Timeline for Characterization of reductive Klein geometries
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 9, 2023 at 17:04 | history | edited | Ben McKay | CC BY-SA 4.0 |
got rid of the warning against the term reductive homogeneous space, since, as Callum points out, it really is standard
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| Jan 9, 2023 at 17:00 | comment | added | Ben McKay | @Callum: thanks, clearly I had the wrong idea there. Indeed Nomizu coined the term reductive homogeneous space, not Sharpe, and it has a very large literature. | |
| Oct 31, 2022 at 14:07 | comment | added | Callum | I'm not sure that I agree that Sharpe's usage is not standard. A quick search for the phrase "reductive homogeneous space" turns up more papers using it in Sharpe's sense than the other as well as Wikipedia, Encylopedia of mathematics and other posts on stack exchange/overflow all using Sharpe's version. | |
| Oct 26, 2022 at 16:53 | comment | added | A. J. Pan-Collantes | Little by little I am putting all the pieces of the puzzle. Thank you | |
| Oct 26, 2022 at 16:46 | comment | added | Ben McKay | @A.J.Pan-Collantes: the choice of $\mathfrak{p}$ is not given a priori: rather, every $G$-invariant connection on $M=G/H$ determines one such $\mathfrak{p}$ and vice versa, so the choices of $\mathfrak{p}$ are arbitrary $\operatorname{Ad} H$-invariant complements to $\mathfrak{h}\subseteq\mathfrak{g}$. | |
| Oct 25, 2022 at 17:47 | vote | accept | A. J. Pan-Collantes | ||
| Oct 25, 2022 at 17:47 | comment | added | A. J. Pan-Collantes | Wow, great answer with the edition. I am really learning a lot thanks to you. Only to confirm: in this kind of geometries (reductive, in Sharpe's sense) the invariant complement $\mathfrak p$ or, equivalently, the admitted invariant connection, is not canonically given a priori, isn't it? That is, you can have several valid decompositions of $\mathfrak g$... | |
| Oct 25, 2022 at 12:25 | history | edited | Ben McKay | CC BY-SA 4.0 |
added 115 characters in body
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| Oct 25, 2022 at 11:52 | history | edited | Ben McKay | CC BY-SA 4.0 |
Rewritten completely to answer the addition question in the comment below from A. J. Pan-Collantes.
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| Oct 23, 2022 at 17:17 | comment | added | A. J. Pan-Collantes | Thank you for your answer. Given a reductive Klein geometry, the $A^{\mathfrak h}$ part of the MC-form is a principal bundle connection, which is "very compatible" with the MC form (a Cartan connection). It is the converse true? That is, if we have a Klein geometry and together with the MC form (or the Cartan connection if we have a Cartan geometry instead) we have a principal bundle connection with this "compatibility", then is it a reductive Klein geometry? | |
| Oct 23, 2022 at 12:28 | history | edited | Ben McKay | CC BY-SA 4.0 |
added 200 characters in body
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| Oct 23, 2022 at 12:21 | history | answered | Ben McKay | CC BY-SA 4.0 |