Right Translation

The diffeomorphism of a Lie group given by multiplying 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 and inverse

Since multiplication is smooth , $R_g$ is a diffeomorphism with inverse $R_{g^{-1}}$ .

For each $h\in G$ , the differential

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

is a linear isomorphism between tangent spaces .

Conjugation by $g$ can be written as a composition of translations:

c_g = L_g \circ R_{g^{-1}},

which underlies the adjoint action .

Right translations are used to define right-invariant vector fields .