Translation-dilation group of the plane (${\mathbb{R}}^2⋊{\mathbb{R}}_{>0}$)

This Lie group acts on the plane by $v\mapsto x+\rho v$ with $\rho >0$. Its multiplication is $(\rho ,x)({\rho }^{\prime },x^{\prime })=(\rho {\rho }^{\prime },x+\rho x^{\prime })$, 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 $[T_1,T_2]=0$.

The Maurer-Cartan form of the translation-dilation group of the plane has these coefficients in its dilation–translation basis, with $a=1,2$. Left multiplication rescales each coordinate differential and $\rho$ equally, proving they are left-invariant differential forms. Their dual left-invariant vector fields are $\rho {\partial }_{\rho }$ and $\rho {\partial }_{x^a}$. The Maurer-Cartan equation is $d{\lambda }^0=0$, $d{\lambda }^a=-{\lambda }^0\wedge {\lambda }^a$. The tensor ${\sum }_j{\lambda }^j\otimes {\lambda }^j$ is a positive definite left-invariant metric.