Timeline for Type of 26-dimensional representation of different real forms of the complex simple Lie algebra $F_4$
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| May 4, 2012 at 16:30 | comment | added | Bruce Westbury | Each entry in the square depends on an unordered pair of composition algebras so you might expect three real forms. | |
| May 4, 2012 at 16:16 | comment | added | José Figueroa-O'Farrill | Yes, I agree: that would settle it for the split real form, but not for $F_4^{-20}$ which, as luck would have it, is what I seem to want to know :( | |
| May 4, 2012 at 16:13 | comment | added | Bruce Westbury | Then the naive candidate for the inner product is $(A,B)=tr(AB)$. This should then give the signature. | |
| May 4, 2012 at 16:10 | comment | added | Bruce Westbury | I think it might be a linguistic convenience. A complex composition algebra has two real forms which are called the compact real form and the split real form. | |
| May 4, 2012 at 15:55 | comment | added | José Figueroa-O'Farrill | And, yes, the 26 dimensional representation is there implicitly. The Jordan algebra of which F_4 is the automorphism algebra is the one of $3\times 3$ traceless, hermitian octonionic matrices, and that's a real vector space of dimension 26. | |
| May 4, 2012 at 15:52 | comment | added | José Figueroa-O'Farrill | Thanks. I believe that the magic square only gives those two real forms, because there are only two "real" forms of the octonions: the usual ones and the so-called split octonions. The usual octonions give the compact real form of $F_4$, whereas in a what is surely only a linguistic coincidence, the split octonions give the split real form of $F_4$. | |
| May 4, 2012 at 15:28 | history | answered | Bruce Westbury | CC BY-SA 3.0 |