Vertex operator algebras and associative algebras
type: Item
instance of: scholarly article
title: Vertex operator algebras and associative algebras
zbMATH Open document ID: 0911.17017
author: Chongying Dong
author: Haisheng Li
author: Geoffrey Mason
published in: Journal of Algebra
publication date: 1999-04-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/q-alg/9612010
review text: Let \(V\) be a vertex operator algebra (VOA). Then there is a one-to-one correspondence between the irreducible admissible \(V\)-modules and the irreducible modules for Zhu's algebra \(A(V)\) [cf. \textit{Y. Zhu}, J. Am. Math. Soc. 9, 237--302 (1996; Zbl 0854.17034)]. When \(V\) is a rational VOA, then \(A(V)\) is a semisimple algebra and the category of the admissible \(V\)-modules is equivalent to the category of \(A(V)\)-modules. In the present paper, the authors construct a series of associative algebras \(A_{n}\) for every \(n \in\mathbb Z_+\) such that \(A_{n} (V)\) is a quotient of \(A_{n+1} (V)\), and \(A_0 = A(V)\). They show that there is a one-to-one correspondence between the irreducible \(A_{n}(V)\)-modules which cannot factor through \(A_{n-1} (V)\) and the irreducible admissible \(V\)-modules. They prove that VOA \(V\) is rational if and only if all \(A_{n} (V)\) are finite-dimensional semisimple algebras. It seems that \(A_n(V)\)-theory can be very useful in the study of induced modules for VOAs, and the representation theory of irrational VOAs.
reviewed by: Dražen Adamović
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 16D99
zbMATH DE Number: 1214143
zbMATH Keywords: vertex operator algebras
zbMATH Keywords: associative algebras
zbMATH Keywords: irreducible representations
zbMATH Keywords: semisimple algebras
MaRDI profile type: Publication
OpenAlex ID: W2023093707
arXiv ID: q-alg/9612010
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cites work: Symmetric invariant bilinear forms on vertex operator algebras
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DOI: 10.1006/JABR.1998.7425
description: scientific article