- #26

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We have fields associated with the U(1) symmetry; we have other fields associated with the SU(2) symmetry, and we have still other fields associated with the SU(3) symmetry. IIRC we have a particular algebra of multiplication between the fields of the U(1) symmetry; we have a different algebra of multiplication between the fields of the SU(2) symmetry, and we have still another algebra of multiplication between the fields of the SU(3) symmetry. Is it true that the algebra of U(1) is complex, the algebra of SU(2) is quaternions, and the algebra of SU(3) is octonions? I think I heard something like this at one time in my studies.

That's an interesting question, and I doubt I'm in a position to answer it, but Geoffry Dixon does work on this very concept. A little numerology that's always fascinated me is

[tex]\begin{align} S^1 \hookrightarrow S^3 \rightarrow S^2 \\

S^3 \hookrightarrow S^7 \rightarrow S^4 \\

S^7 \hookrightarrow S^{15} \rightarrow S^8 \end{align}[/tex]

notice that the fiber of the complex hopf fibration is [tex] S^1 \cong U(1)[/tex] and the fiber of the quatronic hopf fibration is [tex]S^3 \cong SU(2)[/tex] however the fiber of the octonic hopf fibration [tex]S^7[/tex] isn't even a group. Technically it's a Moufang loop. I think what Dixon does is take the automorphism group of the octonions which is G_2 and get's SU(3) out of that. That's according to something I read on John Baez's this weeks finds.