Timeline for Modules over the unipotent subalgebra as direct summands of modules over a semisimple Lie algebra
Current License: CC BY-SA 4.0
4 events
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| Mar 31, 2023 at 6:59 | comment | added | Leonid Positselski | @QixianZhao Thank you! This answers my question for finite-dimensional $\mathfrak g$-modules. Now I am interested in the infinite-dimensional question. | |
| Mar 31, 2023 at 5:32 | comment | added | Qixian Zhao | For example, take the $\mathfrak n$ in $\mathfrak{sl}(3,\mathbb C)$. Its acts on itself by adjoint action, and this is generated by at least two vectors. I suspect the infinite dimensional question also has a negative answer, but I'll have to think about it a little bit. | |
| Mar 31, 2023 at 5:30 | comment | added | Qixian Zhao | The answer is negative if you want finite dimensional representations. WLOG we may assume $N$ is indecomposable. Suppose there exists a $\mathfrak g$-module $M$ containing $N$ with the desired properties. We can decompose $M$ into a direct sum of irreducible $\mathfrak g$-modules. So we may assume $M$ is itself irreducible. Then as an $\mathfrak n$-module $M$ is indecomposable and is generated by a lowest weight vector in $M$. So $M = N$, and in particular $N$ is generated by a single element. But not all indecomposable finite dimensional $\mathfrak n$-module is generated by one element. | |
| Mar 30, 2023 at 22:40 | history | asked | Leonid Positselski | CC BY-SA 4.0 |