Lie Algebra of a Lie Group

Using left translations $L_g(h)=gh$, any $X\in T_eG$ determines a unique left-invariant vector field $\widetilde X$ by

Then the Lie bracket on $\mathfrak{g}$ is defined by

where $[\widetilde X,\widetilde Y]$ is the commutator of vector fields.

• This makes $\mathfrak{g}$ a Lie algebra.
• If $\varphi:G\to H$ is a Lie group homomorphism, then $(d\varphi)_e:\mathfrak{g}\to\mathfrak{h}$ is a Lie algebra homomorphism.

The exponential map $\exp:\mathfrak{g}\to G$ satisfies that $t\mapsto \exp(tX)$ is a one-parameter subgroup for each $X\in\mathfrak{g}$, forming a central part of the Lie correspondence.