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 .
Read more about this topic: Lie Bracket Of Vector Fields
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)