Timeline for On radicals of a lie algebra
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 23, 2013 at 0:27 | vote | accept | Si-Qi Liu | ||
| Nov 20, 2013 at 23:36 | comment | added | Jim Humphreys | @Marguax: As a former student of Seligman, I'm aware that "reductive" is hardly ever used in prime characteristic (unless in the sense of Lie algebra of a reductive algebraic group), whereas the Lie algebras with nondegenerate Killing form are the bdst studied and are close relatives of semisimple Lie algebras in characteristic 0 (with exceptions for some primes). But the label "semisimple" is not at all agreed on, since the characteristic 0 equivalences break down badly. | |
| Nov 20, 2013 at 19:58 | comment | added | Marguax | @Qiaochu Yuan: Seligman's book "modular Lie algebras" presumably provides the appropriate notions for positive characteristic, but I have never looked at it seriously. | |
| Nov 20, 2013 at 19:56 | comment | added | Marguax | @Qiaochu Yuan: The assertions in Bourbaki totally break down in positive characteristic, so it is more serious than a failure of equivalences. I am not aware of a good definition of "reductive Lie algebra" outside characteristic 0, so it was unclear to me if you had a definition. In the setting of Lie algebras (unlike for algebraic groups) it is safest to explicitly assume char. 0 for such things unless one provides a clear definition or reference for the terminology being used. (Bourbaki demands char. 0 in their definition of "reductive Lie algebra"; look at the start of section 6 of Ch. I.) | |
| Nov 20, 2013 at 18:50 | comment | added | Qiaochu Yuan | @Marguax: I don't have one (I assume you're hinting that the equivalence between various possible definitions fails). Is positive characteristic an important issue here? | |
| Nov 20, 2013 at 14:53 | comment | added | Marguax | @Qiaochu Yuan: What is your definition of "reductive Lie algebra" in positive characteristic? | |
| Nov 20, 2013 at 14:52 | comment | added | Marguax | @Si-Qi Yu: Those parts of Bourbaki have a running hypothesis that char$(k)=0$ (look at the start of various sections); e.g., for $\mathfrak{g}=\mathfrak{sl}_p$ in characteristic $p>0$ the Killing form vanishes and the subalgebra ${\rm{Lie}}(\mu_p)$ of diagonal scalars form a nonzero central ideal yet if $p>2$ then $[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}$, etc. | |
| Nov 20, 2013 at 14:45 | answer | added | Jim Humphreys | timeline score: 4 | |
| Nov 20, 2013 at 10:27 | answer | added | Dietrich Burde | timeline score: 4 | |
| Nov 20, 2013 at 5:28 | comment | added | Qiaochu Yuan | $\mathfrak{s}$ vanishes iff $\mathfrak{g}$ is reductive, but it's easy to find reductive Lie algebras for which $\mathfrak{n}$ fails to vanish (e.g. abelian ones). | |
| Nov 20, 2013 at 3:09 | answer | added | Ben Webster♦ | timeline score: 4 | |
| Nov 20, 2013 at 3:09 | review | First posts | |||
| Nov 20, 2013 at 3:46 | |||||
| Nov 20, 2013 at 2:53 | history | asked | Si-Qi Liu | CC BY-SA 3.0 |