Timeline for Pedagogical question on Lie groups vs. matrix Lie groups
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 7, 2015 at 13:33 | comment | added | Allen Knutson | Why a finite cover, Ben? The two examples I think about are $\widetilde{SL_2(\mathbb R)}$, and $\{ \begin{pmatrix} 1&a&z\\&1&b\\&&1\end{pmatrix} \ :\ a,b,z \in \mathbb R \} \big/ \langle z \in \mathbb Z\rangle$. I can't see much benefit in an introductory course in including non-matrix groups. | |
| Dec 4, 2015 at 20:12 | comment | added | Ben Webster♦ | It seems like any reasonable definition you make will ultimately be just "a quotient of a finite cover of a matrix group." In that context, it really is easier to define the exponential (just lift the path $\mathrm{exp}(tX)$), and the challenge of @DavidSpeyer is easy: do it for matrix groups and take a preimage, and then an image. It is all a little awkward, though. | |
| Dec 4, 2015 at 16:30 | comment | added | Noah Snyder | I'll say more after class, but the definition I had in mind was "there's a local homomorphism from G to GL_n whose image is closed." Now that I write that out I see there's a problem in that one can't expect the image of an open neighborhood to be closed. Not sure if that's salvageable or not. | |
| Dec 4, 2015 at 16:12 | comment | added | David E Speyer | It seems to me that defining the exponential map in this setting would be just as hard as in the fully abstract setting. And once you have the exponential map, I don't think the abstract setting is much harder. As a concrete challenge, if $G$ is a locally matrix Lie group with Lie algebra $\mathfrak{g}$, and $\mathfrak{h}$ is a Lie-subalgebra of $\mathfrak{g}$, can you show that there is a corresponding subgroup $H$? | |
| Dec 4, 2015 at 16:07 | comment | added | Paul Siegel | My other reaction is that this might not actually clarify the main examples: for instance, when working with $Spin(n)$ I don't often find it helpful to think of it as "locally $SO(n)$"; instead, it's easier to suppress the matrices and work with Clifford algebras (even though the spin groups have matrix representations!) That said, your idea would certainly make it easier to contemplate the category of Lie groups. | |
| Dec 4, 2015 at 16:00 | comment | added | Paul Siegel | What is the actual definition you have in mind? If it is something like "every point has a neighborhood homeomorphic to a neighborhood in a matrix Lie group" then my immediate reaction is that this isn't essentially different from from the usual definition because the neighborhoods won't be compatible with the group structure in any natural way (Lie groups don't have small open subgroups). It's also not clear to me that the transition functions would be compatible with the group structure. | |
| Dec 4, 2015 at 15:48 | history | asked | Noah Snyder | CC BY-SA 3.0 |