Left translation on a Lie group

Let $G$ be a Lie group and fix an element $g\in G$ .

The left translation by $g$ is the map

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

Smoothness and tangent maps

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

Applying the differential (pushforward) gives linear isomorphisms on tangent spaces:

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

This is a fundamental way to move tangent vectors between the tangent spaces of $G$ . In particular, left translations are used to define left-invariant vector fields and to construct the exponential map via flows.

The family $\{L_g\}_{g\in G}$ satisfies the representation property

L_g\circ L_h=L_{gh},\qquad L_e=\mathrm{id}_G.

For comparison, one can also transport data via right translations.

Examples

$(\mathbb R^n,+)$ . For the additive Lie group, $L_a(x)=a+x$ . The differential $(dL_a)_x$ is the identity map on $\mathbb R^n$ for every $x$ .

Matrix groups. If $G\subseteq GL(n,\mathbb R)$ , then $L_A(B)=AB$ . Identifying $T_BG$ with an appropriate subspace of matrices, $(dL_A)_B$ acts by left multiplication: $(dL_A)_B(V)=AV.$

Circle group $S^1$ . Writing elements as complex numbers of unit modulus, $L_{e^{i\theta}}(e^{it})=e^{i(\theta+t)}$ . Geometrically, left translation rotates the circle by angle $\theta$ .