Classification of Ising vectors in the vertex operator algebra \(V_L^+\)
type: Item
instance of: scholarly article
title: Classification of Ising vectors in the vertex operator algebra \(V_L^+\)
zbMATH Open document ID: 1270.17016
author: Hiroki Shimakura
published in: Pacific Journal of Mathematics
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.6107
full work available at URL: p11.xhtml
reviewed by: Ching Hung Lam
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 6083929
zbMATH Keywords: lattice vertex oeprator algebras
zbMATH Keywords: Ising vectors
zbMATH Keywords: roots
MaRDI profile type: Publication
OpenAlex ID: W2042410054
arXiv ID: 1202.6107
DOI: 10.2140/PJM.2012.258.487
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review text: Let \(V=\bigoplus_{n=0}^\infty V_n\) be a vertex operator algebra with \(\dim V_0=1\). An element \(e\in V_2\) is called an Ising vector if the sub vertex algebra generated by \(e\) is isomorphic to the simple Virasoro vertex operator algebra \(L(1/2,0)\) of central charge \(1/2\).NEWLINENEWLINEIn this article, the author classifies all Ising vectors in the lattice type vertex operator algebra \(V_L^+\) associated to an even positive definite lattice \(L\) with no roots, i.e., the set \(L_2=\{ \alpha \in L| \langle \alpha ,\alpha\rangle =2\}=\emptyset\}\). The main theorem is as follows.NEWLINENEWLINETheorem: Let \(L\) be an even lattice without roots and \(e\) an Ising vector in \(V_L^+\). There is a sublattice \(U\) of \(L\) isomorphic to \(\sqrt{2}A_1\) or \(\sqrt{2}E_8\) such that \(e \in V_U^+\).NEWLINENEWLINEThe main idea of the proof is to show a certain sublattice \(N\) constructed by Ising vector \(e\) is doubly even, i.e., \(N/\sqrt{2}\) is even, and \(e\in V_N^+\). Then the main theorem follows from a result of [\textit{C. H. Lam, S. Sakuma} and \textit{H. Yamauchi}, Math. Z. 255, No. 3, 597--626 (2007; Zbl 1139.17010)].
description: scientific article; zbMATH DE number 6083929