Timeline for The $U({\frak g})v$-module generated by a single element of a $U({\frak g})v$-module
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Jul 25, 2021 at 12:16 | comment | added | Nicolas Hemelsoet | No, for infinite dimensional modules the situation is different than finite-dimensional one. You can look up "Verma modules" to get started. | |
| Jul 25, 2021 at 12:11 | comment | added | Spyros Olympopolous | but maybe it does not . . . . | |
| Jul 25, 2021 at 11:56 | comment | added | Spyros Olympopolous | However since $\frak{g}$ is semisimple by assumption, it must have complete decomposibility (I guess this extends to the infinfite dimensional setting). Hence any $V' \subseteq V$ admits a complement and hence reducible implies decomposable . . . . I hope I am not wrong about complete decomposibility though. | |
| Jul 25, 2021 at 11:51 | comment | added | Spyros Olympopolous | Ok, I see it's the distinction between the existence of a decomposition $V = V' \oplus V''$ and the existence of a submodule $V' \subseteq V$. So irreducible is stronger than indecomposable. | |
| Jul 25, 2021 at 11:44 | comment | added | Spyros Olympopolous | Sorry but what is the distiction between "indecomposable" and "irreducible"? | |
| Jul 25, 2021 at 11:36 | comment | added | Nicolas Hemelsoet | It's not true that if $v$ is highest weight then $U(\mathfrak g)v$ is irreducible : for example the Verma module $M(0)$ for $\mathfrak{sl}_2$ admits $M(-2)$ as submodule. Even if you replace "irreducible" by "indecomposable", for a general $v \in V$ the module $U(\mathfrak g)v$ doesn't need to be incomposable. For example take $M(0)$ generated by $v$ and $M(1)$ generated by $w$. Then, the vector $v+w \in M(0) \oplus M(1)$ will generate $M(0) \oplus M(1)$ which is not indecomposable. If $v$ is highest weight, then $U(\mathfrak g)v$ is always indecomposable. | |
| Jul 25, 2021 at 11:27 | review | First posts | |||
| Jul 25, 2021 at 12:35 | |||||
| Jul 25, 2021 at 11:21 | history | asked | Spyros Olympopolous | CC BY-SA 4.0 |