ABSTRACT
The search for interesting unified theories, like the SU(5) and SO(10) theories, leads naturally to an interesting theory based on the algebra, E6, one of the exceptional Lie algebras. The exceptional algebras are associated with the octonians, a peculiar set of objects of the form https://www.w3.org/1998/Math/MathML">a+bαiα https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429499210/072e8f02-fd02-46a7-9aee-b57ab6049c01/content/eqn1167.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where a and the bα for α = 1 to 7 are real numbers. The iα are a set of seven “imaginary units” which are generalizations of i and of the quaternions, https://www.w3.org/1998/Math/MathML">j→https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429499210/072e8f02-fd02-46a7-9aee-b57ab6049c01/content/ieq0245.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>. The iα have the following multiplication law: https://www.w3.org/1998/Math/MathML">iαiβ=−δαβ+gαβγ iγ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429499210/072e8f02-fd02-46a7-9aee-b57ab6049c01/content/eqn1168.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where gαβγ is completely antisymmetric. In some basis, g is https://www.w3.org/1998/Math/MathML">g123=g247=g451=g562=g634=g375=g716=1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429499210/072e8f02-fd02-46a7-9aee-b57ab6049c01/content/eqn1169.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with all other components either zero or obtainable from (27.3) by antisymmetrization. This multiplication law can be obtained from the following pic-ture: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429499210/072e8f02-fd02-46a7-9aee-b57ab6049c01/content/eq0148.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The seven sets {jkℓ} for which gjkℓ is not zero are obtained by rotating the doted triangle, with the cyclic order of the indices maintained.
