Right Maurer–Cartan form

Let $G$ be a Lie group with Lie algebra $\mathfrak g=\mathrm{Lie}(G)$ (see Lie algebra of a Lie group). The right Maurer–Cartan form is the $\mathfrak g$-valued 1-form $\theta^R\in\Omega^1(G;\mathfrak g)$ defined by

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

Key properties:

• Right invariance: $(R_h)^*\theta^R=\theta^R$ for all $h\in G$, so $\theta^R$ is a canonical example of a right-invariant form.
• Left equivariance: under left translation, $\theta^R$ transforms by the adjoint action:
  $$(L_h)^*\theta^R = \mathrm{Ad}_h\circ \theta^R.$$
• Maurer–Cartan equation (right form):
  $$d\theta^R - \tfrac12[\theta^R,\theta^R]=0,$$
  where the bracket is induced from the Lie bracket on $\mathfrak g$ (compare Maurer–Cartan equation and the left Maurer–Cartan form).

If $X$ is a right-invariant vector field, then $\theta^R(X)$ is constant in $G$ and recovers the corresponding element of $\mathfrak g$. This is one way to see the tight relationship between invariant vector fields, one-parameter subgroups, and the exponential map.