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Commutator
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the commutator gives an indication of the extent to which a certain binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
 fails to be commutative. There are different definitions used in group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 and ring theory
Ring theory

In mathematics, ring theory is the study of ring , algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers....
.

[g, h] = g−1h−1gh
It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg).






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the commutator gives an indication of the extent to which a certain binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
 fails to be commutative. There are different definitions used in group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 and ring theory
Ring theory

In mathematics, ring theory is the study of ring , algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers....
.

Group theory


The commutator of two elements, g and h, of a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
, G, is the element
[g, h] = g−1h−1gh
It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup
Subgroup

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *....
 of generated by all commutators is called the derived group or the commutator subgroup
Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generating set of a group by all the commutators of the group....
 of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent
Nilpotent group

In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian group, through repeated application of the commutator operation, [x,y] = x-1y-1xy....
 and solvable
Solvable group

In the history of mathematics, the origins of group theory lie in the search for a Mathematical_proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory....
 groups.

N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as
[g, h] = ghg−1h−1


Identities


Commutator identities are an important tool in group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
, . The expression ax denotes x−1a x.

  3. and
  4. and
  5. and


Identity 5 is also known as the Hall-Witt identity. It is a group-theoretic analogue of the Jacobi identity
Jacobi identity

In mathematics the Carl Gustav Jakob Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation....
 for the ring-theoretic commutator (see next section).

N.B. The above definition of the conjugate of a by x is used by group theorists. Many other mathematicians define the conjugate of a by x as xax−1. This is usually written . Similar identities hold for these conventions.

A wide range of identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable group
Solvable group

In the history of mathematics, the origins of group theory lie in the search for a Mathematical_proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory....
s and nilpotent group
Nilpotent group

In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian group, through repeated application of the commutator operation, [x,y] = x-1y-1xy....
s. For instance, in any group second powers behave well



If the derived subgroup is central, then

Ring theory


The commutator of two elements a and b of a ring or an associative algebra
Associative algebra

In mathematics, an associative algebra is a vector space which also allows the multiplication of vectors in a distributivity and associativity manner....
 is defined by
[a, b] = ab − ba
It is zero if and only if a and b commute. In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, if two endomorphisms of a space are represented by commuting matrices with respect to one basis, then they are so represented with respect to every basis. By using the commutator as a Lie bracket
Lie bracket

Lie bracket can refer to:*Lie algebra*Lie bracket of vector fields...
, every associative algebra can be turned into a Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
. The commutator of two operators defined on a Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
 is an important concept in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 since it measures how well the two observable
Observable

In physics, particularly in quantum physics, a system observable is a property of the State that can be determined by some sequence of physical operational definition....
s described by the operators can be measured simultaneously. The uncertainty principle
Uncertainty principle

In quantum physics, the Werner Heisenberg uncertainty principle states that certain physical quantities, like the position and momentum, cannot both have precise values at the same time....
 is ultimately a theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
 about these commutators via the Robertson-Schrödinger relation.

Identities

The commutator has the following properties:

Lie-algebra relations:
The second relation is called anticommutativity
Anticommutativity

In mathematics, anticommutativity refers to the property of an Operation being anticommutative, i.e. being non-Commutativity in a precise way....
, while the third is the Jacobi identity
Jacobi identity

In mathematics the Carl Gustav Jakob Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation....
.

Additional relations:


If ' is a fixed element of a ring , the first additional relation can also be interpreted as a Leibniz rule for the map given by . In other words: the map ' defines a derivation
Derivation (abstract algebra)

In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...
 on the ring .

The following identity involving commutators, a special case of the Baker-Campbell-Hausdorff formula
Baker-Campbell-Hausdorff formula

In mathematics, the Baker-Campbell-Hausdorff formula is the solution tofor non-Commutativity X and Y. It links Lie Groups to Lie Algebras, by expressing the logarithm of the product of two Lie group elements as a Lie algebra element in...
, is also useful:

Graded rings and algebras

When dealing with graded algebra
Graded algebra

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
s, the commutator is usually replaced by the graded commutator, defined in homogeneous components as

Derivations

Especially if one deals with multiple commutators, another notation turns out to be useful involving the adjoint representation
Adjoint representation

In mathematics, the adjoint representation of a Lie group G is the natural group representation of G on its own Lie algebra. This representation is the linearized version of the group action of G on itself by conjugation ....
:



Then is a derivation
Derivation (abstract algebra)

In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...
 and is linear, i.e., and , and a Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 homomorphism, i.e, , but it is not always an algebra homomorphism, i.e the identity does not hold in general.

Examples:


Anticommutator

The anticommutator of two elements a and b of a ring or an associative algebra is defined by
= ab + ba
The anticommutator is used a lot less than the commutator, but can be used for example to define the Clifford Algebra
Clifford algebra

In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions....
.

See also

  • Anticommutativity
    Anticommutativity

    In mathematics, anticommutativity refers to the property of an Operation being anticommutative, i.e. being non-Commutativity in a precise way....
  • Derivation (abstract algebra)
    Derivation (abstract algebra)

    In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...
  • Pincherle derivative
    Pincherle derivative

    In mathematics, the Pincherle derivative of a linear operator on the vector space of polynomials in the variable over a field is another linear operator defined as...
  • Poisson bracket
    Poisson bracket

    In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation....
  • Canonical commutation relation
    Canonical commutation relation

    In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Planck's constant ....