bridge

A bridge is the assertion that two sets, given in two theories, are incarnations of one object. Its status records what is known about it: built, type, name or refuted. A bridge is an assertion with a status, and only a built bridge is a theorem.

Number the imaginary units of the octonions $e_x$, $x\in\Z/7$, so that $e_xe_{x+1}=e_{x+3}$, and identify the plane of this table with the Fano plane, the unit lines being its points. For a doubly even binary code $C$ of length 8 and dimension 4, $L_C=\Z^8+\tfrac12C$ is a copy of $E_8$.

(a) There are 30 such codes, in bijection with the 30 Fano planes on the seven units. (b) Exactly seven of the 30 lattices are closed under multiplication: those whose Fano plane shares exactly three lines with the plane of the table, and those three lines pass through one unit $e_c$. (c) The collineations of the table permute the 30 lattices in orbits of sizes 1, 7, 14 and 8, with stabilizer classes $G$, $S_4^a$, $A_4^b$ and $7{:}3$; the lattice of the orbit of size 1 is not closed. (d) The stabilizer of each closed lattice fixes exactly one unit, the $e_c$ of (b).

In the octonions of the cyclic table let $u=\sum_xe_x$, so that $u^2=-7$, let $v=u/\sqrt7$, and let $c\in\Z/7$. (1) The derivations killing $v$ are those commuting with left multiplication by $v$, and they act on the complement of $\langle1,v\rangle$, a complex 3-space, as $\mathfrak{su}(3)$; the same holds for $e_c$. (2) The two copies of $\mathfrak{su}(3)$ meet in a copy of $\mathfrak{su}(2)$ and together generate $\mathfrak g_2$. (3) The automorphism group $G_2$ is transitive on the unit imaginary octonions, so the stabilizers of $v$ and of $e_c$ are conjugate in $G_2$; but no element of the stabilizer $K$ of $v$ carries $e_c$ to $\pm v$. (4) The 21 permutations $e_x\mapsto e_{ax+b}$, $a\in\{1,2,4\}$, are automorphisms fixing $u$, and three of them fix $e_c$.

So $v$ and $e_c$ are two points of one object of $G_2$, the sphere $S^6$, and seams of $G_2$-sets join them, while for $K$ they are not joined: $K$ fixes $v$, and its orbit through $e_c$ has dimension 5. The bridge between the two units is built for $G_2$ and refuted for $K$. The status of a bridge depends on the group in which seams are sought.