In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, ring theory is the study of rings, 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 G is the element This picture illustrates how the hours in a clock form a group. ...

[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 of G generated by all commutators is called the derived group or the commutator subgroup 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 and solvable groups. 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 *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ...

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

In the sequel the expression a^x denotes the conjugated (by x) element x⁻¹a x.

• .
• .
• .
• .

The second identity is also known under the name Hall-Witt identity. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section). The fourth identity follows from the first and third.

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 ^xa (but not by Harish-Chandra). See Harishchandra for the character in Hindu mythology Harish-Chandra (11 October 1923-16 October 1983) was an Indian mathematician, who did fundamental work in representation theory. ...

Ring theory

The commutator of two elements a and b of a ring or an associative algebra is defined by In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...

[a, b] = ab − ba

It is zero if and only if a and b commute. In linear algebra, 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, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about these commutators. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... Fig. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical limit on the accuracy with which it is possible to measure everything there is to know about a physical system. ... Look up theorem in Wiktionary, the free dictionary. ...

Identities

The commutator has the following properties:

Lie-algebra relations:

The second relation is called anticommutativity, while the third is the Jacobi identity. A mathematical operator (typically a binary operator, represented by *) is anticommutative if and only if it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ... In mathematics the 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 A 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 D_A defines a derivation on the ring . At least two results in calculus are called Leibnizs rule or the Leibniz rule, in honor of Gottfried Leibniz. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

Graded Rings and Algebras

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as [ω,η]_gr: = ωη − ( − 1)^degωdegηηω In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

Derivations

Especially if one deals with multiple commutators, another notation turns out to be useful:

Then ad(x) is a derivation and ad is linear, i.e., ad(x + y) = ad(x) + ad(y) and , and a Lie algebra homomorphism, i.e, ad([x,y]) = [ad(x),ad(y)], but it is not always an algebra homomorphism, i.e the identity does not hold in general. There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

Examples:

See also

• Anticommutativity
• Derivation (abstract algebra)
• Lie algebra
• Pincherle derivative
• Poisson bracket
• Canonical commutation relation

A mathematical operator (typically a binary operator, represented by *) is anticommutative if and only if it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the Pincherle derivative of a linear operator T on the space of polynomials in x is another linear operator T′ defined by which means that for any polynomial f(x), This is a derivation satisfying the sum and product rules: (T + S)′ = T′ + S&#8242... 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. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant . ...

References

• Griffiths, David J. (2004). Introduction to Quantum Mechanics, 2nd ed., Prentice Hall. ISBN 0-13-805326-X. 
• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.