Maurer-Cartan coframe of the real affine group
= Maurer-Cartan coframe of the real affine group
{c}
{title2=$\sigma^D=da/a,\quad\sigma^T=db/a$}
In the faithful matrices $g=\begin{pmatrix}a&b\\0&1\end{pmatrix}$ 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 $[D,T]=T$.