Reflective integral lattices
type: Item
instance of: scholarly article
title: Reflective integral lattices
zbMATH Open document ID: 0853.11053
author: Rudolf Scharlau
author: Britta Blaschke
published in: Journal of Algebra
publication date: 1996-07-09T00:00:00Z
review text: The root system \(R(L)\) of a positive definite integral lattice \(L\) consists of the primitive vectors in \(L\) whose associated hyperplane reflection maps \(L\) into itself. Following \textit{E. B. Vinberg} [Math. USSR, Sb. 16, 17-35 (1972); translation from Mat. Sb., Nov. Ser. 87(129), 18-36 (1972; Zbl 0244.20058)], the authors call \(L\) reflective if the sublattice generated by \(R(L)\) has the same rank as \(L\). Reflective lattices are constructed and classified by pairs \((R, C)\) where \(R\) is a ``scaled root system'' and \(C\) a ``glue code'', i.e. some subgroup of \(\langle R\rangle^*/ \langle R\rangle\). The first step in the general classification process is to determine an appropriate set of root systems \(R_0\) of rank \(n\) such that for each indecomposable \(n\)-dimensional reflective lattice \(L\) the root system \(R(L)\) contains \(R_0\). Main results of the paper are descriptions of \(R (L)\) when \(R_0\) is of type \(nA_1\) or \(mA_2\), and the classification of all reflective lattices in dimensions \(n\leq 6\).
Mathematics Subject Classification ID: 11H55
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 883795
zbMATH Keywords: integral lattice
zbMATH Keywords: root systems
zbMATH Keywords: reflective lattice
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
full work available at URL: jabr.1996.0155
OpenAlex ID: W2039463256
DOI: 10.1006/JABR.1996.0155
description: scientific article; zbMATH DE number 883795