Timeline for Lie algebras whose derivation algebra is nilpotent
Current License: CC BY-SA 3.0
15 events
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| Mar 9, 2017 at 15:51 | comment | added | Salvatore Siciliano | If the ground field has positive characteristic and $L$ is not finitely generated, I suspect that this is not the case. However, I do not have the counterexample. | |
| Mar 9, 2017 at 13:55 | comment | added | YCor | By the way, does your condition that $Der(L)$ consists of nilpotent maps imply that $Der(L)$ is nilpotent? I'm not sure about positive characteristic. | |
| Mar 9, 2017 at 10:30 | history | edited | Salvatore Siciliano | CC BY-SA 3.0 |
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| Apr 10, 2016 at 17:11 | history | edited | Salvatore Siciliano | CC BY-SA 3.0 |
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| Apr 10, 2016 at 15:52 | comment | added | Salvatore Siciliano | I just realized that if the derivations of L are nilpotent of bounded index, then Der(L) clairly satisfies an Engel condition and so it is locally nilpotent (indeed nilpotent in case of characteristic 0) by a celebrated result of Zelmanov. | |
| Apr 8, 2016 at 16:03 | comment | added | YCor | In the definition of An-Ca, it's clear that char nilpotent implies that the Lie algebra of derivations is nilpotent (since it acts faithfully on the Lie algebra, preserving a finite filtration and acting by identity on successive quotients). | |
| Apr 8, 2016 at 15:00 | comment | added | Salvatore Siciliano | OK, thanks. What happens by using the definition of characteristically nilpotent Lie algebra as in the survey of Ancochea-Campoamor instead of the pointwise one? Of course, by the Engel-Jacobson Theorem, in the finite-dimensional case these definitions are equivalent, but in the infinite dimensional case the pointwise one is weaker. | |
| Apr 8, 2016 at 14:39 | comment | added | YCor | Well, probably a direct sum of finite-dimensional characteristically nilpotent Lie algebras of nilpotency length tending to infinity can yield a characteristically nilpotent Lie algebra (in the "pointwise" meaning you give) that is not nilpotent modulo its center (and hence has non-nilpotent inner derivation algebra). (This would disprove one implication of the equivalence you mention, but not in the direction of your last question) | |
| Apr 8, 2016 at 14:35 | comment | added | Salvatore Siciliano | Yves: The definition of "characteristically nilpotent" is already included, and I am interested in Lie algebras defined over arbitrary fields. | |
| S Apr 8, 2016 at 13:12 | history | suggested | Martin Sleziak | CC BY-SA 3.0 |
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| Apr 8, 2016 at 13:07 | comment | added | YCor | 1) You haven't defined "characteristically nilpotent", I see 2 possible meanings, in terms of automorphisms, and in terms of derivations (which are equivalent for finite-dimensional Lie algebras in characteristic zero) 2) you haven't specified the ground field, possibly you have in mind the complex field, or arbitrary characteristic... please could you fix this? | |
| Apr 8, 2016 at 12:47 | review | Suggested edits | |||
| S Apr 8, 2016 at 13:12 | |||||
| Apr 8, 2016 at 12:00 | history | edited | Salvatore Siciliano | CC BY-SA 3.0 |
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| Apr 8, 2016 at 11:43 | history | edited | Salvatore Siciliano | CC BY-SA 3.0 |
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| Apr 8, 2016 at 11:19 | history | asked | Salvatore Siciliano | CC BY-SA 3.0 |