Construction of F4 and E8 using triality

[john baez]

Kostant recently told me about a very simple construction of
the Lie algebras F4 and E8 using triality. He said he'd
discovered it but never published it. Does anyone know a
reference for this? Please Cc: your reply to me since it's
a pain reading news from where I am via telnet.

[Geoffrey Mess]

In article <4sjdno$3...@math.ucr.edu> ba...@math.ucr.edu (john baez) writes:
> Kostant recently told me about a very simple construction of
> the Lie algebras F4 and E8 using triality.

In Jacobson's book Exceptional Simple Lie Algebras F4 is constructed by
triality. One statement of triality is that D4 has 3 inequivalent 8 dimensional
irreducible representations and Aut(D4) permutes these. A more explicit
statement is this one: Put a Cayley algebra structure on R^8 and let .
denote the multiplication. Let so(8) be the lie algebra of infinitesimal
automorphism of the orthogonal form determined by the Cayley algebra:
it is < a, b> = Trace a.b. Given A in so(8) there exist uniquely determined
B, C in so(8) such that a.b A = aB . c + a . bC, and A->A, A->B, A->C
are inequivalent 8 dimensional reps of so(8). (From this it's not hard to
see that there's a period 3 automorphism of D4 with fixed point set G2.)

>From this it's not too hard to show that there's a nice Lie algebra structure
on so(8) + R^8 + R^8 + R^8 in which so(8) normalizes each of the three copies
of R^8 acting by one of the three reps. Details in Jacobson. Kostant's
construction is likely to be related.

I've never seen a construction of E8 that I thought was "very simple" or that
used triality. (There's a uniform construction of Lie algebras from root
systems but that's not what we're looking for.) Jacobson's book explains
Freudenthal's "magic quadrangle" nicely.

--
Geoffrey Mess
Department of Mathematics, UCLA. ge...@math.ucla.edu

[Geoffrey Mess]

Correspondents have pointed out that the book Spinor construction of vertex
operator algebras, triality and E_8^(1), Contemporary Mathematics 121 by
Feingold, Frenkel and Ries constructs E_8 using triality and spinors.
John Baez says that Kostant constructs the E8 Lie algebra as
so(8) + so(8) + end(R1) + end(R2) + end(R3).
where R1, R2, R3 are the three 8 dimensional representations of so(8).

There's a chapter in Fulton-Harris, Representation Theory: A First Course
which outlines a nice construction of E_8, and gives pointers to other
constructions in the literature.