Inner Automorphisms of the Octonions
Posted by John Baez
What are the inner automorphisms of the octonions?
Of course this is an odd question. Since the octonions are nonassociative you might fear the map
given by
for some octonion is not even well-defined!
But it is.
The reason is that the octonions are alternative: the unital subalgebra generated by any two octonions is associative. Furthermore, the inverse of is in the unital subalgebra generated by . This follows from
and the fact that is in the unital subalgebra generated by , since we can write where is a real multiple of the identity and is purely imaginary, and then .
It follows that whenever is a nonzero octonion, we have
for all octonions , so we can write either as
However, there is no reason a priori to expect to be an automorphism, meaning
for all . For which octonions does this happen?
Of course it happens when is real, i.e. a real multiple of . But that’s boring—because then is the identity. Can we find more interesting inner automorphisms of the octonions?
A correspondent, Charles Wynn, told me that is an automorphism when
and is any element with . This kind of element is a particular sort of 6th root of unity in the octonions—one that lies at a angle from the positive real axis.
A bit of digging revealed this paper:
- P. J. C. Lamont, Arithmetics in Cayley’s algebra, Glasgow Mathematical Journal 6 no. 2 (1963), 99–106.
In Theorem 2.1, Lamont claims that is an automorphism of the octonions iff and only if is either real or
In other words, is an automorphism iff the octonion lies at an angle of or from the positive real axis. These cases include all 6th roots of unity in the octonions!
I haven’t fully checked the proof, but it seems to use little more than the Moufang identity.
I wonder what this fact means? How do these inner automorphisms sit inside the group of all automorphisms of the octonions, ?
Re: Inner Automorphisms of the Octonions
How many 6th roots of unity in the octonions are there?
Are there any general results about numbers of roots of octonion polynomials?