Left Translation - Knowlpedia

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

L_g:G\to G,\qquad L_g(h)=gh.

Smoothness and inverse

Because group multiplication is smooth, $L_g$ is a diffeomorphism with inverse $L_{g^{-1}}$ .

Differential

For each $h\in G$ , the differential

(dL_g)_h : T_hG \to T_{gh}G

is a linear isomorphism of tangent spaces; see differential.

Why it matters

Left translations let you “move” tangent vectors around the group and are used to define left-invariant vector fields and the canonical identification $\mathfrak{g}=T_eG$ with left-invariant vector fields (see Lie algebra of a Lie group).