Properties
The Lie bracket of vector fields equips the real vector space (i.e., smooth sections of the tangent bundle of ) with the structure of a Lie algebra, i.e., is a map from to with the following properties
- is R-bilinear
- This is the Jacobi identity.
- For functions f and g we have
-
- .
These three properties together also define a Lie algebroid. Note that the infinite dimensional Lie algebra has nice topological properties.
An immediate consequence of the second property is that for any .
The name commutator is used because the Lie bracket is the commutator of the vector fields considered as differentiable operators. We also have the following fact:
Theorem:
iff the commutator of flows is a closed loop. Alternatively the lift of the flows on the universal covering commute .
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