The Okubo Algebra
Posted by John Baez
While studying the algebra of Grand Unified Theories and the role of division algebras in supersymmetric Yang–Mills theory, I bumped into a curious entity called the ‘Okubo algebra’.
If you know anything about it, tell me!
Susumu Okubo discovered his algebra while pondering quarks. For some reason not yet known to me, this led him to seek an 8-dimensional algebra having as its Lie algebra of derivations.
The octonions won’t do: their Lie algebra of derivations is the exceptional Lie algebra , which contains as a subalgebra.
The Okubo algebra can be described as follows. Let be the space of traceless self-adjoint complex matrices. In other words, take and multiply all the matrices in there by . Give the product
where the product at right is just ordinary matrix multiplication, and
Don’t ask me why!
The result is a nonunital, nonassociative algebra. That sounds bad. But, just like and , it’s a composition algebra: it’s equipped with a nondegenerate quadratic form
such that
And it’s power-associative: that is, the subalgebra generated by any one element is associative. This is equivalent to having the identities
and
for every element .
Furthermore, the Okubo algebra is Lie-admissible: that is, the commutator
defines a Lie algebra. For the Okubo algebra, this Lie algebra is .
(Everyone knows that associative algebras are Lie-admissible; fewer people know the converse fails — and fewer still know that every operad gives a Lie-admissible algebra! I learned the last fact from Bill Schmitt; the construction appear in section 4.7 here.)
The Okubo algebra is not alternative, as the octonions are. Remember, an algebra is alternative if the subalgebra generated by any two elements is associative — or equivalently, if any two of these three identities hold:
The Okubo algebra only satisfies the third of these laws:
Such algebras are called flexible.
But what’s going on here, really?
I might consider the whole subject too bizarre to be worth bothering with, except that Elduque has used a -grading on the Okubo algebra to put a similar grading on the Lie algebras and . In the latter case, this relies on an amazing construction of as
Some references I should read in my copious spare time:
- S. Okubo, Pseudo-quaternion and pseudo-octonion algebras, Hadronic J. 1 (1978), 1250–1278.
- S. Okubo, Deformation of the Lie-admissible pseudo-octonion algebra into the octonion algebra, Hadronic J., 1 (1978), 1383–1431.
- S. Okubo, Octonions as traceless matrices via a flexible Lie-admissible algebra, Hadronic J. 1 (1978), 1432–1465.
- S. Okubo, A generalization of Hurwitz theorem and flexible Lie-admissible algebras, Hadronic J. 3 (1978), 1–52.
- S. Okubo, H.C. Myung, Some new classes of division algebras, J. Algebra 67 (1980), 479–490.
- S. Okubo, Introduction to Octonion and other Non-Associative Algebras in Physics, Cambridge Univ. Press, 1995.
The Hadronic Journal is a bit of a mystery to me. I’ve seen a lot of quirky papers on nonassociative algebras and physics in this journal. Are the people who write them followers of Okubo, in some sense?
Re: The Okubo Algebra
Here’s something I learned about the Okubo-product from a nice talk last week by Melanie Raczek. She used it to classify degree 3 central simple algebras, but I’ll restrict to the split case, that is consider 3x3 complex matrices and look at the 8-dimensional subspace V=sl(3) of trace zero matrices.
On this space there is a nondegenerate quadratic form Tr(A2) (here A is any trace zero matrix). Going projective, 3x3 matrices satisfying Tr(A)=Tr(A2)=0, form a 6-dimensional quadric in P7. A nice fact about 6-dimensional quadrics is that any point determines two 3-planes through the point and contained in the quadric (compare this to the two lines through a point on a 2-dimensional quadric). The Okubo product on V describes these.
For any two trace-zero 3x3 matrices A and B define the Okubo product to be
A*B =(1/(1-w))(AB-wBA) - Tr(AB)
where w is a third root of unity.
If a matrix A determines a point on the quadric, then one of the 3-planes through it is the projective space of the 4-dimensional subspace of all Okubo products A*B with B ranging over V (the other coming from the 4-dimensional subspace of all products B*A).