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A computational identity: the exterior derivative of the Maurer–Cartan form is the negative bracket.
Let be a Lie group with Lie algebra , and let be the left Maurer–Cartan form.
This lemma is the workhorse behind computations with invariant forms and is the differential-geometric source of the Lie bracket, complementary to the flow-based viewpoint via one-parameter subgroups as integral curves.
A Lie group is a group equipped with the structure of a smooth manifold such that the group operations are smooth maps:
For each , the left translation and the right translation are diffeomorphisms of , with inverses and . The tangent space at the identity carries a canonical Lie algebra structure, called the Lie algebra of , and the exponential map relates this infinitesimal structure to local group behavior near .
Smooth group homomorphisms between Lie groups are Lie group homomorphisms.
Let be a Lie group with Lie algebra .
Definition (Left Maurer–Cartan form). The left Maurer–Cartan form is the -valued -form defined by
Equivalently, is characterized by:
Let be a Lie group with Lie algebra .
Let denote the left Maurer–Cartan form on , i.e. the -valued 1-form characterized by
Then satisfies the Maurer–Cartan equation
Here denotes the -valued 2-form obtained by combining wedge product with the Lie bracket: for vector fields on ,
Let be a Lie group with Lie algebra .
Fix , and let be the corresponding left-invariant vector field on (obtained by translating via left translations).
where is the exponential map. In particular, solves the ODE with .
There is an analogous statement for the right-invariant vector field , whose integral curves are .