Self-dual lattices of type \(A\)
type: Item
instance of: scholarly article
title: Self-dual lattices of type \(A\)
zbMATH Open document ID: 0848.17034
DOI: 10.1007/BF02392488
author: Xiaoping Xu
published in: Acta Mathematica
publication date: 1995-11-22T00:00:00Z
review text: The author generalizes \textit{I. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman}'s construction [Proc. Natl. Acad. Sci. USA 81, 3256-3260 (1984; Zbl 0543.20016) and Vertex Operator Algebras and the Monster, Academic Press (1988; Zbl 0674.17001)] of the moonshine module for the Monster finite simple group. A key step in those authors' work was the ``triality'' construction, which involves certain modules for a vertex operator algebra corresponding to an integral lattice formed by gluing together finitely many copies of the root lattice of type \(A_1\). This author's investigation of the existence of other vertex operator algebras with finite automorphism groups corresponds to the group-theoretic question of the existence of additional finite groups with moonshine representations analogous to the Monster's. In related papers [Commun. Algebra 22, 1281-1303 (1994; Zbl 0820.17028) and J. Pure Appl. Algebra 99, 53-111 (1995; Zbl 0838.17032)] the author has investigated central extensions of these lattices and has constructed an analogue of ``vertex operator triality'' for each self-orthogonal ternary code (that is, code over \(\mathbb{Z}_3)\) that contains \((1, \dots, 1)\). The author generalizes Frenkel, Lepowsky and Meurman's ``\(\mathbb{Z}_2\)-orbit fold technique'' to \(\mathbb{Z}_n\) for \(n \geq 2\) to produce vertex operator algebras that correspond to lattices related to finitely many copies of a root lattice of type \(A_{n - 1}\). Specifically, the paper develops two gluing techniques for constructing self-dual lattices. Given a rational lattice \(L\) (a free \(\mathbb{Z}\)-module of finite rank with a \(\mathbb{Q}\)-valued symmetric \(\mathbb{Z}\)-bilinear form), its integral dual is \(L^0 = \{y \in L_\mathbb{Q} \mid \langle y,x \rangle \in \mathbb{Z}\) for all \(x \in L\}\). The lattice is self-dual if \(L = L^0\). A linear code of length \(k\) over \(\mathbb{Z}_n\) is a \(\mathbb{Z}_n\)-submodule \({\mathcal C}\) of \(\mathbb{Z}^k_n\). For a symmetric…
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 807539
zbMATH Keywords: monster simple group
zbMATH Keywords: moonshine module
zbMATH Keywords: vertex operator algebras
zbMATH Keywords: gluing
zbMATH Keywords: self-dual lattices
zbMATH Keywords: root lattices
zbMATH Keywords: Coxeter element
reviewed by: James F. Hurley
MaRDI profile type: Publication
cites work: Q4081121
cites work: Vertex algebras, Kac-Moody algebras, and the Monster
cites work: A characterisation of Leech's lattice
cites work: Monstrous Moonshine
cites work: Twenty-three constructions for the Leech lattice
cites work: On the enumeration of lattices of determinant one
cites work: Q3973923
cites work: The algebraic structure of relative twisted vertex operators
cites work: On Integral Representations of Finite Groups
cites work: A natural representation of the Fischer-Griess Monster with the modular function J as character
cites work: Q3994993
cites work: The friendly giant
cites work: Notes on Sphere Packings
cites work: An \(E_ 8-\)approach to the Leech lattice and the Conway group
cites work: A Correlation Between PSU 4 (3), the Suzuki Group, and the Conway Group
cites work: Definite quadratische Formen der Dimension 24 und Diskriminante 1
cites work: Codes over GF(4) and complex lattices
cites work: A gluing technique for constructing relatively self-dual codes
cites work: Central extensions of self-dual type-a complex lattices1
cites work: Q4848314
cites work: Analogue of vertex operator triality for ternary codes
cites work: A lattice theoretical construction of a GAB of the Suzuki sporadic simple group
description: scientific article; zbMATH DE number 807539