Timeline for Why do sl(2) and so(3) correspond to different points on the Vogel plane?
Current License: CC BY-SA 3.0
12 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| May 5, 2011 at 17:22 | comment | added | S. Carnahan♦ | Warning: the above comment uses an unorthodox definition of "characteristic polynomial". | |
| May 5, 2011 at 17:12 | comment | added | S. Carnahan♦ | I'd just like to point out that it might be easier to view your orbifold as a weighted $\mathbb{P}^2$, since you can change coordinates from the unordered triple of eigenvalues to the ordered triple of coefficients of the characteristic polynomial of Casimir (whose roots are the eigenvalues). I'm not sure how $\mathfrak{sl}_2$ fits in to this picture. | |
| May 5, 2011 at 14:46 | vote | accept | Noah Snyder | ||
| May 5, 2011 at 6:53 | comment | added | DamienC | @Noah Snyder: you are right. Thanks! | |
| May 5, 2011 at 0:04 | comment | added | Noah Snyder | @DamienC: Those two "different" points for $\mathfrak{sl}_3$ are actually the same in the quotient $\mathbb{P}^2/S_3$. So that's a different issue. | |
| May 4, 2011 at 20:26 | answer | added | Bruce Westbury | timeline score: 6 | |
| May 4, 2011 at 20:20 | comment | added | DamienC | Even more surprising: $\mathfrak{sl}_3$ and $\mathfrak{so}_8$ appear twice (they also appear in the exceptionnal series in math.tamu.edu/~jml/LMunivpub.pdf). | |
| May 4, 2011 at 19:57 | comment | added | Noah Snyder | Perhaps I'm using the word "metric" wrong here. Vogel says "pseudo-quadratic" rather than "metric." The point is that you have (in addition to the bracket and the crossing) an adjoint pair of maps of representations $\mathfrak{g} \otimes \mathfrak{g} \rightarrow \mathbf{1}$ and $\mathbf{1} \rightarrow \mathfrak{g} \otimes \mathfrak{g}$. But at any rate, its a bilinear form, not a Hermitian one. | |
| May 4, 2011 at 19:36 | comment | added | José Figueroa-O'Farrill | But they are different as metric Lie algebras: the inner product is definite in $\mathfrak{so}_3$ and lorentzian in $\mathfrak{sl}_2$, assuming I'm understanding the question. | |
| May 4, 2011 at 19:33 | comment | added | Ben Webster♦ | Could it have something to do with the fact that over R, they are different? | |
| May 4, 2011 at 19:19 | history | asked | Noah Snyder | CC BY-SA 3.0 |