Root systems and orthogonal groups of root lattices
type: Item
instance of: scholarly article
title: Root systems and orthogonal groups of root lattices
zbMATH Open document ID: 0543.17007
author: Jun Morita
published in: Hokkaido Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: For an indecomposable symmetrizable generalized Cartan matrix of spherical, Euclidean or hyperbolic (more appropriately, minimal hyperbolic) type with corresponding (generalized) root system \(\Delta\), the lattice \(\Gamma\) generated by \(\Delta\) is equipped with a canonical quadratic form. In the spherical case only for type \(C_4\) is the orthogonal group \(O(\Gamma)\) not generated by the Weyl group, the diagram automorphisms and the multiplication by \(-1\), but for the other cases this happens more often. As main theorem it is proved that the union of the images of \(\Delta\) under the elements of \(O(\Gamma)\) is again a root system; for several types the Cartan matrix of this root system is determined.
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B22
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 3863425
zbMATH Keywords: generalized Cartan matrix
zbMATH Keywords: spherical
zbMATH Keywords: Euclidean
zbMATH Keywords: hyperbolic
zbMATH Keywords: root system
zbMATH Keywords: orthogonal group
reviewed by: H. de Vries
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381757735
OpenAlex ID: W2145569187
DOI: 10.14492/HOKMJ/1381757735
description: scientific article; zbMATH DE number 3863425