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4.5 $\E _7$

Next we turn to the 133-dimensional exceptional Lie group $\E _7$. In 1954, Freudenthal [37] described this group as the automorphism group of a 56-dimensional octonionic structure now called a `Freudenthal triple system'. We sketch this idea below, but first we give some magic square constructions. Vinberg's version of the magic square gives

\begin{displaymath}
\e _7 = \Der (\H) \oplus \Der (\O) \oplus \sa _3(\H \tensor \O) .
\end{displaymath}

Tits' version gives
\begin{displaymath}
% latex2html id marker 1732\e _7 \iso \Der (\H) \oplus \Der (\h _3(\O)) \oplus
(\Im (\H) \!\tensor \! \sh _3(\O))
\end{displaymath}

and also
\begin{displaymath}
% latex2html id marker 1732\e _7 \iso \Der (\O) \oplus \Der (\h _3(\H)) \oplus
(\Im (\O) \!\tensor \! \sh _3(\H))
\end{displaymath}

The Barton-Sudbery version gives

\begin{displaymath}
\e _7 \iso \Tri (\O) \oplus \Tri (\H) \oplus (\H \tensor \O)^3
\end{displaymath}

Starting from equation (4) and using the fact that $\Der (\H) \iso \Im (\H)$ is 3-dimensional, we obtain the elegant formula
\begin{displaymath}
% latex2html id marker 1733
\e _7 \iso \Der (\h _3(\O)) \, \oplus \, \h _3(\O)^3 .
\end{displaymath}

This gives an illuminating way to compute the dimension of $e_7$:
\begin{displaymath}\dim(\e _7) = \dim(\Der (\h _3(\O))) + 3 \dim(\h _3(\O)) = 52 + 3 \cdot 27 =
133 .\end{displaymath}

Starting from equation (5) and using the concrete descriptions of $\Tri (\H)$ and $\Tri (\O)$ from equation (3), we obtain
\begin{displaymath}
\e _7 \iso \so (\O) \oplus \so (\H) \oplus \Im (\H) \oplus (\H \tensor \O)^3
\end{displaymath}

Using equation (4.2), we may rewrite this as

\begin{displaymath}
\e _7 \iso \so (\O \oplus \H) \oplus \Im (\H) \oplus (\H \tensor \O)^2.
\end{displaymath}

Though not obvious from what we have done, the direct summand $\so (\O
\oplus \H) \oplus \Im (\H)$ here is really a Lie subalgebra of $e_7$. In less octonionic language, this result can also be found in Adams' book [1]:

\begin{displaymath}
\e _7 \iso \so (12) \oplus \symp (1) \oplus S_{12}^+
\end{displaymath}

He describes the bracket in $e_7$ in terms of natural operations involving $\so (12)$ and its spinor representation $S_{12}^+$. The funny-looking factor of $\symp (1)$ comes from the fact that this representation is quaternionic. The bracket of an element of $\symp (1)$ and an element of $S_{12}^+$ is the element of $S_{12}^+$ defined using the natural action of $\symp (1)$ on this space.

If we let $\E _7$ be the simply connected group with Lie algebra $e_7$, it follows from results of Adams [1] that the subgroup generated by the Lie subalgebra $\so (12) \oplus \symp (1)$ is isomorphic to $(\Spin (12) \times \Sp (1))/\Z_2$. This lets us define the quateroctonionic projective plane by

\begin{displaymath}
% latex2html id marker 1738
(\H \tensor \O)\P^2 = \E _7\, / \,((\Spin (12) \times \Sp (1))/\Z_2) \end{displaymath}

and conclude that the tangent space at any point of this manifold is isomorphic to $S_{12}^+ \iso (\H \tensor \O)^2$. We can put an $\E _7$-invariant Riemannian metric on this manifold by the technique of averaging over the group action. It then turns out [5] that

\begin{displaymath}\E _7 \iso \Isom ((\H \tensor \O)\P^2)
\end{displaymath}

and thus

\begin{displaymath}
\e _7 \iso \isom ((\H \tensor \O)\P^2) .
\end{displaymath}

Summarizing, we have the following 7 octonionic descriptions of $e_7$:

Theorem 7   The compact real form of $e_7$ is given by
\begin{displaymath}
% latex2html id marker 1741\begin{array}{lcl}
\e _7 &\iso ...
... \so (\H) \oplus \Im (\H) \oplus (\H \tensor \O)^3
\end{array}\end{displaymath}

where in each case the Lie bracket of $e_7$ is built from natural bilinear operations on the summands.

Before the magic square was developed, Freudenthal [37] used another octonionic construction to study $\E _7$. The smallest nontrivial representation of this group is 56-dimensional. Freudenthal realized we can define a 56-dimensional space

\begin{displaymath}
% latex2html id marker 1742
F = \{ \left( \begin{array}{cc}...
... \right) : \;
x,y \in \h _3(\O) , \; \alpha , \beta \in \R \}
\end{displaymath}

and equip this space with a symplectic structure

\begin{displaymath}\omega \maps F \times F \to \R \end{displaymath}

and trilinear product

\begin{displaymath}\tau \maps F \times F \times F \to F \end{displaymath}

such that the group of linear transformations preserving both these structures is a certain noncompact real form of $\E _7$, namely $\E _{7(-25)}$. The symplectic structure and trilinear product on $F$ satisfy some relations, and algebraists have made these into the definition of a `Freudenthal triple system' [10,32,67]. The geometrical significance of this rather complicated sort of structure has recently been clarified by some physicists working on string theory. At the end of the previous section, we mentioned a relation between 9-dimensional Euclidean geometry and $\F _4$, and a corresponding relation between 10-dimensional Lorentzian geometry and $\E _{6(-26)}$. Murat Günaydin [44] has extended this to a relation between 10-dimensional conformal geometry and $\E _{7(-25)}$, and in work with Kilian Koepsell and Hermann Nikolai [45] has explicated how this is connected to Freudenthal triple systems.


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© 2001 John Baez

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