Timeline for Real forms of complex Lie algebras with specified semisimple part
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 25, 2022 at 14:38 | comment | added | Claudio Gorodski | I think $\mathfrak g= \mathbb C^2 \rtimes \mathfrak{sl}_2\mathbb C$ is a counter example. The only real form is $\mathfrak g_0=\mathbb R^2 \rtimes \mathfrak{sl}_2\mathbb R$. | |
| Oct 25, 2022 at 14:35 | comment | added | Claudio Gorodski | I asked if there exists $\mathfrak g_0$ such that $\mathfrak s_0$ is a compact Lie algebra. But now I think the result is false. | |
| Oct 25, 2022 at 10:05 | comment | added | Venkataramana | I am not sure I understand. Levi Malcev theorem implies that the real form ${\mathfrak g}_0$ has a Levi decomposition whose complexification gives a Levi decomposition of $\mathfrak g$ and by the uniqueness of the Levi decomposition, must be the original Levi decomposition up to automorphism. | |
| Oct 24, 2022 at 20:44 | comment | added | Claudio Gorodski | Yes, that is what I meant. | |
| Oct 24, 2022 at 20:43 | comment | added | Qiaochu Yuan | To be clear, you mean "does there exist a real form of $\mathfrak{g}$ such that..."? | |
| Oct 24, 2022 at 20:41 | comment | added | Claudio Gorodski | Thanks Qiaochu Yuan, I have edited the question. | |
| Oct 24, 2022 at 20:40 | history | edited | Claudio Gorodski | CC BY-SA 4.0 |
Edited for clarity.
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| Oct 24, 2022 at 19:51 | history | asked | Claudio Gorodski | CC BY-SA 4.0 |