A uniform construction of the root lattices \(E_ 6,E_ 7,E_ 8\) and their dual lattices
type: Item
instance of: scholarly article
title: A uniform construction of the root lattices \(E_ 6,E_ 7,E_ 8\) and their dual lattices
zbMATH Open document ID: 0854.17008
author: Tetsuji Shioda
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-01-22T00:00:00Z
review text: A simple construction of the root lattice \(E_r\) and its dual lattices \(E^*_r\) is given, which works uniformly with respect to the rank \(r= 6, 7, 8\). This construction is inspired by the classical theory of del Pezzo surfaces and the viewpoint of Mordell-Weil lattices. Let \(L_r\) be a free \(\mathbb{Z}\)-module of rank \(r= 6, 7, 8\), generated by \(r\) elements \(u_1, \dots, u_r\), and define a symmetric bilinear pairing on \(L_r\) by the rule \((u_i, u_j)= \delta_{ij}+ {1\over {9-r}}\). Let \(\widetilde {L}_r\) be the \(\mathbb{Z}\)-submodule of \(L_r \otimes Q\) generated by \(L_r\) and element \(v_0= {1\over 3} \sum^r_{i=1} u_i\). Let \(L^0_r\) be the sublattice of \(\widetilde {L}_r\) generated by \(r\) elements \(\alpha_i= u_i- u_j\) \((i=1, \dots, r-1)\) and \(\beta= v_0- u_1- u_2- u_3\). Then the lattice \(L^0_r\) is isomorphic to the root lattice \(E_r\) and \(\widetilde {L}_r\) is isomorphic to the dual lattice \(E^*_r\) of \(E_r\). The set \(\{\alpha_i, \beta\}\) forms a basis of the root system of type \(E_r\). An advantage of this construction is that the description of the minimal vectors in \(E_r\) and \(E^*_r\) is reasonably concise.
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 897671
zbMATH Keywords: construction
zbMATH Keywords: root lattice
zbMATH Keywords: dual lattices
zbMATH Keywords: del Pezzo surfaces
zbMATH Keywords: Mordell-Weil lattices
zbMATH Keywords: root system
reviewed by: Kanat Abdukhalikov
MaRDI profile type: Publication
full work available at URL: pjaa.71.140
OpenAlex ID: W1996007110
cites work: Q3941610
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cites work: Construction of elliptic curves with high rank via the invariants of the Weyl groups
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DOI: 10.3792/PJAA.71.140
description: scientific article; zbMATH DE number 897671