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Let G {\displaystyle G} be a (connected) Lie group. The Maurer-Cartan form of G {\displaystyle G} is a 1-form on G {\displaystyle G} with values in the Lie algebra g {\displaystyle {\mathfrak {g}}} , given as follows.
The tangent vector v {\displaystyle v} at a point a ∈ G {\displaystyle a\in G} , is sent to the unique vector X ∈ g {\displaystyle X\in {\mathfrak {g}}} which is mapped by the differential of g ↦ a g {\displaystyle g\mapsto ag} , to v {\displaystyle v} . In other words, the map is:
v ↦ D f ( v ) {\displaystyle v\mapsto Df(v)}
where:
f := g ↦ a − 1 g {\displaystyle f:=g\mapsto a^{-1}g}
