory of \textit{conformal algebras}. These structures are in many ways closer to Lie algebras rather than vertex operator algebras. Most recently [cf. Vertex algebras and al…
reviewed by: A. Milas
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 2061773
zbMATH Keywords: vertex algebras
zbMATH Keywords: vertex operator algebras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2061773