Today we make the connection between Lie groups and Lie algebras. We begin by explaining how to produce a Lie algebra $\mathfrak{g}$ from a Lie group $G$. We show how the Lie algebra structure on $\mathfrak{g}$ is induced from the Lie subalgebra of left-invariant vector fields on $G$.

We then proved half of the “Lie group-Lie algebra” correspondence: namely that if $G$ is a Lie group with Lie algebra $\mathfrak{g}$ and $H \subset G$ is a Lie subgroup with Lie algebra $\mathfrak{h}$, then $\mathfrak{h}$ is a Lie subalgebra of $\mathfrak{g}$. In Lecture 13 we will prove the converse: if $G$ is a Lie group with Lie algebra $\mathfrak{g}$, then for any Lie subalgebra $\mathfrak{h} \subset \mathfrak{g}$, there is a unique connected Lie subgroup $H$ of $G$ whose Lie algebra is $\mathfrak{h}$.