Right Translation on a Lie Group

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

Definition (right translation). The right translation by $g$ is the map

R_g:G\longrightarrow G,\qquad R_g(h)=hg.

Since multiplication in a Lie group is a smooth map, each $R_g$ is smooth. Moreover, $R_g$ is a diffeomorphism with inverse $R_{g^{-1}}$ .

Its differential (pushforward) gives linear isomorphisms

(dR_g)_h:T_hG\longrightarrow T_{hg}G,

so right translations also transport vectors between tangent spaces. They are used to define right-invariant vector fields.

The family $\{R_g\}_{g\in G}$ satisfies

R_g\circ R_h=R_{hg},\qquad R_e=\mathrm{id}_G,

so the assignment $g\mapsto R_g$ is an antihomomorphism into the diffeomorphism group. For a left-multiplicative analogue, compare with left translations.

Examples

$(\mathbb R^n,+)$ . For the additive Lie group, $R_a(x)=x+a$ . As with left translation, $(dR_a)_x$ is the identity on $\mathbb R^n$ .

Matrix groups. If $G\subseteq GL(n,\mathbb R)$ , then $R_A(B)=BA$ . On tangent vectors represented by matrices $V$ , the differential acts by right multiplication: $(dR_A)_B(V)=VA.$

Circle group $S^1$ . Writing elements as complex numbers, $R_{e^{i\theta}}(e^{it})=e^{i(t+\theta)}$ . For the abelian group $S^1$ , left and right translations agree.