Geometry of planes over nonassociative algebras
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)].
scientific article; zbMATH DE number 1139654