Right Maurer–Cartan form

Let $G$ be a Lie group with Lie algebra Lie algebra $\mathfrak{g}=T_eG$. The right Maurer–Cartan form is the $\mathfrak{g}$-valued 1-form $\theta^R$ on $G$ defined at each $g\in G$ by

where $R_{g^{-1}}$ is right translation by $g^{-1}$.

It is characterized by:

• for each $g$, $\theta^R_g$ is the inverse of $(\mathrm{d}R_g)_e:\mathfrak{g}\to T_gG$;
• $\theta^R$ is right-invariant: $(R_g)^*\theta^R=\theta^R$ for all $g\in G$.

This form provides a canonical “right trivialization” of the tangent bundle $TG\cong G\times\mathfrak{g}$.

Examples

1. Matrix Lie groups. For $G\subset \mathrm{GL}(n,\mathbb{R})$, the right Maurer–Cartan form is $\theta^R=\mathrm{d}g\,g^{-1}$.
2. Abelian groups. If $G$ is abelian (e.g. $\mathbb{R}^n$ or $T^n$), then left and right Maurer–Cartan forms agree under the natural identifications because left and right translations coincide.
3. Circle group. For $G=S^1$ with $\mathfrak{g}\cong\mathbb{R}$, $\theta^R$ is again the angular 1-form; on a standard angle chart it is $\mathrm{d}\theta$.