Maurer-Cartan form

In mathematics, the Maurer–Cartan form for a Lie group $G$ is a distinguished differential one-form on $G$ that carries the basic infinitesimal information about the structure of $G$ . It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer.

As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the Lie algebra associated to the Lie group $G$ . The Lie algebra is identified with the tangent space of $G$ at the identity, denoted $T e G$ . The Maurer–Cartan form $ω$ is thus a one-form defined globally on $G$ which is a linear mapping of the tangent space $T g G$ at each $g \in G$ into $T e G$ . It is given as the pushforward of a vector in $T g G$ along the left-translation in the group:

A Lie group acts on itself by multiplication under the mapping

A question of importance to Cartan and his contemporaries was how to identify a principal homogeneous space of $G$ . That is, a manifold $P$ identical to the group $G$ , but without a fixed choice of unit element. This motivation came, in part, from Felix Klein's Erlangen programme where one was interested in a notion of symmetry on a space, where the symmetries of the space were transformations forming a Lie group. The geometries of interest were homogeneous spaces $G / H$ , but usually without a fixed choice of origin corresponding to the coset $eH$ .

...
Wikipedia