Ternary composition algebras II. Automorphism groups and subgroups

Abstract

We study an eight-dimensional ternary composition algebra ε = ( E , ⟨ ⟩, { }) of signature (8, 0) or (4, 4). Such an algebra is associated with an involutary outer automorphism M of the Lie algebra so ( E )of ⟨, ⟩. The automorphisms and (in the (4, 4) case) counter-automorphisms of ε are obtained in terms of the d = 7 spin group determined by M . Using M we derive a ‘principle of duplicity’ for the ternary multiplication { } (related to the well-known principle of triality for the associated octonionic binary multiplication). However, rather than construct ε out of the octonions as done (in effect) by previous authors, we give a four-dimensional ‘complex’ construction of ε . This non-octonionic view of ε highlights certain (15-dimensional) d = 6 spin groups, rather than the customary (14-dimensional) octonionic automorphism groups. In the (4, 4) case we can choose to interpret ‘complex’ in terms of the split complex numbers, and are thereby led to consider the subgroup chain SO + (4, 4) ⊃ Spin + (3, 4) ⊃ SL (4; ℝ) (⋍ Spin + (3, 3)) instead of the chain SO + (4, 4) ⊃ Spin + (3, 4) ⊃ SU (2, 2) (⋍ Spin + (2, 4)).