Translation-dilation group of the plane
= Translation-dilation group of the plane
{title2=$\mathbb R^2\rtimes\mathbb R_{>0}$}
This <Lie group> acts on the plane by $v\mapsto x+\rho v$ with $\rho>0$. Its multiplication is $(\rho,x)(\rho',x')=(\rho\rho',x+\rho x')$, and its inverse is $(\rho^{-1},-\rho^{-1}x)$. It is a <semidirect product> in which positive dilation acts on the translation subgroup. A dilation generator $D$ and translation generators $T_1,T_2$ have <Lie brackets> $[D,T_a]=T_a$ and $[T_1,T_2]=0$.