> I guess I'm trying to undestand what it means to have > a (non) division algebra over the reals. Does that > mean that fields that comply with the algebra do or > do not have values for continuous values of the real > numbers? I think this is a math question.
The term "algebra" is slightly ambiguous, I think. It is usually used to mean "associative algebra", ie the associative law is assumed to hold. The octonions are not associative.
I think this property is probably more important in the context of quantum theory. You can consider groups of matrices over the reals, the complex numbers or the quaternions, giving various Lie groups as you have said. But if the algebra is not associative then the matrices over the algebra do not satisfy the associative law and so do not define a group.
-- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College Dublin