Maurer-Cartan form

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 [1]. The forms $ \omega _ {i} $ were introduced by E. Cartan in 1904 (see [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.

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