Lie Algebra of a Lie Group

The tangent space at the identity of a Lie group, equipped with a canonical bracket from invariant vector fields.

Lie Algebra of a Lie Group

Let $G$ be a Lie group with identity element $e$ . The Lie algebra of $G$ is the tangent space

\mathfrak{g} := T_eG.

How the bracket is defined

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

\widetilde X_g := (dL_g)_e(X)\in T_gG.

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

[X,Y] := \big[\widetilde X,\widetilde Y\big]_e,

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 .