Relation between the Poincaré Group and the Exceptional Lie Group G₂

Abstract

THERE is a correspondence, L₃↔P₅, between the line geometry on a three-manifold M₃ and that of planes on M₅ (ref. 1). Because, essentially, of the role of quaternions and octonions as co-ordinate rings here, this correspondence is unique². It can be expressed in terms of the groups which act on manifolds, with dimensions, respectively, of the following elements: and is (the middle group being in each case the conformal group): Thus, instead of B₂ (de Sitter group and, by restriction to four-space, Poincaré group), on the right-hand side one finds the exceptional G₂. Behind this is the fact that B₃, D₄, though classical groups, admit exceptionally an outer automorphism of order three, generally known as ‘triality’³.