Triality - General Formulation

General Formulation

A duality between two vector spaces over a field F is a nondegenerate bilinear map

i.e., for each nonzero vector v in one of the two vector spaces, the pairing with v is a nonzero linear functional on the other.

Similarly, a triality between three vector spaces over a field F is a nondegenerate trilinear map

i.e., each nonzero vector in one of the three vector spaces induces a duality between the other two.

By choosing vectors e_i in each V_i on which the trilinear map evaluates to 1, we find that the three vector spaces are all isomorphic to each other, and to their duals. Denoting this common vector space by V, the triality may be reexpressed as a bilinear multiplication

where each e_i corresponds to the identity element in V. The nondegeneracy condition now implies that V is a division algebra. It follows that V has dimension 1, 2, 4 or 8. If further F = R and the identification of V with its dual is given by positive definite inner product, V is a normed division algebra, and is therefore isomorphic to R, C, H or O.

Conversely, the normed division algebras immediately give rise to trialities by taking each V_i equal to the division algebra, and using the inner product on the algebra to dualize the multiplication into a trilinear form.

An alternative construction of trialities uses spinors in dimensions 1, 2, 4 and 8. The eight dimensional case corresponds to the triality property of Spin(8).