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,
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 $\mathfrak{g}$?
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.