Timeline for Moving frames method for non-matrix Lie group
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Aug 2, 2019 at 15:31 | answer | added | Deane Yang | timeline score: 0 | |
| Jul 31, 2019 at 18:51 | comment | added | Deane Yang | By the way, since I prefer to write a homogeneous space as the set of left cosets, the action of $H$ on frames is on the right. | |
| Jul 31, 2019 at 18:51 | comment | added | Deane Yang | In the examples I know, the bundle of frames over a homogeneous space $G/H$ is the principal bundle $G$ with each fiber (of frames over a point) is isomorphic to $H$. This, for example, is true for orthonormal frames of Euclidean space or the unit sphere. Once you see that this holds when $G$ is a matrix group, it raises the question of whether it can be done for an abstract Lie group. I don't know who first did this. I believe, however, the abstract formulation became more widely used when people started studying spin groups, which have no simple linear representation. | |
| Jul 31, 2019 at 18:27 | comment | added | CNS709 | @DeaneYang No, I am talking about moving frames for homegenous spaces, not for principal bundles (of course there is a principal bundle associated to an homogenous space but I don't care in this post) . The generalization I mean is just from matrix Lie group to general Lie group. | |
| Jul 31, 2019 at 18:11 | comment | added | Deane Yang | Are you talking about the generalization of the bundle of frames in the tangent bundle or a vector bundle, on which a matrix group $G$ acts, to that of an abstract principal bundle on which an abstract Lie group $G$ acts? | |
| Jul 31, 2019 at 17:56 | comment | added | CNS709 | @DeaneYang Thank you for the reply, but this does not answer my question. I mean: I understand that all the modern stuff, if specialized to matrix group, is the same as the old; the fact is that I cannot realize how one has thought to generalized it in this way. Please see my new edits for see what I mean. | |
| Jul 31, 2019 at 17:56 | history | edited | CNS709 | CC BY-SA 4.0 |
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| Jul 31, 2019 at 9:26 | answer | added | Ben McKay | timeline score: 6 | |
| Jul 31, 2019 at 3:07 | comment | added | Deane Yang | 3. Matrix multiplication doesn't exist on an abstract Lie group, but the Lie bracket does. The insight is that almost everything you do with a matrix group requires only the Lie bracket, especially if you want invariance or equivariance under the group action. From there, it makes sense to try to extend everything you know for matrix groups to abstract Lie groups. | |
| Jul 31, 2019 at 3:05 | comment | added | Deane Yang | 2. The Lie bracket is in the matrix Maurer-Cartan equation, because the wedge product does the skew-symmetrization of the matrix multiplication in the definition of the Lie bracket for a matrix group. Written using the Lie bracket now allows the equation to be stated for the left invariant Maurer-Cartan form on an abstract Lie groups. The proof that the form really does satisfy the equation does require a little more work now. | |
| Jul 31, 2019 at 2:45 | comment | added | Deane Yang | 1. The point is that if $G$ is a matrix group, then $g^{-1}dg$ is the unique left invariant $\mathfrak{g}$-valued $1$-form equal to the identity at $1$ and that's enough to extend everything from matrix groups to abstract Lie groups. | |
| Jul 30, 2019 at 23:00 | review | Close votes | |||
| Aug 4, 2019 at 3:05 | |||||
| Jul 30, 2019 at 22:44 | history | edited | CNS709 | CC BY-SA 4.0 |
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| Jul 30, 2019 at 22:40 | review | First posts | |||
| Jul 31, 2019 at 1:22 | |||||
| Jul 30, 2019 at 22:35 | history | asked | CNS709 | CC BY-SA 4.0 |