Left translation

Let $G$ be a Lie group and fix $g\in G$. The left translation by $g$ is the map

The group operations are smooth, so $L_g$ is a smooth map. Its inverse is $L_{g^{-1}}$, hence $L_g$ is a diffeomorphism .

For each $h\in G$, the differential $(\mathrm{d}L_g)_h:T_hG\to T_{gh}G$ is a linear isomorphism. In particular, at the identity it gives a canonical identification $T_eG\cong T_gG$ via $(\mathrm{d}L_g)_e$.

Examples

1. Additive group. If $G=\mathbb{R}^n$ with addition, then $L_a(x)=a+x$ is ordinary translation in Euclidean space.
2. Matrix multiplication. If $G=\mathrm{GL}(n,\mathbb{R})$, then $L_A(B)=AB$ is left multiplication by a fixed invertible matrix $A$.
3. Transporting tangent vectors. On any Lie group, $(\mathrm{d}L_g)_e$ sends an element of the Lie algebra $\mathfrak{g}=T_eG$ to the corresponding value at $g$ of the associated left-invariant vector field.