Timeline for Lie groups vs. algebraic groups and deformations
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| May 14, 2012 at 2:15 | vote | accept | Earthliŋ | ||
| May 14, 2012 at 2:14 | comment | added | Earthliŋ | Many thanks for your detailed answer & comments. | |
| May 13, 2012 at 21:37 | comment | added | Misha | The product bundle $\tilde{M}\times G$ has obvious flat connection, where horizontal leaves are of the form $\tilde{M}\times g$. This connection is preserved under the action of $\pi$ and, hence, descends to the flat connection on the principal bundle $P_\phi\to M$, $P_\phi=(\tilde{M}\times G)/\pi$. Conversely, every flat principal $G$-bundle $P\to M$ bundle has monodromy representation $\phi$, so that $P\cong P_\phi$ as a flat $G$-bundle. This relation of flat connections and representations is explained in many places, e.g. Steenrod's book on fiber bundles. | |
| May 13, 2012 at 21:31 | comment | added | Misha | @ s.barmeier: You do not need to assume that $\phi$ is a discrete embedding. The relation between representations of a group $\pi$ (endowed with discrete topology) to a Lie group $G$ and flat principal $G$-bundles over $M$ (where $M$ is any CW-complex with fundamental group $\pi$) is very general and standard. For instance, if you have a representation $\phi: \pi\to G$, form the associated bundle over $M$ by taking $(\tilde{M}\times G)/\pi \to M$, where $\pi$ acts on the first factor by covering transformations and on the second factor via left multiplication $\phi(\gamma)(g)$... | |
| May 13, 2012 at 4:03 | comment | added | Earthliŋ | I'm missing the "very simple" bit. I can't quite see the holonomy group. Is the flatness somehow related to $\pi$ being discrete (and $\phi$ being a discrete embedding)? I'm also stumbling a bit over the notation. What is the base manifold in this case, and what is $A$? | |
| May 13, 2012 at 3:10 | history | edited | Misha | CC BY-SA 3.0 |
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| May 13, 2012 at 2:59 | comment | added | Misha | For your last question, see the reference in the Addendum. | |
| May 13, 2012 at 2:58 | comment | added | Misha | Yes, this functor is equivalent to the functor defined by DGLAs. The origin of this result is very simple: $Hom(\pi, G)$ can be identified with the space of (normalized) flat connections on the base-manifold. Flat connections lead to DGLAs, since a connection can be locally written as $d +A$: The exterior differential $d$ (with values in an appropriate Lie algebra) is responsible for differential in DGLA, while Lie comes from the fact that everything is taking values in a Lie algebra, so you have the Lie bracket, graded comes from natural grading on differential forms. | |
| May 13, 2012 at 2:52 | history | edited | Misha | CC BY-SA 3.0 |
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| May 13, 2012 at 0:59 | comment | added | Earthliŋ | Oh, and why do we know that singularities can be arbitrarily complicated? | |
| May 13, 2012 at 0:53 | comment | added | Earthliŋ | Maybe this should be a new question, but I had a look at the paper of Goldman & Millson. It seems like they end up studying differential graded Lie algebras. How exactly does this tie in with the example you give? I guess $\mathrm{Hom}_\rho(\pi,\underline{G})$ as a functor is somehow equivalent to a deformation functor of the DGLA? | |
| May 13, 2012 at 0:13 | history | edited | Misha | CC BY-SA 3.0 |
relation to Artin local rings
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| May 12, 2012 at 16:47 | history | edited | Misha | CC BY-SA 3.0 |
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| May 12, 2012 at 16:39 | history | edited | Misha | CC BY-SA 3.0 |
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| May 12, 2012 at 16:23 | comment | added | Misha | See "Addendum." | |
| May 12, 2012 at 14:38 | history | edited | Misha | CC BY-SA 3.0 |
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| May 12, 2012 at 13:41 | comment | added | Earthliŋ | Great, that is really interesting. Are there any references for these character varieties $\mathrm{Hom}(\Gamma,G)$ and their quotients? I'm not quite following your point near the end. Are you saying that for discrete and faithful representations the right quotient can always be isolated? Or is that a particular feature of Teichmüller theory, which uses particularly nice features of $\mathrm{SL}_2\,\mathbb{R}$? I would be really grateful for a reference for this process of taking slices as well. | |
| May 12, 2012 at 12:59 | history | answered | Misha | CC BY-SA 3.0 |