Concept

Forme de Maurer-Cartan

Résumé
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 TeG. The Maurer–Cartan form ω is thus a one-form defined globally on G which is a linear mapping of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in TgG along the left-translation in the group: Lie group action 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. A principal homogeneous space of G is a manifold P abstractly characterized by having a free and transitive action of G on P. The Maurer–Cartan form gives an appropriate infinitesimal characterization of the principal homogeneous space. It is a one-form defined on P satisfying an integrability condition known as the Maurer–Cartan equation. Using this integrability condition, it is possible to define the exponential map of the Lie algebra and in this way obtain, locally, a group action on P. Let g ≅ TeG be the tangent space of a Lie group G at the identity (its Lie algebra).
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