Left and right translations on a Lie group

Let $G$ be a Lie group and fix $g\in G$. The left translation by $g$ and right translation by $g$ are respectively

$L_g:G⟶G,L_g(h)=gh,$

and

$R_g:G⟶G,R_g(h)=hg\mathrm{.}$

Both maps are smooth: each is obtained from the smooth multiplication $m:G\times G\to G$ in Lie group by holding one argument fixed. The notation here is the ordinary right-translation convention. Kirillov writes a right action as $h\mapsto h{g}^{-1}$; therefore this page's $R_g$ is that source's right action by $g^{-1}$.

The definition applies in dimensions zero and one and uses no metric or nondegeneracy condition. A Lie group is nonempty, its manifold is boundaryless by the page convention, and the one supplied element $g$ involves no choice from a family.