TITLE: Octonionic Unitary Group?
QUESTION [8 upvotes]: Hi all.
I was wondering if anyone has any references on work related to the Octonionic Unitary group.  I would imagine that such a group would be generated by Octonionic skew-Hermitian matrices (at the lie-algebra level) in analogy to the Complex Unitary and the Quaternion Unitary (aka Compact Symplectic) groups.  Some hints of the construction appear on page 28 of the paper "The Octonions" by Baez.
http://arxiv.org/abs/math/0105155
Thanks

REPLY [5 votes]: As others have pointed out, the nonassociativity of the octonions prevents one from constructing a group.  For example, any subgroup of the octonions lives inside of a quaternion subalgebra.  Having said that, the Clifford algebra $Cl(\mathbb{R}^7)$ has two inequivalent irreducible representations which are each as real vector spaces isomorphic to the octonions (i.e., they are eight-dimensional) and provided that we identify $\mathbb{R}^7$ with the imaginary octonions, the action of $Cl(\mathbb{R}^7)$ is given by left and right octonionic multiplications.  This is analogous to what happens with $Cl(\mathbb{R}^3)$ substituting octonion for quaternion in what I said above.  Now the Spin group $Spin(3)$ is the one-dimensional quaternionic unitary group and lives naturally inside $Cl(\mathbb{R}^3)$, so one could think of the group $Spin(7)$ as being the analogue of the one-dimensional octonionic unitary group.
By the same token, and given the low-dimensional isomorphisms
$$Spin(2,1) \cong SL(2,\mathbb{R})$$
$$Spin(3,1) \cong SL(2,\mathbb{C})$$
$$Spin(5,1) \cong SL(2,\mathbb{H})$$
one would be tempted to think of $Spin(9,1)$ as $SL(2,\mathbb{O})$, even though such a group as written does not exist.