2.4 Spinors and Trialities

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
bilinear map

that is nondegenerate, meaning that if we fix either argument to any nonzero value, the linear functional induced on the other vector space is nonzero. Similarly, given vector spaces and , a

that is nondegenerate in the sense that if we fix any two arguments to any nonzero values, the linear functional induced on the third vector space is nonzero.

Dualities are easy to come by. Trialities are much rarer. For suppose
we have a triality

By dualizing, we can turn this into a bilinear map

which we call `multiplication'. By the nondegeneracy of our triality, left multiplication by any nonzero element of defines an isomorphism from to . Similarly, right multiplication by any nonzero element of defines an isomorphism from to . If we choose nonzero elements and , we can thereby identify the spaces , and with a single vector vector space, say . Note that this identifies all three vectors , , and with the same vector . We thus obtain a product

for which is the left and right unit. Since left or right multiplication by any nonzero element is an isomorphism, is actually a division algebra! Conversely, any division algebra gives 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
with

and such that for all there exists for which this bound is attained -- and similarly for cyclic permutations of . Given a normed triality, picking unit vectors in any two of the spaces allows us to identify all three spaces and get a normed division algebra. Conversely, any normed division algebra gives a normed triality.

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

consisting of all linear combinations of products of an even number of vectors in . Thus the irreps of restrict to representations of , which turn out to be still irreducible. These are called

In fact, there is an isomorphism

given as follows:

where is an orthonormal basis for . Thus spinors in dimensions are the same as pinors in dimensions! Table 3 therefore yields the following table, where we use similar notation but with `' instead of `':

irreps of | ||

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:

and similarly for right-handed and left-handed spinors.

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
inclusion

Using this, we can restrict the action of the Clifford algebra on pinors to a map

This map is actually an intertwining operator between representations of . If we restrict the vector representation to the subgroup , it remains irreducible. This is not always true for the pinor representations, but we can always decompose them as a direct sum of spinor representations. Applying this decomposition to the map , we get a map

All the spinor representations appearing here are self-dual, so we can dualize the above maps and reinterpret them as trilinear maps

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:

In higher dimensions, the spinor representations become bigger than the vector representation, so we get no more trialities this way -- and of course, none exist.

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
such that

for all . These automorphisms form a group we call . If we construct a normed division algebra from by choosing unit vectors , we have

In particular, it turns out that:

are the unit spheres in , and , respectively -- the only spheres that are Lie groups. is just another name for the automorphism group of the octonions; we shall study this group in Section 4.1. The bigger group acts as automorphisms of the triality that gives the octonions, and it does so in an interesting way. Given any element , there exist unique elements such that

for all and . Moreover, the maps

are outer automorphisms of . In fact is the permutation group on 3 letters, and there exist outer automorphisms that have the effect of permuting the vector, left-handed spinor, and right-handed spinor representations any way one likes; and are among these.

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 [14], who
called these symmetries **triality**. The more general notion of
`triality' we have been discussing here came later, and is apparently
due to Adams [1].

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:

where the straight lines denote spinors and the wiggly one denotes a vector. The most familiar example is the process whereby an electron emits or absorbs a photon.

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 [28].

© 2001 John Baez