I'm going through some notes on group theory for physics. After introducing the concept of Lie group and Lie algebra the writer makes the connection between the two.
Let $G$ be a Lie group of endomorphism of a vector space $V$, he defines the set of all curves $\gamma(t)$ of $G$ through the identity as $\mathcal{C}_G$ such that $\gamma(0)=1_V$. Then shows that the derivative of a curve in $\mathcal{C}_G$ evaluated at $0$ is still an endomorphism.
Then define a set as $$ \mathfrak{g}=\left\{X \in \text{End}(V)\text{ s.t. }\exists\gamma \in \mathcal{C}_G \text{ with }X=d\gamma(0)/dt\right\} $$ and then shows that this set $\mathfrak{g}$ is a Lie subalgebra of the general linear algebra $\mathfrak{gl}$. With this last theorem the chapter finishes and moves to the next topic.
I'm getting confused on two points:
- can the set $\mathfrak{g}$ be identified as the tangent space to the identity?
- the last theorem shows that $\mathfrak{g}$ is a Lie subalgebra of the general linear algebra $\mathfrak{gl}$. Being a subalgebra it is also an algebra, but I don't get why it's the Lie algebra of the group $G$.
I hope I've been clear enough, otherwise I'm happy to add details.
Edit: I've had some details on how the curve is defined, I marked them in italic so it's easier to spot them, and rephrased the question.