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I'm working through a course on representation theory and I wanted to get a clearer picture of how the various spaces relevant to this topic mesh together. The following sketch,

Lie Algebra Rough Sketch

shows the group $G$ being mapped to the space $GL(V)$ by the representation $\varphi$ which sits inside the Lie algebra $\mathfrak{g}$ (the mapping $\varphi$ creates a local copy of $G$ inside $\mathfrak{g}$), is this the correct way to think about it? How does the vector space $V$ from $GL(V)$ relate to the Lie algebra $\mathfrak{g}$?

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The group $S^1$ is a Lie group; its Lie Algebra is the tangent space at the identity element $(1 \in \Bbb C)$, i.e., the vertical line $x = 1$.

I don't see how a representation $\phi$ of $S^1$, such as $$ \phi : S^1 \to SO(2) : \theta \mapsto \pmatrix{\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta } $$ "creates a local copy of $S^1$ inside $\Bbb R$" (your words, made specific in this case). I think there's possibly a fundamental misunderstanding here, or maybe it's just a typo.

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  • $\begingroup$ Thanks John, I see how your simple example shows this logic as flawed. My confusion stems partially from this post on the physics stack exchange (physics.stackexchange.com/questions/65141/…) where the top answer states that "basis vectors of the Lie algebra will, through the exponential map, produce a local copy of the Lie group". I'm wondering how this "local copy" can be visualised in the context of the Lie algebra. $\endgroup$ Nov 22, 2019 at 19:48
  • $\begingroup$ Yeah....I have no idea what that sentence means. Then again, I have only a single course's knowledge of Lie Groups... $\endgroup$ Nov 22, 2019 at 20:01
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    $\begingroup$ @MatthewWard: The answer you linked to is a bit imprecise -- I added a comment there -- but also, it does not talk about representations at all, but about the exponential map from the Lie algebra to the Lie group. What is true is that in the Lie group, there is a subgroup around $1_G$ which (I would say) is parametrized by the Lie algebra. So in a way, the Lie group rather contains a local copy of the Lie algebra (where on that copy, the group operation is given by the Baker-Campbell-Hausdorff formula). But, as said, so far this has nothing to do with representations. $\endgroup$ Nov 23, 2019 at 3:48

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