Timeline for Homology of solvable (nilpotent) Lie algebras
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 10, 2019 at 17:09 | answer | added | Simon Wadsley | timeline score: 3 | |
| May 7, 2019 at 20:22 | answer | added | Simon Wadsley | timeline score: 3 | |
| May 7, 2019 at 15:07 | vote | accept | Boris Bilich | ||
| May 7, 2019 at 15:06 | vote | accept | Boris Bilich | ||
| May 7, 2019 at 15:07 | |||||
| May 7, 2019 at 15:06 | vote | accept | Boris Bilich | ||
| May 7, 2019 at 15:06 | |||||
| May 7, 2019 at 14:57 | answer | added | YCor | timeline score: 4 | |
| May 7, 2019 at 13:59 | comment | added | Simon Wadsley | If you want more details I can supply them another time. You can find the general idea in section 3 in one of my papers dpmms.cam.ac.uk/~sjw47/Euler.pdf but for another similar context. | |
| May 7, 2019 at 13:58 | comment | added | Simon Wadsley | Using this paper numdam.org/article/CM_1984__53_3_347_0.pdf of Brown I think you can easily get sufficient conditions for all homology groups to vanish. The idea I have in mind is that you can localise $U(\mathfrak{g})$ at the maximal Ore set $S$ contained in the complement of the augmentation ideal (described in the paper) and then compute homology by taking $(\mathbb{C}_S\otimes^L_{U(\mathfrak{g})_S} (\mathbb{C}_{\lambda})_S)$. $(\mathbb{C}_{\lambda})_S$ will already be zero except for on an explicitly parameterised set of $\lambda$. | |
| May 7, 2019 at 13:13 | comment | added | Boris Bilich | I'm also interested in the fact about nilpotent algebras, you mentioned. Where can I find the proof? | |
| May 7, 2019 at 13:06 | comment | added | Boris Bilich | I have looked through the Guichardet's book and haven't found anything about nilpotent or solvable lie algebras. | |
| May 7, 2019 at 11:25 | comment | added | YCor | About the definition of weight (written "eigenvalue" in the inital post) of a representation $V$: I guess that the correct meaning in this context would be $\lambda$ such that the 1-dimensional $V_\lambda$ appears as subquotient of $V$. Equivalently this means that $\lambda$ appears in a 1-dim diagonal block, for some block-triangulation of the rep. (There's a notion of strong weight, which I initially referred as weight, namely $\lambda$ such that $V_\lambda$ appears as subrepresentation of $V$. For $\mathfrak{g}$ nilpotent this is the same as weight, but not for $\mathfrak{g}$ solvable.) | |
| May 7, 2019 at 10:51 | comment | added | YCor | I guess that Guichardet's book "Cohomologie des groupes topologiques et des algèbres de Lie" might contain relevant information. Unfortunately it's not easy to find. For instance it probably includes the fact that $H_*(\mathfrak{g},V_\lambda)=0$ for $\mathfrak{g}$ nilpotent and all $\lambda\neq 0$. | |
| May 6, 2019 at 22:35 | answer | added | YCor | timeline score: 6 | |
| May 6, 2019 at 18:05 | comment | added | Boris Bilich | How did you get the criterion? | |
| May 6, 2019 at 17:27 | comment | added | YCor | Weight of the adjoint rep. in the metabelianization | |
| May 6, 2019 at 16:40 | comment | added | Boris Bilich | What do you mean by "weight of the metabelianization"? And what is the idea of proving this result? | |
| May 6, 2019 at 14:03 | history | edited | YCor | CC BY-SA 4.0 |
fixed English
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| May 6, 2019 at 13:40 | review | First posts | |||
| May 6, 2019 at 13:48 | |||||
| May 6, 2019 at 13:40 | history | asked | Boris Bilich | CC BY-SA 4.0 |