Lie group - Lie algebra correspondence

There is a direct correspondence between Lie algebras and Lie groups. In particular, we defined the Lie algebra $\mathfrak{g}$ of a Lie group $G$ to be the tangent space of $G$ at the identity, together with the bilinear alternating map given by the Commutator, i.e. $[x,y]\in \mathfrak{g}\forall x,y\in \mathfrak{g}$

$$\begin{array}{rcl}\text{Lie group}& \iff & \text{Lie algebra}\\ & \text{Universal Properties}& \\ G& \Rightarrow & \{X:e^{tX}\in G\forall t\in \mathbb{F}\}\\ XY\in G& & [x,y]\in \mathfrak{g}\\ & \text{Results}& \\ G\text{ Abelian}& \Rightarrow & [x,y]=0\\ & \text{Examples}& \\ {\mathbb{R}}^n& \Rightarrow & {\mathbb{R}}^n\\ G{L}_n(\mathbb{C})& \Rightarrow & M_n(\mathbb{C})\\ {\mathbb{S}}^1& \Rightarrow & \mathbb{C}\end{array}$$

• Confirm that the Lie group of the Circle group ${\mathbb{S}}^1$ equals $Lie({\mathbb{S}}^1)=\mathbb{C}$.