Right translation

The diffeomorphism of a Lie group given by multiplication on the right by a fixed element.

Right translation

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

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

Smoothness of multiplication implies $R_g$ is smooth, and $R_{g^{-1}}$ is its inverse; thus $R_g$ is a diffeomorphism .

For each $h\in G$ , the differential $(\mathrm{d}R_g)_h:T_hG\to T_{hg}G$ is a linear isomorphism. Right translations are used to define right-invariant vector fields and right-trivializations of the tangent bundle.

Examples

Additive group. On $G=\mathbb{R}^n$ under addition, $R_a(x)=x+a$ , which coincides with left translation because the group is abelian.
Matrix multiplication. On $G=\mathrm{GL}(n,\mathbb{R})$ , $R_A(B)=BA$ is right multiplication by a fixed invertible matrix $A$ .
Right-invariant fields. If $X\in T_eG$ , the assignment $h\mapsto (\mathrm{d}R_h)_e(X)$ defines a right-invariant vector field with value $X$ at the identity.