Maurer–Cartan form

The Maurer - Cartan form is a commonly used in differential geometry and mathematical physics Lie algebra -valued differential form on Lie groups.

Definition

Let be a Lie group, its Lie algebra. For induces the left- multiplication

The differential

The Maurer - Cartan form is defined by

For.

Maurer - Cartan equation

The Maurer - Cartan form satisfies the equation

Here, the commutator Lie algebra -valued differential forms is by

And the exterior derivative by

Defined.