Maurer-Cartan form

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Definition

Let G be a (connected) Lie group. The Maurer-Cartan form of G is a 1-form on G with values in the Lie algebra \mathfrak{g}, given as follows.

The tangent vector v at a point a \in G, is sent to the unique vector X \in \mathfrak{g} which is mapped by the differential of g \mapsto ag, to v. In other words, the map is:

v \mapsto Df(v)

where:

f := g \mapsto a^{-1}g