Timeline for Computing the inner automorphism group of a finite Lie algebra
Current License: CC BY-SA 3.0
4 events
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| Feb 21, 2018 at 18:24 | comment | added | Russ Woodroofe | I guess it's a matter of taste and convention as to whether inner automorphism groups are defined outside of linear algebraic groups. To me, the definition of an inner automorphism should be an automorphism arising from exponentiation (subject to suitable restrictions) of a Lie algebra element. There are still some problems with definition in this case, as I'll shortly put in an 'UPDATE2' to my post, but as long as p is large wrt n, one does get a consistent notion. | |
| Feb 13, 2018 at 8:53 | comment | added | Russ Woodroofe | Also, let me try to explain my comment on G_2. The exceptional group G_2(3) is the 'inner automorphism group', as I compute above, of sl(3,3) mod its center. (Or so the StructureDescription command in GAP tell me.) I don't completely understand why this should be the answer, but as G_2 has a nice action on 7-dimensional space, it doesn't seem like complete nonsense. One possible story: I haven't checked carefully, but maybe sl(3,3) is cryptomorphic to a mod 3 version of the octonions. | |
| Feb 13, 2018 at 8:23 | comment | added | Russ Woodroofe | Thank you for your thoughtful reply. I'll need to think a little to absorb it, but will respond immediately to a couple of things. First, unless I'm mistaken, sl(n,p) will have trivial center unless p divides n. (This is the situation in which scalar matrices have nonzero trace.) I should probably be embarrassed that I needed a GAP computation to realize that :-). | |
| Feb 13, 2018 at 2:33 | history | answered | Jim Humphreys | CC BY-SA 3.0 |