Automorphism group of parafermion vertex operator algebras
type: Item
instance of: scholarly article
title: Automorphism group of parafermion vertex operator algebras
zbMATH Open document ID: 1395.17070
author: Qing Wang
published in: Journal of Pure and Applied Algebra
publication date: 2015-10-27T00:00:00Z
review text: Fix an integer \(k\geq 2\). Let \(\mathcal{L}(k,0)\) be the level \(k\) integrable highest weight module for the affine Kac-Moody algebra \(\widehat{\mathfrak{sl}}_3\). Recall that \(\mathcal{L}(k,0)\) possesses a vertex operator algebra structure and that the Cartan subalgebra \(\mathfrak{h}\) of \(\mathfrak{sl}_3\) generates the Heisenberg vertex operator subalgebra \(M(k)\) of \(\mathcal{L}(k,0)\). The parafermion vertex operator algebra \(K(\mathfrak{sl}_3,k)\) associated with \(\mathcal{L}(k,0)\) is defined as the commutant of \(M(k)\) in \(\mathcal{L}(k,0)\), i.e. \[ K(\mathfrak{sl}_3,k) =\left\{v\in \mathcal{L}(k,0)\,:\,h(n)v=0\text{ for all }h\in\mathfrak{h},\,n\geq 0\right\}. \] In this paper, the automorphism group \(\Aut(K(\mathfrak{sl}_3,k))\) of the parafermion vertex operator algebra \(K(\mathfrak{sl}_3,k)\) is determined. More precisely, it is proved that \(\Aut(K(\mathfrak{sl}_3,k))\) is isomorphic to the automorphism group of the root system of the simple Lie algebra \(\mathfrak{sl}_3\), i.e. \[ \text{Aut}(K(\mathfrak{sl}_3,k))\cong S_3\rtimes \mathbb{Z}/2\mathbb{Z}\quad \text{for all }k\geq 3. \] The proof relies on the integrable representation theory of the affine Kac-Moody algebras.
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B65
zbMATH DE Number: 6499129
zbMATH Keywords: parafermion vertex operator algebra
zbMATH Keywords: vertex operator
zbMATH Keywords: vertex operator algebra
zbMATH Keywords: affine Kac-Moody algebra
zbMATH Keywords: highest weight module
reviewed by: Slaven Kožić
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.06.001
OpenAlex ID: W1422971662
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DOI: 10.1016/J.JPAA.2015.06.001
description: scientific article; zbMATH DE number 6499129