Orientation-preserving affine group of the real line (${\mathrm{Aff}}^{+}(\mathbb{R})$)

The real affine group used here is the orientation-preserving subgroup of the affine group on the real line. It acts by $x\mapsto e^{a}x+b$ and has multiplication $(a,b)(a^{\prime },b^{\prime })=(a+a^{\prime },b+e^a{b}^{\prime })$. The positive dilation distinguishes this connected component from the full group, which also contains negative dilations.

In exponential scale coordinates, the real affine group has matrices $\left(\begin{array}{cc}{e}^{\alpha }& \beta \\ 0& 1\end{array}\right)$. The left and right Maurer-Cartan forms are respectively $Dd\alpha +T{e}^{-\alpha }d\beta$ and $Dd\alpha +T(d\beta -\beta d\alpha )$. 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.

In the faithful matrices $g=\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right)$ with $a>0$, the Maurer-Cartan form has entries $da/a$ and $db/a$. Thus the left-invariant differential forms ${\sigma }^D=da/a$, ${\sigma }^T=db/a$ obey $d{\sigma }^D=0$ and $d{\sigma }^T=-{\sigma }^D\wedge {\sigma }^T$. The Lie algebra generators satisfy $[D,T]=T$.