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, Poincare 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.