# 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 [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.

#### References

 [1] L. Maurer, Sitzungsber. Bayer. Akad. Wiss. Math. Phys. Kl. (München) , 18 (1879) pp. 103–150 [2] E. Cartan, "Sur la structure des groupes infinis de transformations" Ann. Ecole Norm. , 21 (1904) pp. 153–206 [3] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) [4] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) [5] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
How to Cite This Entry:
Maurer–Cartan form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maurer%E2%80%93Cartan_form&oldid=22800