On the triality of the Hamming code vertex operator algebra
type: Item
instance of: scholarly article
title: On the triality of the Hamming code vertex operator algebra
zbMATH Open document ID: 0978.17022
author: Mika Matsuo
published in: Journal of Algebra
publication date: 2002-02-07T00:00:00Z
review text: \textit{M. Miyamoto} introduced the notion of code vertex operator algebra [J. Algebra 181, 207-222 (1996; Zbl 0857.17026)]. The code vertex operator algebra \(V_D\) associated to an even binary linear code \(D\) is a sum of tensor products of the Ising model, the irreducible unitary highest weight representations of the Virasoro algebra of central charge \(1/2\), where the number of the factors is the length of the code \(D\), and the summands are in one-to-one correspondence with \(D\) by forming the tensor product of \(L(1/2,0)\)'s and \(L(1/2, 1/2)\)'s according codewords. Moreover, Miyamoto proved that there is an extra involution automorphism for each conformal vector in such an algebra. Denote by \(H_8\) the \([8,4,4]\) extended binary Hamming code and by \(V(D_4)\) the vertex operator algebra associated with the root lattice of the simple Lie algebra of type \(D_4\). The author of this paper constructs an explicit embedding of \(V_{H_8}\) into \(V(D_4)\) so that the extra involutions are obtained by restricting the automorphism of \(V(D_4)\) that corresponds to the classical triality of \(D_4\).
reviewed by: Xiaoping Xu
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 94B99
zbMATH DE Number: 1526419
zbMATH Keywords: \(D_4\)-root lattice
zbMATH Keywords: extended binary Hamming code
zbMATH Keywords: vertex operator algebra
zbMATH Keywords: involutions
zbMATH Keywords: automorphism
zbMATH Keywords: triality
MaRDI profile type: Publication
full work available at URL: jabr.1999.8254
OpenAlex ID: W2042782645
cites work: Vertex algebras, Kac-Moody algebras, and the Monster
cites work: Framed vertex operator algebras, codes and the moonshine module
cites work: Q4308239
cites work: Q3994993
cites work: Q5185738
cites work: Vertex operator algebras associated to representations of affine and Virasoro algebras
cites work: Unitary representations of the Virasoro and super-Virasoro algebras
cites work: Binary codes and vertex operator (super)algebras
cites work: Q4277369
DOI: 10.1006/JABR.1999.8254
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description: scientific article; zbMATH DE number 1526419