A Moufang Loop, the Exceptional Jordan Algebra, and a Cubic Form in 27 Variables

Let # be the exceptional 27-dimensional Jordan algebra over C. Its automorphism group is the Lie group F4(@) and this group is known to have a finite subgroup AL, where A is a self centralizing elementary abelian of order 27, L g SL(3, 3), and L normalizes A. As an A-module, 2 decomposes into a direct sum of l-dimensional spaces jrwhich afford the 27 distinct linear characters x E A n := Hom(A, @ x ). These spaces satisfy X$ = 2X.r. Let o = ezn@. There are a basisofdoftheforme,,forxEA”,andafunctiong:A^xA^~IF,suchthat(*) e,e, =(-2)-d’. l)“R(.h.b)e,,, where c(x, y) = 0 if x and y are linearly dependent and c(x, y)= 1 otherwise. .Identifying A^ with F:, we write X=(X,, x2, xX) and y = (y,, yZ, y3). A function g which has the above properties is g(x, y) = -~,x~~,-x~~,~~+x~x,~,+x,y,~~. The elements {e,lxeA^} generate the infinite commutative loop 4p := { (-2)” w”e, 1 m E Z, II E Z,, x E A” } under Jordan multiplication. The loop $P is not Moufang but has as quotient a Moufang loop & of order 81 and exponent 3. Conversely, the loop d may be constructed from scratch (using g) and used to define the Jordan algebra 2 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 n ). We discuss some finite groups associated to & and the Lie groups F4(@) and 3&,(C) and compare the analogous situation with the loop 0 ,6, the Cayley numbers, and Lie groups G2(c) and D.,(c). We also get a new construction of the cubic form in 27 variables whose group is 3E,(C) and an easy and natural construction of the exotic 3-local subgroup 3” 3+? SL(3,3). I(“ 1990 Academic Press, Inc