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There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can deal with very concrete notions of matrix exponential and logarithms and avoid the much more abstract exponential for general Lie groups, while the main disadvantages are that the universal cover of a matrix Lie group need not be matrix and that it's difficult to show that analytic subgroups of a matrix Lie group are matrix Lie groups.

It seems to me that if instead one considered locally matrix Lie groups, then you get the advantages of the concrete matrix group approach without any of the disadvantages. One obvious objection is that locally matrix Lie groups might seem like an unmotivated definition, but it should be possible to motivate them nicely using the fact that manifolds are themselves defined by local properties so it's counterintuitive to focus on matrix Lie groups which are defined by a global property.

My question is twofold:

  1. Are there any introductory texts that take this approach of looking only at locally matrix Lie groups?
  2. Are there any problems with this approach that I've overlooked?
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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Dec 4, 2015 at 16:00
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    $\begingroup$ 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. $\endgroup$
    – Paul Siegel
    Commented Dec 4, 2015 at 16:07
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    $\begingroup$ 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 $H$? $\endgroup$
    – David E Speyer
    Commented Dec 4, 2015 at 16:12
  • $\begingroup$ 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. $\endgroup$
    – Noah Snyder
    Commented Dec 4, 2015 at 16:30
  • $\begingroup$ 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. $\endgroup$
    – Ben Webster ♦
    Commented Dec 4, 2015 at 20:12

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