Knowlpedia

Right Maurer–Cartan form

The canonical g-valued 1-form on a Lie group obtained by translating tangent vectors to the identity on the right.

Let GG be a Lie group with Lie algebra g=Lie(G)\mathfrak g=\mathrm{Lie}(G) (see ). The right Maurer–Cartan form is the g\mathfrak g-valued 1-form θR∈Ω1(G;g)\theta^R\in\Omega^1(G;\mathfrak g) defined by

θgR  :=  (dRg−1)g:TgG⟶TeG≅g,\theta^R_g \;:=\; (dR_{g^{-1}})_g : T_gG \longrightarrow T_eG \cong \mathfrak g,

where Rg−1R_{g^{-1}} is by g−1g^{-1}.

Properties

Key properties:

  • Right invariance: (Rh)∗θR=θR(R_h)^*\theta^R=\theta^R for all h∈Gh\in G, so θR\theta^R is a canonical example of a .
  • Left equivariance: under left translation, θR\theta^R transforms by the :
    (Lh)∗θR=Adh∘θR.(L_h)^*\theta^R = \mathrm{Ad}_h\circ \theta^R.
  • Maurer–Cartan equation (right form):
    dθR−[θR,θR]=0,d\theta^R - [\theta^R,\theta^R]=0,
    where, in the convention used by the , [θR,θR](X,Y)=[θR(X),θR(Y)][\theta^R,\theta^R](X,Y)=[\theta^R(X),\theta^R(Y)].

If XX is a , then θR(X)\theta^R(X) is constant in GG and recovers the corresponding element of g\mathfrak g. This is one way to see the tight relationship between invariant vector fields, , and the .