A Moufang loop, the exceptional Jordan algebra, and a cubic form in 27 variables

AbstractLet I be the exceptional 27-dimensional Jordan algebra over C. Its automorphism group is the Lie group F4C and this group is known to have a finite subgroup AL, where A is a self centralizing elementary abelian of order 27, L≅SL(3, 3), and L normalizes A. As an A-module, I decomposes into a direct sum of 1-dimensional spaces Ix which afford the 27 distinct linear characters xϵA^:=Hom(A, Cx). These spaces satisfy IxIy = Ixy. Let ω = e2πi3. There are a basis of I of the form ex, for xϵA^, and a function g:A^×A^→F3 such that (∗) exey = (−2)e(x, y) ωg(x, y)exy, where c(x, y) = 0 if x and y are linearly dependent and c(x,y) = 1 otherwise. Identifying A^ with F33, we write x = (x1, x2, x3) and y = (y1, y2, y3). A function g which has the above properties is g(x,y) = −x1,x2,x3 − x3,y1,y2 + x2,x3,y1 + x1,y2,y3. The elements ex|x ϵ A^ generate the infinite commutative loop L:={(−2)mωnex|mϵZ, nϵZ3, xϵA^} under Jordan multiplication. The loop S is not Moufang but has as quotient a Moufang loop M of order 81 and exponent 3. Conversely, the loop M may be constructed from scratch (using g) and used to define the Jordan algebra I using the formula (∗); this gives a new existence proof for a simple 27-dimensional Jordan algebra over fields of characteristic not 2 or 3 with a primitive cube root of unity (in characteristic 3, we get the group algebra of A^). We discuss some finite groups associated to M and the Lie groups F4(C) and 3E6(C) and compare the analogous situation with the loop O16, the Cayley numbers, and Lie groups G2(C) and D4(C). We also get a new construction of the cubic form in 27 variables whose group is 3E6(C) and an easy and natural construction of the exotic 3-local subgroup 31+3+3:SL(3, 3)