Spinors and Octonions

Octonions are introduced through some spin representations. The groups G2, F4 and the E series appear in a natural manner; one way to understand octonions is as the “second coming” of the reals, but with the spinors instead of vectors. Some physical applications in M and F -theory as putative “theories of everything” are suggested. 1 The Seven Sphere . Consider the real, complex and quaternion numbers R, C, H . Identify the normed vector spaces R ∼= C ∼= H, and write the natural inclusion of the isometry groups O(8k) ⊃ U(4k) ⊃ Sp(2k) (1) Recall now the Spin groups, which cover the rotation groups twice, Spin(n)/Z2 = SO(n); remind only that Spin(7) has a real 8-dim irreducible representation; as SO(n) is the maximal isometry group for spheres, the previous sequence becomes for k = 1 Sp(2) ⊂ SU(4) ⊂ Spin(7) ⊂ SO(8) H C (O) R (2) where the adscription of division algebras other than (O) is clear. It is fairly easy to see that all these group act trans on the seven sphere of constant norm vectors in R; therefore after finding the stabilizers we get ∗To Alberto Galindo in his seventieth birthday