Reflective Lorentzian lattices of signature \((5,1)\)
type: Item
instance of: scholarly article
title: Reflective Lorentzian lattices of signature \((5,1)\)
zbMATH Open document ID: 1415.11090
author: Ivica Turkalj
published in: Journal of Algebra
publication date: 2018-09-24T00:00:00Z
full work available at URL: http://hdl.handle.net/2003/36087
review text: A \(\mathbb{Z}\)-lattice of signature \((n,m)\) is a pair \((L,b)\), where \(L\) is a free \(\mathbb{Z}\)-module of rank \(n+m\) equipped with a symmetric bilinear form \(b : L \times L \to \mathbb{Q}\) of signature \((n,m)\). By a Lorentzian lattice people usually mean \(\mathbb{Z}\)-lattices of signature \((n,1)\). These notions can also be extended to lattices over rings of integers of totally real number fields. A lattice \(E\) like this is called reflective if its isometry group \(O^+(E)\) is generated up to finite index by reflections. Further, a positive definite lattice is called totally-reflective if every lattice in its genus has a root system of full rank. The paper under review is concerned with classification of reflective lattices of signature \((5,1)\). The author shows that every such lattice can be written as orthogonal direct sum \(^{\alpha} \mathbb{H} \perp L\), where \(^{\alpha} \mathbb{H} \cong \begin{pmatrix} 0 & \alpha \\ \alpha & 0 \end{pmatrix}\) and \(L\) is totally-reflective. The author then uses this result to prove that up to isometry there are \(80\) strongly square-free (a certain technical local condition on Jordan decomposition blocks) reflective \(\mathbb{Z}\)-lattices of signature \((5, 1)\). The author is also interested in totally-reflective lattices as objects in their own right, and obtains some classification results for these positive definite lattices as well.
reviewed by: Lenny Fukshansky
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 6942907
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: integral lattices
zbMATH Keywords: reflective lattices
zbMATH Keywords: hyperbolic reflection groups
zbMATH Keywords: Vinberg algorithm
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2754949534
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Wikidata QID: Q129617507
DOI: 10.1016/J.JALGEBRA.2018.06.013
description: scientific article; zbMATH DE number 6942907