Is any lie group element corresponding the element in lie algebra, if so, why need the small motion around identity?

Question

I'm reading on Lie group and Lie algebras, but I am confused with the correspondence between them. Does any element in the Lie group correspond to an element in the Lie algebra (tangent space at identity)? If so (such as two-dimensional rotation), why is it only valid near the tangent point? Or is it just a convenience for the first-order approximation?

Answer 1

Perhaps this introduction helps? Let G be a Lie group and $g \in G$. Then this defines a map $G \rightarrow Aut(G)$ by $g\mapsto (\Psi_g: h\mapsto ghg^{-1})$. Note that this action fixes $h=e$. Taking differentials at the identity $e$ gives a map $Ad_g=d(\Psi_g)_e$ acting on $\mathfrak{g}=T_eG$. This is a linearisation of G and is called the ajoint representation $Ad:G \rightarrow Aut(\mathfrak{g})$ sending $g \mapsto Ad_g$. So elements of the Lie group can be represented by matrices acting on the Lie algebra.
So answer your question: every element of $G$ acts on $\mathfrak{g}$ via the adjoint representation. You look at this “small motion around the identity” because that’s what you do very often in mathematics: You try to linearize problems and phrase them in terms of vector spaces and matrices.