4.4 $\E _6$

We begin with the 78-dimensional exceptional Lie group $\E _6$. As we mentioned in Section 3.4, there is a nice description of a certain noncompact real form of $\E _6$ as the group of collineations of $\OP^2$, or equivalently, the group of determinant-preserving linear transformations of $\h _3(\O)$. But before going into these, we consider the magic square constructions of the Lie algebra $e_6$. Vinberg's construction gives

$$\e _6 = \Der (\O) \oplus \sa _3(\C \tensor \O) .$$

Tits' construction, which is asymmetrical, gives

$$\e _6 \iso \Der (\h _3(\O)) \oplus \sh _3(\O)$$

and also

$$% latex2html id marker 1718\e _6 \iso \Der (\O) \oplus \Der (\h _3(\C)) \oplus (\Im (\O) \!\tensor \! \sh _3(\C)) .$$

The Barton-Sudbery construction gives

$$\e _6 \iso \Tri (\O) \oplus \Tri (\C) \oplus (\C \tensor \O)^3 .$$

We can use any of these to determine the dimension of $e_6$. For example, we have

$$\dim(\e _6) = \dim(\Der (\h _3(\O))) + \dim(\sh _3(\O)) = 52 + 26 = 78.$$

Starting from the Barton-Sudbery construction and using the concrete descriptions of $\Tri (\O)$ and $\Tri (\C)$ from equation (3), we obtain

$$\e _6 \iso \so (\O) \oplus \so (\C) \oplus \Im (\C) \oplus (\C \tensor \O)^3$$

Using equation (4.2), we may rewrite this as

$$\e _6 \iso \so (\O \oplus \C) \oplus \Im (\C) \oplus (\C \tensor \O)^2$$

and it turns out that the summand $\so (\O \oplus \C) \oplus \Im (\C)$ is actually a Lie subalgebra of $e_6$. This result can also be found in Adams' book [1], phrased as follows:

$$% latex2html id marker 1723\e _6 \iso \so (10) \oplus \u (1) \oplus S_{10}$$

In fact, he describes the bracket in $e_6$ in terms of natural operations involving $\so (10)$ and its spinor representation $S_{10}$. The funny-looking factor of $\u (1)$ comes from the fact that this spinor representation is complex. The bracket of an element of $\u (1)$ and an element of $S_{10}$ is another element of $S_{10}$, defined using the obvious action of $\u (1)$ on this complex vector space.

If we define $\E _6$ to be the simply connected group with Lie algebra $e_6$, it follows from results of Adams that the subgroup generated by the Lie subalgebra $\so (10) \oplus \u (1)$ is isomorphic to $(\Spin (10) \times \U (1))/\Z_4$. This lets us define the bioctonionic projective plane by

$$% latex2html id marker 1724 (\C \tensor \O)\P^2 = \E _6\, / \, ((\Spin (10) \times \U (1))/\Z_4)$$

and conclude that the tangent space at any point of this manifold is isomorphic to $S_{10} \iso (\C \tensor \O)^2$.

Since $\E _6$ is compact, we can put an $\E _6$-invariant Riemannian metric on the bioctonionic projective plane by averaging any metric with respect to the action of this group. It turns out [5] that the isometry group of this metric is exactly $\E _6$, so we have

$$\E _6 \iso \Isom ((\C \tensor \O)\P^2).$$

It follows that

$$\e _6 \iso \isom ((\C \tensor \O)\P^2) .$$

Summarizing, we have 6 octonionic descriptions of $e_6$:

Theorem 6.   The compact real form of $e_6$ is given by

$$% latex2html id marker 1727\begin{array}{lcl} \e _6 &\iso... ... \so (\C) \oplus \Im (\C) \oplus (\C \tensor \O)^3 \end{array}$$

where in each case the Lie bracket of $e_6$ is built from natural bilinear operations on the summands.

The smallest nontrivial representations of $\E _6$ are 27-dimensional: in fact it has two inequivalent representations of this dimension, which are dual to one another. Now, the exceptional Jordan algebra is also 27-dimensional, and in 1950 this clue led Chevalley and Schafer [18] to give a nice description of $\E _6$ as symmetries of this algebra. These symmetries do not preserve the product, but only the determinant.

More precisely, the group of determinant-preserving linear transformations of $\h _3(\O)$ turns out to be a noncompact real form of $\E _6$. This real form is sometimes called $\E _{6(-26)}$, because its Killing form has signature $-26$. To see this, note that any automorphism of $\h _3(\O)$ preserves the determinant, so we get an inclusion

$$\F _4 \hookrightarrow \E _{6(-26)} .$$

This means that $\F _4$ is a compact subgroup of $\E _{6(-26)}$. In fact it is a maximal compact subgroup, since if there were a larger one, we could average a Riemannian metric group on $\OP^2$ with respect to this group and get a metric with an isometry group larger than $\F _4$, but no such metric exists. It follows that the Killing form on the Lie algebra $\e _{6(-26)}$ is negative definite on its 52-dimensional maximal compact Lie algebra, $\f _4$ and positive definite on the complementary 26-dimensional subspace, giving a signature of $26 - 52 = -26$.

We saw in Section 3.4 that the projective plane structure of $\OP^2$ can be constructed starting only with the determinant function on the vector space $\h _3(\O)$. It follows that $\E _{6(-26)}$ acts as collineations on $\OP^2$, that is, line-preserving transformations. In fact, the group of collineations of $\OP^2$ is precisely $\E _{6(-26)}$:

$$% latex2html id marker 1729 \E _{6(-26)} \iso {\rm Coll}(\OP^2).$$

Moreover, just as the group of isometries of $\OP^2$ fixing a specific point is a copy of $\Spin (9)$, the group of collineations fixing a specific point is $\Spin (9,1)$. This fact follows with some work starting from equation (3.4), and it gives us a commutative square of inclusions:

$$% latex2html id marker 1730\begin{array}{ccl} \Spin (9) ... ...ngrightarrow & {\rm Coll}(\OP^2) \iso \E _{6(-26)} \end{array}$$

where the groups on the top are maximal compact subgroups of those on the bottom. Thus in a very real sense, $\F _4$ is to 9-dimensional Euclidean geometry as $\E _{6(-26)}$ is to 10-dimensional Lorentzian geometry.

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