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Triality

In mathematics, triality is a relationship between three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry.

Of all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram. The diagram has four nodes with one node located at the center, and the other three attached symmetrically. The symmetry group of the diagram is the symmetric group S3 which acts by permuting the three legs. This gives rise to an S3 group of outer automorphisms of Spin(8). This automorphism group permutes the three 8-dimensional irreducible representations of Spin(8); these being the vector representation and two chiral spin representations. These automorphisms do not project to automorphisms of SO(8).

Roughly speaking, symmetries of the Dynkin diagram lead to automorphisms of the Bruhat-Tits building associated with the group. For special linear groups, one obtains projective duality. For Spin(8), one finds a curious phenomenon involving 1, 2, and 4 dimensional subspaces of 8-dimensional space, historically known as "geometric triality".

The exceptional 3-fold symmetry of the $D_4$ diagram also gives rise to the Steinberg group $\left\{\right\}^3 D_4$.

General formulation

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

$V_1times V_2to mathbb F,$
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

$V_1times V_2times V_3to mathbb F,$
i.e., each nonzero vector in one of the three vector spaces induces a duality between the other two.

By choosing vectors ei in each Vi 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

$V times V to V$
where each ei 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 Vi 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).

References

• John Frank Adams (1981), Spin(8), Triality, F4 and all that, in "Superspace and supergravity", edited by Stephen Hawking and Martin Roček, Cambridge University Press, pages 435-445.
• John Frank Adams (1996), Lectures on Exceptional Lie Groups (Chicago Lectures in Mathematics), edited by Zafer Mahmud and Mamora Mimura, University of Chicago Press, ISBN 0-226-00527-5.