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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 of $G$ through the identity as $\mathcal{C}_G$, then shows that the derivative of a curve in $\mathcal{C}_G$ evaluated at $0$ is still an endomorphism.

The 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 the meaning of the set $\mathfrak{g}$, which to me seems the tangent space $T_eG$ to the identity of $G$, and how the last theorem shows that the set $\mathfrak{g}$ is indeed the Lie algebra of the group $G$.

I hope I've been clear enough, otherwise I'm happy to add details.

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    $\begingroup$ This is, in terms of focus, applicability, and culture, solidly mathematical and belonging to MSE . I would urge you to move it there. $\endgroup$
    – Cosmas Zachos
    Mar 13 at 15:26
  • $\begingroup$ Thank you, I'll post it there too. $\endgroup$
    – john
    Mar 13 at 15:35
  • $\begingroup$ Lie groups and Lie algebras are heavy math but uniquely physics related. Basically, the algebra is the tangent space to the group. There are more details than that, which are in the definition given by the OP. $\endgroup$
    – Boba Fit
    Mar 13 at 16:43

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