Timeline for Can all G-connections on a Riemann surface X be induced by maps from X to G
Current License: CC BY-SA 2.5
13 events
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| Mar 21, 2010 at 7:20 | history | edited | Bo Peng | CC BY-SA 2.5 |
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| Mar 21, 2010 at 7:18 | comment | added | Bo Peng | Thank everyone for the quick and detailed responses! I went to sleep last night and then I realized "pulling-back" the Maurer-Cartan form is essentially "differentiating" the map from X to G, hence it became a very standard integrablity problem with local and global obstructions, when G is abelian. In the non-abelian case, it seems one have to use the flatness of the Maurer-Cartan form for a quick argument. | |
| Mar 21, 2010 at 7:02 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Mar 21, 2010 at 4:57 | vote | accept | Bo Peng | ||
| Mar 21, 2010 at 1:34 | answer | added | Paul | timeline score: 1 | |
| Mar 20, 2010 at 23:38 | answer | added | Tom Church | timeline score: 7 | |
| Mar 20, 2010 at 21:16 | answer | added | José Figueroa-O'Farrill | timeline score: 2 | |
| Mar 20, 2010 at 21:08 | comment | added | Theo Johnson-Freyd | Oh, and of course the statement of the question does not require that $G$ be semisimple: any group has a left-invariant MC 1-form. | |
| Mar 20, 2010 at 21:07 | comment | added | Theo Johnson-Freyd | When I think about my second comment more, it must be the latter meaning that you mean. In general, without trivializing a connection on a principle $G$-bundle is a fairly complicated thing. But connections on the trivial $G$-bundle $G\times X \to X$ are in natural bijection with $\mathfrak g$-valued $1$-forms. Note that the MC 1-form is a $\mathfrak g$-valued $1$-form on $G$, which acts on a tangent vector by sending it to its left- (say) invariant vector field, and interpreting this as an element of $\mathfrak g$ (or equivalent translating the vector to $T_eG$ via right translations). | |
| Mar 20, 2010 at 21:03 | comment | added | Theo Johnson-Freyd | I wonder what you mean by "$G$-connection". Do you mean "connection on some principle $G$-bundle"? In that case, I think you'd be hard pressed to recover the bundle just from the map $X \to G$. More likely, you mean "connection on the trivial $G$-bundle". | |
| Mar 20, 2010 at 21:00 | comment | added | Theo Johnson-Freyd | Incidentally, MathOverflow understands LaTeX just fine, provided you put dollar signs around things the way you would in TeX. (Well, there are a few exceptions.) | |
| Mar 20, 2010 at 20:17 | history | edited | Bo Peng |
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| Mar 20, 2010 at 20:10 | history | asked | Bo Peng | CC BY-SA 2.5 |