 Commutator

This article is about the mathematical concept. For the relation between canonical conjugate entities, see canonical commutation relation. For the type of electrical switch, see commutator (electric).
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.
Contents
Group theory
The commutator of two elements, g and h, of a group, G, is the element
 [g, h] = g^{−1}h^{−1}gh.
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.
N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as
 [g, h] = ghg^{−1}h^{−1}.
Identities
Commutator identities are an important tool in group theory, (McKay 2000, p. 4). The expression a^{x} denotes the conjugate of a by x, defined as x^{−1}a x.
 and
 and
 and
Identity 5 is also known as the HallWitt identity. It is a grouptheoretic analogue of the Jacobi identity for the ringtheoretic 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 often written ^{x}a. 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 groups and nilpotent groups. 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 is defined by
 [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 via the RobertsonSchrödinger relation.
Identities
The commutator has the following properties:
Liealgebra relations:
 [A,A] = 0
 [A,B] = − [B,A]
 [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0
The second relation is called anticommutativity, while the third is the Jacobi identity.
Additional relations:
 [A,BC] = [A,B]C + B[A,C]
 [AB,C] = A[B,C] + [A,C]B
 [ABC,D] = AB[C,D] + A[B,D]C + [A,D]BC
 [AB,CD] = A[B,CD] + [A,CD]B = A[B,C]D + AC[B,D] + [A,C]DB + C[A,D]B
 [[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] = [[A,C],[B,D]]
 [AB,C] = A{B,C} − {A,C}B, where {A,B}=AB+BA is the anticommutator defined below
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 .
The following identity involving nested commutators, underlying the CampbellBakerHausdorff expansion, is also useful:
Graded rings and algebras
When dealing with graded algebras, 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:
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.
Examples:
Anticommutator
The anticommutator of two elements a and b of a ring or an associative algebra is defined by
 {a, b} = ab + ba.
The anticommutator is used less often than the commutator, but can be used for example to define Clifford algebras and Jordan algebras.
See also
 Anticommutativity
 Derivation (abstract algebra)
 Pincherle derivative
 Poisson bracket
 Moyal bracket
 Canonical commutation relation
 Associator
References
 Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 013805326X
 Liboff, Richard L. (2002), Introductory Quantum Mechanics, AddisonWesley, ISBN 0805387145
 McKay, Susan (2000), Finite pgroups, Queen Mary Maths Notes, 18, University of London, ISBN 9780902480179, MR1802994
External links
Categories: Abstract algebra
 Group theory
 Binary operations
 Mathematical identities
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