3.4 and the Exceptional Jordan Algebra

The octonions are fascinating in themselves, but the magic really starts
when we use them to construct the exceptional Jordan algebra
and its associated projective space, the octonionic projective plane.
The symmetry groups of these structures turn out to be exceptional Lie
groups, and triality gains an eerie pervasive influence over the
proceedings, since an element of consists of 3 octonions and
3 real numbers. Using the relation between normed division algebras and
trialities, we get an isomorphism

where and . Examining the Jordan product in then reveals a wonderful fact: while superficially this product is defined using the -algebra structure of , it can actually be defined using only the natural maps

together with the inner products on these 3 spaces. All this information is contained in the normed triality

so any automorphism of this triality gives an automorphism of . In Section 2.4 we saw that . With a little thought, it follows that

However, this picture of in terms of 8-dimensional Euclidean
geometry is just part of a bigger picture — a picture set in
10-dimensional Minkowski spacetime! If we regard as sitting
in the lower right-hand corner of , we get an isomorphism

We saw in Section 3.3 that and can be identified with a vector and a spinor in 10-dimensional Minkowski spacetime, respectively. Similarly, is a scalar.

This picture gives a representation of as linear
transformations of . Unfortunately, most of these
transformations do not preserve the Jordan product on . As we
shall see, they only preserve a lesser structure on : the *determinant*. However, the transformations coming from the subgroup
do preserve the Jordan product. We can
see this as follows. As a representation of ,
splits into 'space' and 'time':

with the two pieces corresponding to the traceless elements of and the real multiples of the identity, respectively. On the other hand, the spinor representation of splits as when we restrict it to , so we have

We thus obtain an isomorphism

where has vanishing trace and is a real multiple of the identity. In these terms, one can easily check that the Jordan product in is built from invariant operations on scalars, vectors and spinors in 9 dimensions. It follows that

For more details on this, see Harvey's book [50].

This does not exhaust all the symmetries of , since there are other automorphisms coming from the permutation group on 3 letters, which acts on and in an obvious way. Also, any matrix acts by conjugation as an automorphism of ; since the entries of are real, there is no problem with nonassociativity here. The group is 36-dimensional, but the full automorphism group is much bigger: it is 52-dimensional. As we explain in Section 4.2, it goes by the name of .

However, we can already do something interesting with the automorphisms we have: we can use them to diagonalize any element of . To see this, first note that the rotation group, and thus , acts transitively on the unit sphere in . This means we can use an automorphism in our subgroup to bring any element of to the form

where is real. The next step is to apply an automorphism that makes and real while leaving alone. To do this, note that the subgroup of fixing any nonzero vector in is isomorphic to . When we restrict the representation to this subgroup, it splits as , and with some work [50] one can show that acts on in such a way that any element can be carried to an element with both components real. The final step is to take our element of with all real entries and use an automorphism to diagonalize it. We can do this by conjugating it with a suitable matrix in .

To understand , we need to understand projections in .
Here is where our ability to diagonalize matrices in via
automorphisms comes in handy. Up to automorphism, every projection in
looks like one of these four:

Now, the trace of a matrix in is invariant under automorphisms, because we can define it using only the Jordan algebra structure:

where is left multiplication by . It follows that the trace of any projection in is 0,1,2, or 3. Furthermore, the rank of any projection equals its trace. To see this, first note that , since implies , and the trace goes up by integer steps. Thus we only need show . For this it suffices to consider the four projections shown above, as both trace and rank are invariant under automorphisms. Since , it is clear that for these projections we indeed have .

It follows that the points of the octonionic projective plane are
projections with trace 1 in , while the lines are projections
with trace 2. A calculation [50] shows that any projection
with trace 1 has the form

where has

On the other hand, any projection with trace 2 is of the form where has trace 1. This sets up a one-to-one correspondence between points and lines in the octonionic projective plane. If we use this correspondence to think of both as trace-1 projections, the point lies on the line if and only if . Of course, iff . The symmetry of this relation means the octonionic projective plane is self-dual! This is also true of the real, complex and quaternionic projective planes. In all cases, the operation that switches points and lines corresponds in quantum logic to the 'negation' of propositions [91].

