In the mathematical field of differential topology, the **Lie bracket of vector fields**, also known as the **Jacobi–Lie bracket** or the **commutator of vector fields**, is an operator that assigns to any two vector fields *X* and *Y* on a smooth manifold *M* a third vector field denoted [*X*, *Y*].

- Definitions
- Vector fields as derivations
- Flows and limits
- In coordinates
- Properties
- Examples
- Applications
- Generalizations
- References

Conceptually, the Lie bracket [*X*, *Y*] is the derivative of *Y* along the flow generated by *X*, and is sometimes denoted * ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by **X*.

The Lie bracket is an **R**-bilinear operation and turns the set of all smooth vector fields on the manifold *M* 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.^{ [1] }

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

Each smooth vector field on a manifold *M* may be regarded as a differential operator acting on smooth functions (where and of class ) when we define to be another function whose value at a point is the directional derivative of *f* at *p* in the direction *X*(*p*). In this way, each smooth vector field *X* becomes a derivation on *C*^{∞}(*M*). Furthermore, any derivation on *C*^{∞}(*M*) arises from a unique smooth vector field *X*.

In general, the commutator of any two derivations and is again a derivation, where denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:

Let be the flow associated with the vector field *X*, and let D denote the tangent map derivative operator. Then the Lie bracket of *X* and *Y* at the point *x*∈*M* can be defined as the Lie derivative:

This also measures the failure of the flow in the successive directions to return to the point *x*:

Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold *M*), in practice one often wants to compute the bracket in terms of a specific coordinate system . We write for the associated local basis of the tangent bundle, so that general vector fields can be written and for smooth functions . Then the Lie bracket can be computed as:

If *M* is (an open subset of) **R**^{n}, then the vector fields *X* and *Y* can be written as smooth maps of the form and , and the Lie bracket is given by:

where and are *n* × *n* Jacobian matrices ( and respectively using index notation) multiplying the *n* × 1 column vectors *X* and *Y*.

The Lie bracket of vector fields equips the real vector space of all vector fields on *M* (i.e., smooth sections of the tangent bundle ) with the structure of a Lie algebra, which means [ • , • ] is a map with:

**R**-bilinearity- Anti-symmetry,
- Jacobi identity,

An immediate consequence of the second property is that for any .

Furthermore, there is a "product rule" for Lie brackets. Given a smooth (scalar-valued) function *f* on *M* and a vector field *Y* on *M*, we get a new vector field *fY* by multiplying the vector *Y _{x}* by the scalar

where we multiply the scalar function *X*(*f*) with the vector field *Y*, and the scalar function *f* with the vector field [*X*, *Y*]. This turns the vector fields with the Lie bracket into a Lie algebroid.

Vanishing of the Lie bracket of *X* and *Y* means that following the flows in these directions defines a surface embedded in *M*, with *X* and *Y* as coordinate vector fields:

**Theorem:** iff the flows of *X* and *Y* commute locally, meaning for all *x*∈*M* and sufficiently small *s*, *t*.

This is a special case of the Frobenius integrability theorem.

For a Lie group *G*, the corresponding Lie algebra is the tangent space at the identity , which can be identified with the vector space of left invariant vector fields on *G*. The Lie bracket of two left invariant vector fields is also left invariant, which defines the Jacobi–Lie bracket operation .

For a matrix Lie group, whose elements are matrices , each tangent space can be represented as matrices: , where means matrix multiplication and *I* is the identity matrix. The invariant vector field corresponding to is given by , and a computation shows the Lie bracket on corresponds to the usual commutator of matrices:

The Jacobi–Lie bracket is essential to proving small-time local controllability (STLC) for driftless affine control systems.

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.

In vector calculus, the **gradient** of a scalar-valued differentiable function *f* of several variables is the vector field whose value at a point is the vector whose components are the partial derivatives of at . That is, for , its gradient is defined at the point in *n-*dimensional space as the vector:

In mathematics, a **Lie algebra** is a vector space together with an operation called the **Lie bracket**, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative.

In mathematics, the **tangent space** of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

**Hamiltonian mechanics** emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) *momenta*. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

In mathematics and classical mechanics, the **Poisson bracket** is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called *canonical transformations*, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.

In mathematics, the **adjoint representation** of a Lie group *G* is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if *G* is , the Lie group of real *n*-by-*n* invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible *n*-by-*n* matrix to an endomorphism of the vector space of all linear transformations of defined by: .

In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.

In differential geometry, a **Poisson structure** on a smooth manifold is a Lie bracket on the algebra of smooth functions on , subject to the Leibniz rule

In mathematics, a **Killing vector field**, named after Wilhelm Killing, is a vector field on a Riemannian manifold that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the **Killing vector** will not distort distances on the object.

In mathematical analysis, the **smoothness** of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered smooth if it is differentiable everywhere. At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be **infinitely differentiable** and referred to as a **C-infinity function**.

In mathematics, the **jet** is an operation that takes a differentiable function *f* and produces a polynomial, the truncated Taylor polynomial of *f*, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In physics, a **sigma model** is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or a symmetric space. The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the Skyrmion for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by Ginzburg–Landau theory. This article is primarily devoted to the classical field theory of the sigma model; the corresponding quantized theory is presented in the article titled "non-linear sigma model".

In differential geometry, a discipline within mathematics, a **distribution** on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .

The following are important identities involving derivatives and integrals in vector calculus.

In mathematics, the **Frölicher–Nijenhuis bracket** is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold.

In mathematics, the **classical groups** are defined as the special linear groups over the reals **R**, the complex numbers **C** and the quaternions **H** together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the **complex classical Lie groups** are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The **compact classical groups** are compact real forms of the complex classical groups. The finite analogues of the classical groups are the **classical groups of Lie type**. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph *The Classical Groups*.

A **Representation up to homotopy** has several meanings. One of the earliest appeared in the `physical' context of constrained Hamiltonian systems. The essential idea is lifting a non-representation on a quotient to a **representation up to strong homotopy** on a resolution of the quotient. As a concept in differential geometry, it generalizes the notion of representation of a Lie algebra to Lie algebroids and nontrivial vector bundles. As such, it was introduced by Abad and Crainic.

In mathematics, **Lie group–Lie algebra correspondence** allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Isomorphic Lie groups have isomorphic Lie algebras but the converse is not necessarily true. One obvious counter example is and which are non-isomorphic as Lie groups but their Lie algebras are isomorphic. However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.

**Computational anatomy** is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.

- ↑ Isaiah 2009 , pp. 20–21, nonholonomic systems; Khalil 2002 , pp. 523–530, feedback linearization.

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*IEEE Control Systems Magazine*,**29**(3): 17–21, 132, doi:10.1109/MCS.2009.932394, S2CID 42908664 - Khalil, H.K. (2002),
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*Natural operations in differential geometry*, Springer-VerlagCS1 maint: multiple names: authors list (link) Extensive discussion of Lie brackets, and the general theory of Lie derivatives. - Lang, S. (1995),
*Differential and Riemannian manifolds*, Springer-Verlag, ISBN 978-0-387-94338-1 For generalizations to infinite dimensions. - Lewis, Andrew D.,
*Notes on (Nonlinear) Control Theory*(PDF)^{[ permanent dead link ]} - Warner, Frank (1983) [1971],
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