4.2

The second smallest of the exceptional Lie groups is the 52-dimensional group . The geometric meaning of this group became clear in a number of nearly simultaneous papers by various mathematicians. In 1949, Jordan constructed the octonionic projective plane using projections in . One year later, Armand Borel [8] noted that is the isometry group of a 16-dimensional projective plane. In fact, this plane is none other than than . Also in 1950, Claude Chevalley and Richard Schafer [18] showed that is the automorphism group of . In 1951, Freudenthal [35] embarked upon a long series of papers in which he described not only but also the other exceptional Lie groups using octonionic projective geometry. To survey these developments, one still cannot do better than to read his classic 1964 paper on Lie groups and the foundations of geometry [38].

Let us take Chevalley and Schafer's result as the definition of :

Its Lie algebra is thus

As we saw in Section 3.4, points of correspond to trace-1 projections in the exceptional Jordan algebra. It follows that acts as transformations of . In fact, we can equip with a Riemannian metric for which is the isometry group. To get a sense of how this works, let us describe as a quotient space of .

In Section 3.4 we saw that the exceptional Jordan algebra can
be built using natural operations on the scalar, vector and spinor
representations of . This implies that is a
subgroup of . Equation (3.4) makes it clear that
is precisely the subgroup fixing the element

Since this element is a trace-one projection, it corresponds to a point of . We have already seen that acts transitively on . It follows that

This fact has various nice spinoffs. First, it gives an easy way to
compute the dimension of :

Second, since is compact, we can take any Riemannian metric on and average it with respect to the action of this group. The isometry group of the resulting metric will automatically include as a subgroup. With more work [5], one can show that actually

and thus

Equation (4.2) also implies that the tangent space of our
chosen point in is isomorphic to . But we already
know that this tangent space is just , or in other words, the
spinor representation of . We thus have

as vector spaces, where is a Lie subalgebra. The bracket in is built from the bracket in , the action , and the map obtained by dualizing this action. We can also rewrite this description of in terms of the octonions, as follows:

This last formula suggests that we decompose further using the
splitting of into and .
It is easily seen by looking at matrices that for all we have

Moreover, when we restrict the representation to , it splits as a direct sum . Using these facts and equation (4.2), we see

This formula emphasizes the close relation between and triality: the Lie bracket in is completely built out of maps involving and its three 8-dimensional irreducible representations! We can rewrite this in a way that brings out the role of the octonions:

While elegant, none of these descriptions of gives a convenient
picture of all the derivations of the exceptional Jordan algebra. In
fact, there is a nice picture of this sort for whenever
is a normed division algebra. One way to get a derivation of the
Jordan algebra is to take a derivation of and let it
act on each entry of the matrices in . Another way uses
elements of

Given , there is a derivation of given by

In fact [4], every derivation of can be uniquely expressed as a linear combination of derivations of these two sorts, so we have

as vector spaces. In the case of the octonions, this decomposition says that

In equation (4.2), the subspace is always a Lie
subalgebra, but is not unless is commutative and
associative -- in which case vanishes. Nonetheless, there
is a formula for the brackets in
which applies in every
case [70]. Given
and
, we
have

where acts on componentwise, is the trace-free part of the commutator , and is the derivation of defined using equation (4.1).

Summarizing these different descriptions of , we have:

**Theorem 5.**
The compact real form of
is given by

where in each case the Lie bracket is built from natural bilinear operations on the summands.

© 2001 John Baez