= Left-invariant coframe of the translation-dilation group
{title2=$\lambda^0=d\rho/\rho,\quad\lambda^a=dx^a/\rho$}
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>.
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