Geometry of planes over nonassociative algebras
type: Item
instance of: scholarly article
title: Geometry of planes over nonassociative algebras
zbMATH Open document ID: 0893.22006
DOI: 10.1023/A:1005871202247
published in: Acta Applicandae Mathematicae
publication date: 1998-07-26T00:00:00Z
review text: It is a well known fact that the real simple Lie groups of type \(A_n,\dots,D_n\) admit interpretations as collineation groups of geometries defined over associative algebras. In 1951, H. Freudenthal gave a geometric interpretation of the compact real Lie group of type \(F_4\) and of the non-compact real Lie group of type \(E_6(-26)\): the latter group turns out to be the group of collineations of the octonion projective plane \(P_2O\), while the first group is the group of motions of \(P_2O\) (when regarded as an elliptic plane). Freudenthal achieved this result by using the non-associative Jordan algebra \textbf{J}\(_3\) in order to describe the set of points of \(P_2O\). In the paper under review the author gives geometric interpretations in the sense of Freudenthal for the exceptional compact real Lie groups of type \(E\) and \(F\) using geometries defined over non-associative algebras. No proofs are presented. However, proofs are included in two other papers of the author [Akad. Nauk Azerbajdzh. SSR, Dokl. 10, 829-833 (1954; Zbl 0059.14002); Dokl. Akad. Nauk SSSR 106, 600-603 (1956; Zbl 0074.15301)].
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 51N30
Mathematics Subject Classification ID: 51C05
zbMATH DE Number: 1139654
zbMATH Keywords: octonions
zbMATH Keywords: octonion plane
zbMATH Keywords: exceptional Lie group
zbMATH Keywords: real simple Lie groups
zbMATH Keywords: collineation groups of geometries
author: Boris A. Rosenfeld
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1139654