The commutator of two elements, g and h, of a group 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 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.
The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as
Identities (group theory)
Identity 5 is also known as the Hall–Witt identity. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).
N.B. The above definition of the conjugate of a by x is used by some group theorists. Many other group theorists define the conjugate of a by x as xax−1. This is often 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 groups and nilpotent groups. For instance, in any group second powers behave well,
If the derived subgroup is central, then
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 anticommutator of two elements a and b of a ring or an associative algebra is defined by
Sometimes the brackets [ ]+ are also used to denote anticommutators. The anticommutator is used less often than the commutator, but can be used for example to define Clifford algebras, Jordan algebras and is utilised to derive the Dirac equation in particle physics.
The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation. In phase space, equivalent commutators of function star-products are called Moyal brackets, and are completely isomorphic to the Hilbert-space commutator structures mentioned.
Identities (ring theory)
The commutator has the following properties:
An additional identity may be found for this last expression, in the form:
If A is a fixed element of a ring R, the first additional identity can be interpreted as a Leibniz rule for the map given by . In other words, the map adA defines a derivation on the ring R. The second and third identities represent Leibniz rules for more than two factors that are valid for any derivation. Identities 4-6 can also be interpreted as Leibniz rules for a certain derivation.
The following useful identity ("Hadamard Lemma") involves nested commutators and underlies the Baker-Campbell–Hausdorff expansion of log (exp A exp B):
Use of the same expansion expresses the above Lie group commutator in terms of a series of nested Lie bracket (algebra) commutators,
These identities differ slightly for the anticommutator (defined above), for instance
Graded rings and algebras
When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as
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,
and, crucially, a Lie algebra homomorphism,
By contrast, it is not always an algebra homomorphism, i.e., a relation does not hold in general.
General Leibniz rule
The general Leibniz rule (general product rule) can be written abstractly using the adjoint representation:
By substituting x = ∂ and appending a test function to both sides of the equation, one recovers the general Leibniz rule.
- Canonical commutation relation
- Centralizer a.k.a. commutant
- Derivation (abstract algebra)
- Moyal bracket
- Pincherle derivative
- Poisson bracket
- Ternary commutator
- Three subgroups lemma
- Baker–Campbell–Hausdorff formula
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