Timeline for Is $R(su_{4})\cong R(so_{6})$?
Current License: CC BY-SA 3.0
21 events
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| May 10, 2012 at 9:26 | history | edited | Willie Wong |
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| Apr 15, 2012 at 1:02 | comment | added | Kerry | @Gjergji Zaimi: Could you answer me? (or should I email you?) | |
| Apr 14, 2012 at 2:43 | comment | added | Kerry | I do not get why the $[xyz]^{1/2}$ term appeared though this appeared in part of the sums in $R(g)$, and I could not write it down in the form $x^{\pm a}y^{\pm b}z^{\pm c}$ since $x,y,z$ bore no relations. But I believe with that base ring we could achieve the isomorphism I wanted. | |
| Apr 14, 2012 at 1:40 | comment | added | Gjergji Zaimi | I wasn't explaining myself properly. Explicitly, $R(h_{so_6})$ is $\mathbb Z[x^{\pm 1},y^{\pm 1},z^{\pm 1},(xyz)^{1/2}]$. This is also isomorphic to $\mathbb Z[x^{\pm 1},y^{\pm 1},z^{\pm 1}]$, which is where your confusion lies, I believe. | |
| Apr 13, 2012 at 23:42 | history | edited | Kerry | CC BY-SA 3.0 |
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| Apr 13, 2012 at 23:35 | history | edited | Kerry | CC BY-SA 3.0 |
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| Apr 13, 2012 at 23:30 | comment | added | Kerry | Well, you wrote "I'm pretty sure that's not the right..". The standard representation is clear, and the spin representation comes from viewing $so_{6}$ inside of $\wedge V$ by clifford algebra. | |
| Apr 13, 2012 at 23:27 | comment | added | Gjergji Zaimi | what you added is a proof that $R(h_{so})$ is isomorphic to $\mathbb Z[\cdots]$. I asked that you derive $R(so)$ and R(h_{so}) in parallel so that the inclusion of the first in the second becomes clear. | |
| Apr 13, 2012 at 23:12 | comment | added | Kerry | @Gjergji Zaimi: Sure. Added. | |
| Apr 13, 2012 at 23:11 | history | edited | Kerry | CC BY-SA 3.0 |
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| Apr 13, 2012 at 22:36 | comment | added | Gjergji Zaimi | I don't have time to do the calculation right now, but I'm pretty sure that's not the right presentation for $R(h_{SO_6})$, though I agree it is isomorphic to the Laurent ring you say. Why don't you add an explanation here on how you found that? | |
| Apr 13, 2012 at 15:05 | answer | added | BS. | timeline score: 2 | |
| Apr 13, 2012 at 14:27 | history | edited | Kerry | CC BY-SA 3.0 |
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| Apr 13, 2012 at 3:57 | comment | added | Kerry | The first one is the standard representation, the second and the third are the spin representations. | |
| Apr 13, 2012 at 3:56 | comment | added | Kerry | Yes, that is what I am expecting. But I do not know what is wrong so I really need some help. | |
| Apr 13, 2012 at 3:44 | comment | added | Gjergji Zaimi | Hmm, shouldn't $R(h_{so_6})$ contain $R(so_6)$ as a subring? | |
| Apr 13, 2012 at 2:56 | history | edited | Kerry | CC BY-SA 3.0 |
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| Apr 13, 2012 at 2:30 | comment | added | Kerry | @ARupinski: $h$ is the Cartan subalgebra of $G$, to distinguish the two rings I use $R_{h_{G}}$. | |
| Apr 13, 2012 at 2:16 | comment | added | ARupinski | I have worked with $R(G)$ quite a bit, but since I've never seen the notation before, what is $R(h_G)$ for a group/lie algebra $G$? | |
| Apr 13, 2012 at 1:24 | history | edited | Kerry | CC BY-SA 3.0 |
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| Apr 13, 2012 at 1:15 | history | asked | Kerry | CC BY-SA 3.0 |