Timeline for Lie algebra action from the action of $\mathfrak{sl}_2$-triple operators
Current License: CC BY-SA 4.0
13 events
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| Aug 12, 2021 at 21:33 | comment | added | Paul Levy | @Callum The centraliser of a regular $\mathfrak{sl} _2$ in a simple Lie algebra is trivial. If $V$ is a simple ${\mathfrak g}$-module then it equals $L(\lambda)$ for some dominant weight $\lambda$, and the action of your regular $h$ tells you the height of $\lambda$. You also know the dimension of $V$, and there can't be too many possibilities for $\lambda$ given this information. However, the action of a regular $\mathfrak{sl} _2$ can't tell the difference between $V$ and $V^*$. | |
| Aug 10, 2021 at 22:14 | comment | added | richrow | @JeffreyAdams What exactly I should looking for? As far as I can see they are mostly working with real $\operatorname{SL}_2$. | |
| Aug 10, 2021 at 18:54 | comment | added | Jeffrey Adams | See the book Non-Abelian Harmonic Analysis by Roger Howe and Eng-Chye Tan. | |
| Aug 9, 2021 at 14:36 | comment | added | Callum | (Very rough musings so take with a pinch of salt). I'm going to assume we know what kind of representation it is (i.e. is it a single irrep or a sum of irreps of certain dimension etc.) . For high enough dimension this is not at all obvious. From there you'll at least need to know the action of the centraliser of $\mathfrak{sl}_2$ since its action will commute. If it is regular this centraliser will be small (indeed I think it is just the orthocomplement of $h$ in the Cartan subalgebra it defines). I don't know if that is enough to fix the action though. | |
| Aug 9, 2021 at 6:33 | history | edited | richrow | CC BY-SA 4.0 |
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| Aug 8, 2021 at 16:38 | history | edited | richrow | CC BY-SA 4.0 |
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| Aug 8, 2021 at 16:37 | comment | added | richrow | @BenMcKay Indeed. I think we should restrict ourselves to the case of simple Lie algebra (actually, the initial motivation was about particular case $\mathfrak{g}=\mathfrak{sl}_{n}$). I will edit the question accordingly. | |
| Aug 8, 2021 at 16:32 | history | edited | richrow | CC BY-SA 4.0 |
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| Aug 8, 2021 at 16:28 | comment | added | Ben McKay | The Lie algebra $\mathfrak{g}$ could be a sum of some $\mathfrak{sl}_2$ and some arbitrary semisimple Lie algebra, with an arbitrary action on $V$. | |
| Aug 8, 2021 at 16:17 | comment | added | LSpice | I think that an example from @DaveWitteMorris is relevant. | |
| Aug 8, 2021 at 16:15 | history | edited | LSpice | CC BY-SA 4.0 |
Missing 'answer'
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| Aug 8, 2021 at 16:14 | comment | added | LSpice | Are you given in advance that the action does extend to $\mathfrak g$, or are you interested in determining whether it extends? (Also, are you working over $\mathbb C$?) | |
| Aug 8, 2021 at 15:40 | history | asked | richrow | CC BY-SA 4.0 |