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I have several basic questions about matrix Lie groups. Suppose $G$ is a matrix Lie group, say $\mathrm{GL}(n)$ or $\mathrm{SO}(n)$.

  1. I understand that elements of $G$ can be looked at points in $\mathbb{R}^{n^2}$. Does this make the study of these matrix Lie groups easier in any sense? Please elaborate.
  2. What are "natural" choices for charts on such matrix Lie groups (particularly, $\mathrm{GL}(n)$ and $\mathrm{SO}(n)$). Can you provide an example atlases for these two specific matrix Lie groups?
  3. Is it possible to define valid charts based on the tangent space and exponential/logarithm maps?

Thanks!

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1 Answer 1

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(1) It is just the observation that an $n \times n$ matrix has $n^2$ entries, and so you can think of it as a vector in $n^2$-dimensional space. Going along with question #2, this can help find coordinates.

(2) Going along with point (1) above, for $\mathrm{GL}_n$, you can take the $n^2$ matrix entries as a set of global coordinates, since $\mathrm{GL}_n$ is an open subset of $\Bbb{R}^{n^2}$.

(3) Near the identity element of $G$, the exponential map is a diffeomorphism, so yes, the tangent space can be used to get local coordinates.

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  • $\begingroup$ Thanks! Sorry if my questions are dumb/obvious :) (2) Formally, does this imply $\phi : g \to \mathrm{vec}(g)$ is a chart for any matrix Lie group $G \ni g$? If so (probably not), then we can cover the entire manifold $G$ with a single chart? I guess this only holds for $\mathrm{GL}_n$ and not $\mathrm{SO}_n$ (which we know doesn't have dimension $n^2$). In that case, is there a special way to create charts for matrix Lie groups that works also for e.g. $\mathrm{SO}_n$. (3) This holds for any matrix Lie group? "Near identity" means such a chart doesn't cover the entire manifold? $\endgroup$
    – newbie777
    Commented Oct 2, 2019 at 15:57
  • $\begingroup$ (3) ... I was asking this question because charts are supposed to map open subsets of a $d$-dimensional manifold to open sets in $\mathbb{R}^d$. But tangent spaces (at any point) already provide a $d$-dimensional vector space (isomorphic to $\mathbb{R}^d$ - correct?). I was wondering if in general (and especially for matrix Lie groups) this can be leveraged to construct an Atlas? (e.g., by defining chart maps similar to exp/log maps at any point). $\endgroup$
    – newbie777
    Commented Oct 2, 2019 at 16:05
  • $\begingroup$ can you please see the above questions? Thanks! $\endgroup$
    – newbie777
    Commented Oct 5, 2019 at 19:47
  • $\begingroup$ (2) No, $g \to \mathrm{vec}(g)$ is not a global chart all the time. Just for $\mathrm{GL}_n$. (3) "Near identity" means an open set containing the identity. Maybe not the entire manifold. The exponential map may just identify an open set of the tangent space (containing the zero vector) with an open set of the group containing the identity. $\endgroup$
    – Nick
    Commented Oct 6, 2019 at 2:57
  • $\begingroup$ Thanks @Nick. (2) Understood. I was asking this because of another question I asked here: what structure allows us to talk about $\frac{d}{dt} \gamma(t)$ for curves that live on a matrix Lie group (not necessarily $\mathrm{GL}_n$) where $\frac{d}{dt}$ is the derivative in the usual sense? According to you, this is not because $\mathrm{vec}$ is a chart (which makes sense) - but still this puzzles me - e.g., can we do this for curves on the sphere? $\endgroup$
    – newbie777
    Commented Oct 6, 2019 at 3:11

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