Timeline for Sign conventions for a Chevalley basis of a simple complex Lie algebra
Current License: CC BY-SA 3.0
12 events
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| Jul 15 at 18:24 | comment | added | Skip | Someone coming across this question may want to know about the relevant paper M. Geck, On the construction of semisimple Lie algebras and Chevalley groups, Proc. AMS 145 (2017), 3233-3247. | |
| Jan 6, 2012 at 5:01 | vote | accept | brandyn | ||
| Dec 19, 2011 at 16:45 | comment | added | Chuck Hague | I've also had this issue come up when trying to do by-hand computations; in my experience, the best thing to do is to find a computer algebra package that implements some choice of these signs and then just let the computer handle it. I use Sage with a MAGMA interface and it's worked quite nicely, since MAGMA has all of this Lie-theoretic information built in already. (Feel free to drop me an email if you want to know more). | |
| Dec 18, 2011 at 11:54 | answer | added | Denis Chaperon de Lauzières | timeline score: 7 | |
| Dec 18, 2011 at 6:57 | history | edited | brandyn | CC BY-SA 3.0 |
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| Dec 18, 2011 at 4:29 | history | edited | brandyn | CC BY-SA 3.0 |
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| Dec 18, 2011 at 2:37 | vote | accept | brandyn | ||
| Jan 6, 2012 at 5:01 | |||||
| Dec 17, 2011 at 23:33 | comment | added | brandyn | I'm working on a calculation in the exterior algebra $\bigwedge(g\oplus g)$ where $g$ is type $F_4$. I was hoping to do a small part of the calculation via a computer program, but getting a working model of $g$ is the first step. | |
| Dec 17, 2011 at 23:31 | answer | added | Jim Humphreys | timeline score: 7 | |
| Dec 17, 2011 at 23:19 | comment | added | fherzig | How can the knowledge of these signs be useful? | |
| Dec 17, 2011 at 23:16 | comment | added | fherzig | The Chevalley basis is only unique up to automorphisms of the Lie algebra and sign changes of the $X_\alpha$. (You can replace $(X_\alpha, X_{-\alpha})$ above by $(-X_\alpha, -X_{-\alpha})$ for any root $\alpha$ and all the axioms remain true.) I'm not sure whether a canonical choice is possible. However, it's possible that someone wrote down a possible choice for $F_4$ somewhere... (I suppose you can fix $X_\alpha$ for $\alpha$ simple, define the other positive ones using fixed instances of () with your favourite signs, and then work out the rest of () with a lot of patience.) | |
| Dec 17, 2011 at 22:57 | history | asked | brandyn | CC BY-SA 3.0 |