Analogue of vertex operator triality for ternary codes
type: Item
instance of: scholarly article
title: Analogue of vertex operator triality for ternary codes
zbMATH Open document ID: 0838.17032
DOI: 10.1016/0022-4049(93)E0185-7
author: Xiaoping Xu
published in: Journal of Pure and Applied Algebra
publication date: 1996-06-04T00:00:00Z
review text: For each self-orthogonal ternary code of length \(k\) which contains \((1, \dots, 1)\) the author constructs an integral even lattice \(L_0\) and a lattice \(L = L_{-1} \cup L_0 \cup L_1\) which is a union of cosets for \(L_0\). For each coset the author constructs one untwisted space \(V_{L_1}\) and two twisted spaces \(V_{T_{L_1}}\) and \(V_{\widetilde T_{L_1}}\); let us denote these nine spaces by \(V_0, \dots, V_8\). The author also constructs the corresponding untwisted and twisted vertex operators on \(V_{L_0} = V_0\) with coefficients in \(\text{End} V_i\), say \(Y_{V_i} (-,z)\). All nine untwisted and twisted spaces \(V_i\) are irreducible for the corresponding vertex operators and they provide different realizations of the affine Lie algebra \(\text{sl} (3, \mathbb{C})^\wedge\) [\textit{X. Xu}, Algebra Colloq. 2, 241-268 (1995; Zbl 0838.17030; see the preceding review)]. The main result of the paper under review is a construction of an order 4 linear automorphism \(\sigma\) of \(V_0 \oplus \cdots \oplus V_8\), where \(\sigma (V_0) = V_0\) and \(\sigma (V_i) = V_j\) \((j\) depends on \(i)\), such that \(\sigma Y_{V_i} (u,z) \sigma^{-1} = Y_{\sigma^{-1} V_i} (\sigma u,z)\). This is an analogue of the vertex operator triality found by \textit{I. B. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman} [Vertex operator algebras and the Monster, Pure Appl. Math. 134, Academic Press (1988; Zbl 0674.17001)].
reviewed by: Mirko Primc
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 762338
zbMATH Keywords: self-orthogonal ternary code
zbMATH Keywords: integral even lattice
zbMATH Keywords: vertex operators
MaRDI profile type: Publication
cites work: Vertex algebras, Kac-Moody algebras, and the Monster
cites work: Monstrous Moonshine
cites work: Twenty-three constructions for the Leech lattice
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cites work: A natural representation of the Fischer-Griess Monster with the modular function J as character
cites work: Q3792881
cites work: Calculus of twisted vertex operators
cites work: Q4793326
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description: scientific article; zbMATH DE number 762338