Timeline for Replacement for Lie-algebra complements
Current License: CC BY-SA 3.0
15 events
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| Feb 22, 2020 at 10:30 | history | edited | YCor |
edited tags
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Feb 21, 2016 at 19:02 | vote | accept | LSpice | ||
| Feb 21, 2016 at 13:08 | answer | added | David Stewart | timeline score: 1 | |
| Feb 20, 2016 at 23:49 | history | edited | LSpice | CC BY-SA 3.0 |
Highlighted a precise question; clarified desire for a characteristic-free answer
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| Feb 20, 2016 at 23:38 | comment | added | LSpice | @JimHumphreys, I thought that the result about complements in large characteristic was in Steinberg's endomorphisms … or torsion … paper, but now I don't find it. Maybe he even just stated it as a hypothesis. As to the restrictions on $p$, I see that my post was misleading; I did mean to ask for a characteristic-free answer (which, as you say, I would expect to be either very hard or very weak). I will edit my post accordingly. | |
| Feb 19, 2016 at 0:34 | comment | added | Jim Humphreys | I just meant the source of your quoted result. Anyway, my proposed answer does require restrictions on $p$ in your situation, which may or may not be essential for your question. But there probably isn't an easy answer. | |
| Feb 18, 2016 at 23:50 | comment | added | LSpice | @JimHumphreys, I'm sorry that I'm so obtuse, but I don't know where I referred to an argument about complements. I did cite what I thought were some facts about complements, but didn't mean to suggest that I proved them (and I might have misstated). Do you mean where I said that my desired surjectivity statement follows from the existence of complements? | |
| Feb 18, 2016 at 23:41 | comment | added | Jim Humphreys | P.S. I still don't know what argument about complements you are referring to. | |
| Feb 18, 2016 at 23:37 | answer | added | Jim Humphreys | timeline score: 1 | |
| Feb 18, 2016 at 1:35 | comment | added | LSpice | Also @JimHumphreys, I was intentionally a little vague because I thought that someone might know the 'right' replacement for complements in any setting (whatever the characteristic) where they don't exist; but the specific formulation I propose is in the last paragraph: is it true that $\operatorname{Lie}(G')$ surjects onto $\mathrm N_{\operatorname{Lie}(G)}(\operatorname{Lie}(G'))/\mathrm C_{\operatorname{Lie}(G)}(\operatorname{Lie}(G'))$? | |
| Feb 18, 2016 at 1:33 | comment | added | LSpice | @JimHumphreys, I am not citing complete reducibility in large characteristic, only the existence of complements, which I thought still held. (Of course, one might have to fix the root datum before deciding how large is 'large'.) It doesn't change my question, but I'd be glad to know if that's wrong. | |
| Feb 17, 2016 at 23:15 | comment | added | Jim Humphreys | Your formulation seems unhelpful, in terms of small vs. large primes. It's true that Jacobson/Morosov embeddings exist for "large" enough characteristic $p$ (as in Carter's 1985 book, for example). But this isn't a useful model for representation theory, where complete reducibility of finite dimensional representations often fails for reductive groups over fields of arbitrarily large characteristic (though it does in fact tend to hold for representations of dimension less than $p$). Can you word the question more precisely? | |
| Feb 17, 2016 at 16:45 | history | edited | LSpice | CC BY-SA 3.0 |
Added simplifying hypothesis
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| Feb 17, 2016 at 16:25 | history | asked | LSpice | CC BY-SA 3.0 |