Timeline for Longest element of a Weyl group
Current License: CC BY-SA 3.0
7 events
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| Sep 14, 2011 at 13:25 | comment | added | Jim Humphreys | In prime characteristic, your description of the Lie algebra of a reductive group depends on knowing virtually all of the Borel-Chevalley structure theory for the groups. But the bigger complication is using information about subalgebras of the Lie algebra to get back to the group. There is no machine that does this directly, only a lot of partial results which often depend on the prime or Lie type. Chevalley abandoned the Lie algebra approach to get uniform results including classification of semisimple groups. But finding the root system (etc.) in the group is not easy. | |
| Sep 13, 2011 at 23:04 | comment | added | Faisal | @Jim: My outline does betray the fact that I'm much more accustomed to working over an alg. closed field of char. 0. However, I did check my references to make sure what I wrote still held true in the case of positive characteristic. Did I overlook something? I would appreciate any comments on the matter, as I am ignorant of this aspect of the theory. In any case, as you mention in your answer, the outline does rely on some nontrivial structure theory, but that is to be expected. | |
| Sep 13, 2011 at 22:45 | comment | added | Jim Humphreys | @Faisal: What you outline over an algebraically closed field of characteristic 0 for the Lie algebra is the right blueprint for passing to the algebraic (or Lie) group using the classical dictionary. But in prime characteristic the Lie algebra becomes too cumbersome to work with, as Chevalley found, even though his end results are remarkably close to the classical ones. | |
| Sep 12, 2011 at 19:21 | vote | accept | th.ng | ||
| Sep 12, 2011 at 19:21 | comment | added | th.ng | Thank you ! I was trying to do the same thing but on the group-level, meaning that I was trying to look at the $U_\alpha$ (whose Lie algebras are the $\mathfrak{g}_\alpha$). However, I was trying to explicitely determine the image $w_0(\alpha)$ and I couldn't figure it out... I'll look at the global image of the positive roots then. | |
| Sep 12, 2011 at 18:15 | history | edited | Faisal | CC BY-SA 3.0 |
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| Sep 12, 2011 at 17:52 | history | answered | Faisal | CC BY-SA 3.0 |