Representation theory of the vertex algebra \(W_{1+\infty}\)
type: Item
instance of: scholarly article
title: Representation theory of the vertex algebra \(W_{1+\infty}\)
zbMATH Open document ID: 0862.17023
DOI: 10.1007/BF02587735
author: Victor G. Kac
author: Andrey Radul
published in: Transformation Groups
publication date: 1996-11-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/9512150
review text: Representations of the (unique) central extension \(\widehat {\mathcal D}\) of the Lie algebra of differential operators (with a finite number of Fourier modes) on the circle are studied. The authors together with other collaborators began these studies in the articles [\textit{V. Kac} and \textit{A. Radul}, Commun. Math. Phys. 157, 429-457 (1993; Zbl 0826.17027)] and [\textit{E. Frenkel}, \textit{V. Kac}, \textit{A. Radul} and \textit{W. Wang}, Commun. Math. Phys. 170, 337-357 (1995; Zbl 0838.17028)]. The representations are studied with the help of irreducible highest weight representations of the central extension \(\widehat {gl} (\infty)\) of the Lie algebra of infinite matrices with only finitely many diagonals and with the help of vertex algebras. For every central charge \(c\in \mathbb{C}\) there is an induced \(\widehat {\mathcal D}\)-module \(M_c\). This module admits a canonical vertex algebra structure. The unique irreducible quotient is denoted by \(W_{1+ \infty}\). The highest weight representations of the vertex algebra \(M_c\) are in canonical \(1-1\) correspondence to the highest weight representations of \(\widehat {\mathcal D}\) with central charge \(c\). The goal is to describe the representations of the simple vertex algebra \(W_{1+ \infty}\). By suitable normalisation of the defining cocycle for \(\widehat {\mathcal D}\) the representation \(M_c\) is already irreducible for \(c \notin \mathbb{Z}\). The representations with \(c= N \in \mathbb{N}_0\) were studied in the above mentioned articles. The case \(c = -N\), \(N\in \mathbb{N}\) is considered in the article under review. The main result is a decomposition of the vertex algebra of \(N\) charged free bosons with respect to the commuting pair of Lie algebras \(gl(N)\) and \(\widehat {gl} (\infty)\). In this way a large class of irreducible modules are produced. The authors conjecture that all irreducible modules can be obtained by applying certain constructions to them. Explicit character form…
reviewed by: M. Schlichenmaier
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 934932
zbMATH Keywords: Verma modules
zbMATH Keywords: central extension
zbMATH Keywords: Lie algebra of differential operators
zbMATH Keywords: highest weight representations
zbMATH Keywords: vertex algebras
MaRDI profile type: Publication
arXiv ID: hep-th/9512150
cites work: Character and determinant formulae of quasifinite representation of the \(W_{1+\infty}\) algebra
cites work: From the \(w_\infty\)-algebra to its central extension: a \(\tau\)-function approach
cites work: Semi-infinite cohomology in conformal field theory and 2D gravity
cites work: Vertex algebras, Kac-Moody algebras, and the Monster
cites work: Monstrous moonshine and monstrous Lie superalgebras
cites work: Q4065692
cites work: Q3798824
cites work: \({\mathcal W}_{1+\infty}\) and \({\mathcal W}(gl_ N)\) with central charge \(N\)
cites work: Q3994993
cites work: Q4011453
cites work: Q3223062
cites work: Remarks on Classical Invariant Theory
cites work: Q4040877
cites work: Spin and wedge representations of infinite-dimensional Lie algebras and groups
cites work: Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
cites work: On the Segal-Shale-Weil representations and harmonic polynomials
cites work: 2-cocycles on the algebra of differential operators
cites work: Q3947818
cites work: Free fields and quasi-finite representations of \(W_{1+{}\infty}\) algebra
cites work: Q5635639
description: scientific article