A left-invariant -form on a Lie group , i.e. a differential form of degree 1 on satisfying the condition for any left translation , . The Maurer–Cartan forms on are in one-to-one correspondence with the linear forms on the tangent space at the point ; specifically, the mapping which sends each Maurer–Cartan form to its value is an isomorphism of the space of Maurer–Cartan forms onto . The differential of a Maurer–Cartan form is a left-invariant -form on , defined by the formula
where are arbitrary left-invariant vector fields on . Suppose that is a basis in and let , , be Maurer–Cartan forms such that
where are the structure constants of the Lie algebra of consisting of the left-invariant vector fields on , with respect to the basis determined by
The equalities (2) (or (1)) are called the Maurer–Cartan equations. They were first obtained (in a different, yet equivalent form) by L. Maurer . The forms were introduced by E. Cartan in 1904 (see ).
Let be the canonical coordinates in a neighbourhood of the point determined by the basis . Then the forms are written in the form
in which the matrix
is calculated by the formula
where and is the adjoint representation of the Lie algebra .
Furthermore, let be the -valued -form on which assigns to each tangent vector to the unique left-invariant vector field containing this vector ( is called the canonical left differential form). Then
which is yet another way of writing the Maurer–Cartan equations.
|||L. Maurer, Sitzungsber. Bayer. Akad. Wiss. Math. Phys. Kl. (München) , 18 (1879) pp. 103–150|
|||E. Cartan, "Sur la structure des groupes infinis de transformations" Ann. Ecole Norm. , 21 (1904) pp. 153–206|
|||C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)|
|||N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)|
|||S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)|
Maurer-Cartan form. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Maurer-Cartan_form&oldid=22799