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, that 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:
ω ( v ) = ( L g − 1 ) ∗ v , v ∈ T g G . {\displaystyle \omega (v)=(L_{g^{-1}})_{*}v,\quad v\in T_{g}G.}
Motivation and interpretation
A Lie group acts on itself by multiplication under the mapping
G × G ∋ ( g , h ) ↦ g h ∈ G . {\displaystyle G\times G\ni (g,h)\mapsto gh\in G.}
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.
Construction
Intrinsic construction Let g ≅ TeG be the tangent space of a Lie group G at the identity (its Lie algebra). G acts on itself by left translation
L : G × G → G {\displaystyle L:G\times G\to G}
such that for a given g ∈ G we have
L g : G → G where L g ( h ) = g h , {\displaystyle L_{g}:G\to G\quad {\mbox{where}}\quad L_{g}(h)=gh,}
and this induces a map of the tangent bundle to itself:
( L g ) ∗ : T h G → T g h G . {\displaystyle (L_{g})_{*}:T_{h}G\to T_{gh}G.}
A left-invariant vector field is a section X of TG such that
( L g ) ∗ X = X ∀ g ∈ G . {\displaystyle (L_{g})_{*}X=X\quad \forall g\in G.}
The Maurer–Cartan form ω is a g-valued one-form on G defined on vectors v ∈ TgG by the formula
ω g ( v ) = ( L g − 1 ) ∗ v . {\displaystyle \omega _{g}(v)=(L_{g^{-1}})_{*}v.}
Extrinsic construction
If G is embedded in GL(n) by a matrix valued mapping g =(gij), then one can write ω explicitly as
ω g = g − 1 d g . {\displaystyle \omega _{g}=g^{-1}\,dg.}
In this sense, the Maurer–Cartan form is always the left logarithmic derivative of the identity map of G.
Characterization as a connection If we regard the Lie group G as a principal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique principal connection on the principal bundle G. Indeed, it is the unique g = TeG valued 1-form on G satisfying
ω e = i d : T e G → g , and {\displaystyle \omega _{e}=\mathrm {id} :T_{e}G\rightarrow {\mathfrak {g}},{\text{ and}}}
… excerpt ends here. Continue reading the full article.
