In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the JacobiLie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields on a smooth manifold.

along the flow, and is sometimes denoted ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field.

The Lie bracket is an R-bilinear operation and turns the set of all smooth into an (infinite-dimensional) Lie algebra.

The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems.

V. I. Arnold refers to this as the "fisherman derivative", as one can imagine being a fisherman, holding a fishing rod, sitting in a boat. Both the boat and the float. The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.

Definitions

There are three conceptually different but equivalent approaches to defining the Lie bracket:

Vector fields as derivations

may be regarded as a differential operator is the directional derivative becomes a derivation.

In general, the commutator denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

Flows and limits

be the flow denote the tangent map derivative operator can be defined as the Lie derivative:

:

In coordinates

Though the above definitions of Lie bracket are intrinsic. Then the Lie bracket can be computed as:

is given by:

Jacobian matrices.

Properties

) with the structure of a Lie algebra with:

• R-bilinearity

• Jacobi identity

.

Furthermore, there is a "product rule. Then:

. This turns the vector fields with the Lie bracket into a Lie algebroid.

as coordinate vector fields:

Theorem:.

This is a special case of the Frobenius integrability theorem.

Examples

For a Lie group, the corresponding Lie algebra, which can be identified with the vector space of left invariant.

corresponds to the usual commutator of matrices:

Generalizations

As mentioned above, the Lie derivative can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to vector-valued differential forms) is the Frölicher–Nijenhuis bracket.

References

• Extensive discussion of Lie brackets, and the general theory of Lie derivatives.

• For generalizations to infinite dimensions.