A nonassociative division algebra may seem like a strange thing to
bother with, but the notion of triality makes it seem a bit more
natural. The concept of duality is important throughout linear algebra.
The concept of triality is similar, but considerably subtler. Given
vector spaces and , we may define a duality to be a
Dualities are easy to come by. Trialities are much rarer. For suppose
we have a triality
It follows from Theorem 3 that trialities only occur in dimensions 1, 2, 4, or 8. This theorem is quite deep. By comparison, Hurwitz's classification of normed division algebras is easy to prove. Not surprisingly, these correspond to a special sort of triality, which we call a 'normed' triality.
To be precise, a normed triality consists of inner product
spaces equipped with a trilinear map
But where do normed trialities come from? They come from the theory of spinors! From Section 2.3, we already know that any -dimensional normed division algebra is a representation of , so it makes sense to look for normed trialities here. In fact, representations of give certain representations of , the double cover of the rotation group in dimensions. These are called 'spinors'. As we shall see, the relation between spinors and vectors gives a nice way to construct normed trialities in dimensions 1, 2, 4 and 8.
To see how this works, first let be the group sitting inside that consists of all products of unit vectors in . This group is a double cover of the orthogonal group , where given any unit vector , we map both to the element of that reflects across the hyperplane perpendicular to . Since every element of is a product of reflections, this homomorphism is indeed onto.
Next, let be the subgroup consisting of all elements that are a product of an even number of unit vectors in . An element of has determinant 1 iff it is the product of an even number of reflections, so just as is a double cover of , is a double cover of . Together with a French dirty joke which we shall not explain, this analogy is the origin of the terms '' and 'pinor'.
Since sits inside , the irreps of
restrict to representations of , which turn out to be still
irreducible. These are again called pinors, and we know what
they are from Table 3. Similarly, sits inside the subalgebra
In fact, there is an isomorphism
Table 4 — Spinor Representations
We call and the right-handed and left-handed
spinor representations. For we can work out the spinor
representations using Bott periodicity:
Now, besides its pinor representation(s), the group also has an
irrep where we first apply the 2-1 homomorphism
and then use the obvious representation of on . We call
this the vector representation, . As a vector space
is just , and is generated by , so we have an
These trilinear maps are candidates for trialities! However, they can
only be trialities when the dimension of the vector representation
matches that of the relevant spinor representations. In the range of
the above table this happens only for . In these cases we
actually do get normed trialities, which in turn give normed division algebras:
Of the four normed trialities, the one that gives the octonions
has an interesting property that the rest lack. To see this property,
one must pay careful attention to the difference between a normed triality
and a normed division algebra. To construct a normed division
algebra from the normed triality
we must arbitrarily choose unit vectors in two of the three spaces, so
the symmetry group of is smaller than that of . More precisely,
let us define a automorphism of the normed triality
to be a triple of norm-preserving maps
In particular, it turns out that:
In general, outer automorphisms of simple Lie groups come from symmetries of their Dynkin diagrams. Of all the simple Lie groups, has the most symmetrical Dynkin diagram! It looks like this:
Here the three outer nodes correspond to the vector, left-handed spinor and right-handed spinor representations of , while the central node corresponds to the adjoint representation — that is, the representation of on its own Lie algebra, better known as . The outer automorphisms corresponding to the symmetries of this diagram were discovered in 1925 by Cartan , who called these symmetries triality. The more general notion of 'triality' we have been discussing here came later, and is apparently due to Adams .
The construction of division algebras from trialities has tantalizing links to physics. In the Standard Model of particle physics, all particles other than the Higgs boson transform either as vectors or spinors. The vector particles are also called 'gauge bosons', and they serve to carry the forces in the Standard Model. The spinor particles are also called 'fermions', and they correspond to the basic forms of matter: quarks and leptons. The interaction between matter and the forces is described by a trilinear map involving two spinors and one vector. This map is often drawn as a Feynman diagram:
It is fascinating that the same sort of mathematics can be used both to construct the normed division algebras and to describe the interaction between matter and forces. Could this be important for physics? One prima facie problem with this speculation is that physics uses spinors associated to Lorentz groups rather than rotation groups, due to the fact that spacetime has a Lorentzian rather than Euclidean metric. However, in Section 3.3 we describe a way around this problem. Just as octonions give the spinor representations of , pairs of octonions give the spinor representations of . This is one reason so many theories of physics work best when spacetime is 10-dimensional! Examples include superstring theory [26,42], supersymmetric gauge theories [31,60,78], and Geoffrey Dixon's extension of the Standard Model based on the algebra , in which the 3 forces arise naturally from the three factors in this tensor product .
© 2001 John Baez