= Invariant frames of the real affine group
{title2=$D_L=\partial_\alpha,\ T_L=e^\alpha\partial_\beta,\ D_R=\partial_\alpha+\beta\partial_\beta,\ T_R=\partial_\beta$}
In exponential scale coordinates, the <real affine group> has matrices $\begin{pmatrix}e^\alpha&\beta\\0&1\end{pmatrix}$. The left and right <Maurer-Cartan forms> are respectively $D\,d\alpha+T\,e^{-\alpha}d\beta$ and $D\,d\alpha+T(d\beta-\beta\,d\alpha)$. Their dual <vector fields> obey $[D_L,T_L]=T_L$ and $[D_R,T_R]=-T_R$, displaying the opposite bracket sign for <right-invariant vector fields>.
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