We use to stand for the set of points in the octonionic projective plane. Given any nonzero element with , we can normalize it and then use equation (3.4) to obtain a point . Copying the strategy that worked for , we can make into a smooth manifold by covering it with three coordinate charts:

- one chart containing all points of the form ,
- one chart containing all points of the form ,
- one chart containing all points of the form .

We thus obtain the following picture of the octonionic projective plane. As a manifold, is 16-dimensional. The lines in are copies of , and thus 8-spheres. For any two distinct points in , there is a unique line on which they both lie. For any two distinct lines, there is a unique point lying on both of them. There is a 'duality' transformation that maps points to lines and vice versa while preserving this incidence relation. In particular, since the space of all points lying on any given line is a copy of , so is the space of all lines containing a given point!

To dig more deeply into the geometry of one needs another
important structure on the exceptional Jordan algebra: the determinant.
We saw in Section 3.3 that despite noncommutativity and
nonassociativity, the determinant of a matrix in is a
well-defined and useful concept. The same holds for ! We
can define the **determinant** of a matrix in by

We can express this in terms of the trace and product via

This shows that the determinant is invariant under all automorphisms of . However, the determinant is invariant under an even bigger group of linear transformations. As we shall see in Section 4.4, this group is 78-dimensional: it is a noncompact real form of the exceptional Lie group . This extra symmetry makes it worth seeing how much geometry we can do starting with just the determinant and the vector space structure of .

The determinant is a cubic form on , so there is a unique symmetric
trilinear form

such that

By dualizing this, we obtain the so-called

Explicitly, this is given by

Despite its name, this product is commutative.

We have already seen that points of correspond to trace-1
projections in . Freudenthal [36] noticed that
these are the same as elements
with and
. Even better, we can drop the equation as
long as we promise to work with *equivalence classes* of nonzero
elements satisfying
, where two such elements are
equivalent when one is a nonzero real multiple of the other. Each
such equivalence class corresponds to a unique point of ,
and we get all the points this way.

Given two points and , their cross product is well-defined up to a nonzero real multiple. This suggests that we define a 'line' to be an equivalence class of elements , where again two such elements are deemed equivalent if one is a nonzero real multiple of the other. Freudenthal showed that we get a projective plane isomorphic to if we take these as our definitions of points and lines and decree that the point lies on the line if and only if . Note that this equation makes sense even though and are only well-defined up to nonzero real multiples.

One consequence of all this is that one can recover the structure of
as a projective plane starting from just the determinant on
: we did not need the Jordan algebra structure! However, to
get a 'duality' map switching points and lines while preserving the
incidence relation, we need a bit more: we need the nondegenerate
pairing

on . This sets up an isomorphism

This isomorphism turns out to map points to lines, and in fact, it sets up a one-to-one correspondence between points and lines. We can use this correspondence to think of both points and lines in as equivalence classes of elements of . In these terms, the point lies on the line iff . This relationship is symmetrical! It follows that if we switch points and lines using this correspondence, the incidence relation is preserved.

We thus obtain a very pretty setup for working with . If we
use the isomorphism between and its dual to reinterpret
the cross product as a map

then not only is the line through distinct points and given by , but also the point in which two distinct lines and meet is given by . A triple of points and is collinear iff , and a triple of lines , , meets at a point iff . In addition, there is a delightful bunch of identities relating the Jordan product, the determinant, the cross product and the inner product in .

For more on octonionic geometry, the reader is urged to consult the original papers of Freudenthal [35,36,37,38], as well as those of Jacques Tits [87,88] and Tonny Springer [80,81,82]. Unfortunately, we must now bid goodbye to this subject and begin our trip through the exceptional groups. However, we shall return to study the symmetries of and the exceptional Jordan algebra in Sections 4.2 and 4.4.

© 2001 John Baez