View source for Maurer-Cartan form

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A left-invariant  $  1 $-
form on a Lie group  $  G $, 
i.e. a differential form  $  \omega $
of degree 1 on  $  G $
satisfying the condition  $  l _ {g}  ^  \star  \omega = \omega $
for any left translation  $  l _ {g} :  x \rightarrow gx $, 
$  g, x \in G $. 
The Maurer–Cartan forms on  $  G $
are in one-to-one correspondence with the linear forms on the tangent space  $  T _ {e} ( G) $
at the point  $  e $; 
specifically, the mapping which sends each Maurer–Cartan form  $  \omega $
to its value  $  \omega _ {e} \in T _ {e} ( G)  ^  \star  $
is an isomorphism of the space of Maurer–Cartan forms onto  $  T _ {e} ( G)  ^  \star  $. 
The differential of a Maurer–Cartan form  $  \omega $
is a left-invariant  $  2 $-
form on  $  G $, 
defined by the formula

$$ \tag{1 }
d \omega ( X, Y)  =  - \omega ([ X, Y]),
$$

where  $  X, Y $
are arbitrary left-invariant vector fields on  $  G $. 
Suppose that  $  X _ {1} \dots X _ {n} $
is a basis in  $  T _ {e} ( G) $
and let  $  \omega _ {i} $, 
$  i = 1 \dots n $, 
be Maurer–Cartan forms such that

$$ 
( \omega _ {i} ) _ {e} ( X _ {j} )  =  \delta _ {ij} ,\ \ 
j = 1 \dots n.
$$

Then

$$ \tag{2 }
d \omega _ {i}  =  - \sum _ { j,k= } 1 ^ { n }  c _ {jk}  ^ {i}
\omega _ {j} \wedge \omega _ {k} ,
$$

where  $  c _ {jk}  ^ {i} $
are the structure constants of the Lie algebra  $  \mathfrak g $
of  $  G $
consisting of the left-invariant vector fields on  $  G $, 
with respect to the basis  $  \widetilde{X}  _ {1} \dots \widetilde{X}  _ {n} $
determined by

$$ 
( \widetilde{X}  _ {i} ) _ {e}  =  X _ {i} ,\ \ 
i = 1 \dots n.
$$

The equalities (2) (or (1)) are called the Maurer–Cartan equations. They were first obtained (in a different, yet equivalent form) by L. Maurer [[#References|[1]]]. The forms  $  \omega _ {i} $
were introduced by E. Cartan in 1904 (see [[#References|[2]]]).

Let  $  x _ {1} \dots x _ {n} $
be the canonical coordinates in a neighbourhood of the point  $  e \in G $
determined by the basis  $  X _ {1} \dots X _ {n} $. 
Then the forms  $  \omega _ {i} $
are written in the form

$$ 
\omega _ {i}  =  \sum _ { j= } 1 ^ { n }  A _ {ij} ( x _ {1} \dots x _ {n} )  dx _ {j} ,
$$

in which the matrix

$$ 
A( x _ {1} \dots x _ {n} )  = \ 
( A _ {ij} ( x _ {1} \dots x _ {n} ))
$$

is calculated by the formula

$$ 
A( x _ {1} \dots x _ {n} )  = \ 

\frac{1- e ^ {-  \mathop{\rm ad}  X } }{ \mathop{\rm ad}  X }
 ,
$$

where  $  X = \sum _ {i=} 1  ^ {n} x _ {i} \widetilde{X}  _ {i} $
and  $   \mathop{\rm ad} $
is the adjoint representation of the Lie algebra  $  \mathfrak g $.

Furthermore, let  $  \theta $
be the  $  \mathfrak g $-
valued  $  1 $-
form on  $  G $
which assigns to each tangent vector to  $  G $
the unique left-invariant vector field containing this vector ( $  \theta $
is called the canonical left differential form). Then

$$ 
\theta  =  \sum _ { i= } 1 ^ { n }  \widetilde{X}  _ {i} \omega _ {i}  $$

and

$$ 
d \theta + 
\frac{1}{2}
 [ \theta , \theta ]  =  0,
$$

which is yet another way of writing the Maurer–Cartan equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Maurer,   ''Sitzungsber. Bayer. Akad. Wiss. Math. Phys. Kl. (München)'' , '''18'''  (1879)  pp. 103–150</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Cartan,   "Sur la structure des groupes infinis de transformations"  ''Ann. Ecole Norm.'' , '''21'''  (1904)  pp. 153–206</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>