4.6 $\E _8$

With 248 dimensions, $\E _8$ is the biggest of the exceptional Lie groups, and in some ways the most mysterious. The easiest way to understand a group is to realize it as as symmetries of a structure one already understands. Of all the simple Lie groups, $\E _8$ is the only one whose smallest nontrivial representation is the adjoint representation. This means that in the context of linear algebra, $\E _8$ is most simply described as the group of symmetries of its own Lie algebra! One way out of this vicious circle would be to describe $\E _8$ as isometries of a Riemannian manifold. As already mentioned, $\E _8$ is the isometry group of a 128-dimensional manifold called $(\O \tensor \O)\P^2$. But alas, nobody seems to know how to define $(\O \tensor \O)\P^2$ without first defining $\E _8$. Thus this group remains a bit enigmatic.

At present, to get our hands on $\E _8$ we must start with its Lie algebra. We can define this using any of the three equivalent magic square constructions explained in the previous section. Vinberg's construction gives

$$\e _8 = \Der (\O) \oplus \Der (\O) \oplus \sa _3(\O \tensor \O) .$$

Tits' construction gives

$$% latex2html id marker 1746\e _8 \iso \Der (\O) \oplus \Der (\h _3(\O)) \oplus (\Im (\O) \!\tensor \! \sh _3(\O)) .$$

The Barton-Sudbery construction gives

We can use any of these to count the dimension of $\e _8$; for example, the last one gives

$$\dim \e _8 = 28 + 28 + 3 \cdot 8^2 = 248.$$

To emphasize the importance of triality, we can rewrite the the Barton-Sudbery description of $\e _8$ as

$$\e _8 \iso \so (8) \oplus \so (8) \oplus (V_8 \tensor V_8) \oplus (S_8^+ \tensor S_8^+) \oplus (S_8^- \tensor S_8^-).$$

Here the Lie bracket is built from natural maps relating $\so (8)$ and its three 8-dimensional irreducible representations. In particular, $\so (8) \oplus \so (8)$ is a Lie subalgebra, and the first copy of $\so (8)$ acts on the first factor in $V_8 \tensor V_8$, $S_8^+ \tensor S_8^+$, and $S_8^- \tensor S_8^-$, while the second copy acts on the second factor in each of these. This has a pleasant resemblance to the triality description of $\f _4$ that we found in Section 4.2:

$$\f _4 \iso \so (8) \oplus V_8 \oplus S_8^+ \oplus S_8^-$$

Now let us turn from 8-dimensional geometry to 16-dimensional geometry. On the one hand, we have

$$\so (16) \iso \so (8) \oplus \so (8) \oplus (V_8 \tensor V_8) .$$

On the other hand, we have seen that

$$\e _8 \iso \so (8) \oplus \so (8) \oplus (V_8 \tensor V_8) \oplus (S_8^+ \tensor S_8^+) \oplus (S_8^- \tensor S_8^-).$$

Comparing these, it is natural to hope that $\e _8$ contains $\so (16)$ as a Lie subalgebra. In fact this is true! Even better, if we restrict the right-handed spinor representation of $\so (16)$ to $\so (8) \oplus \so (8)$, it decomposes as

$$S^+_{16} \iso (S_8^+ \tensor S_8^+) \oplus (S_8^- \tensor S_8^-),$$

so we obtain

$$\e _8 \iso \so (16) \oplus S^+_{16}$$

or in more octonionic language,

$$\e _8 \iso \so (\O \oplus \O) \oplus (\O \otimes \O)^2$$

where we use $\so (V)$ to mean the Lie algebra of skew-adjoint real-linear transformations of the real inner product space $V$.

The really remarkable thing about the isomorphism

$$\e _8 \iso \so (16) \oplus S^+_{16}$$

is that the Lie bracket in $\e _8$ is entirely built from natural maps involving $\so (16)$ and $S^+_{16}$:

$$\so (16) \tensor \so (16) \to \so (16) , \qquad \so (16) \t... ...to S^+_{16} , \qquad S^+_{16} \tensor S^+_{16} \to \so (16) .$$

The first of these is the Lie bracket in $\so (16)$, the second is the action of $\so (16)$ on its right-handed spinor representation, and the third is obtained from the second by duality, using the natural inner product on $\so (16)$ and $S^+_{16}$ to identify these spaces with their duals. In fact, this is a very efficient way to define $\e _8$. If we take this approach, we must verify the Jacobi identity:

$$[[a,b],c]] = [a,[b,c]] - [b,[a,c]] .$$

When all three of $a,b,c$ lie in $\so (16)$ this is just the Jacobi identity for $\so (16)$. When two of them lie in $\so (16)$, it boils down to fact that spinors indeed form a representation of $\so (16)$. Thanks to duality, the same is true when just one lies in $\so (16)$. It thus suffices to consider the case when $a,b,c$ all lie in $S_{16}^+$. This is the only case that uses anything special about the number 16. Unfortunately, at this point a brute-force calculation seems to be required. For two approaches that minimize the pain, see the books by Adams [1] and by Green, Schwarz and Witten [42]. It would be nice to find a more conceptual approach.

Starting from $\e _8$, we can define $\E _8$ to be the simply-connected Lie group with this Lie algebra. As shown by Adams [1], the subgroup of $\E _8$ generated by the Lie subalgebra $\so (16) \subset \e _8$ is $\Spin (16)/\Z_2$. This lets us define the octooctonionic projective plane by

$$% latex2html id marker 1754 (\O \tensor \O)\P^2 = \E _8\,/\,(\Spin (16)/\Z_2) .$$

The tangent space at any point of this manifold is isomorphic to $S_{16}^+ \iso (\O \tensor \O)^2$. This partially justifies calling it 'octooctonionic projective plane', though it seems not to satisfy the usual axioms for a projective plane.

We can put an $\E _8$-invariant Riemannian metric on the octooctonionic projective plane by the technique of averaging over the group action. It then turns out [5] that

$$\E _8 \iso \Isom ((\O \tensor \O)\P^2)$$

and thus

$$\e _8 \iso \isom ((\O \tensor \O)\P^2) .$$

Summarizing, we have the following octonionic descriptions of $\E _8$:

Theorem 8.   The compact real form of $\e _8$ is given by

$$% latex2html id marker 1757\begin{array}{lcl} \e _8 &\iso ... ... \so (\O) \oplus \so (\O) \oplus (\O \tensor \O)^3 \end{array}$$

where in each case the Lie bracket on $\e _8$ is built from natural bilinear operations on the summands.

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