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

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

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

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

Examples

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

2. 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:

   $$(dR_A)_B(V)=VA.$$

3. 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.