The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by all commutators is closed and is called the derived group or the commutator subgroup of G. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group.
The definition of the commutator above is used throughout this article, but many group theorists define the commutator as Using the first definition, this can be expressed as [g−1, h−1].
Commutator identities are an important tool in group theory.
[3] The expression ax denotes the conjugate of a by x, defined as x−1ax.
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.
[4] Many other group theorists define the conjugate of a by x as xax−1.
Similar identities hold for these conventions.
Many identities that are true modulo certain subgroups are also used.
For instance, in any group, second powers behave well: If the derived subgroup is central, then Rings often do not support division.
In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis.
The anticommutator of two elements a and b of a ring or associative algebra is defined by Sometimes
[6] The anticommutator is used less often, but can be used to define Clifford algebras and Jordan algebras and in the derivation of 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.
[7] In phase space, equivalent commutators of function star-products are called Moyal brackets and are completely isomorphic to the Hilbert space commutator structures mentioned.
The commutator has the following properties: Relation (3) is called anticommutativity, while (4) is the Jacobi identity.
If A is a fixed element of a ring R, identity (1) can be interpreted as a Leibniz rule for the map
In other words, the map adA defines a derivation on the ring R. Identities (2), (3) represent Leibniz rules for more than two factors, and are valid for any derivation.
Identities (4)–(6) can also be interpreted as Leibniz rules.
From identity (9), one finds that the commutator of integer powers of ring elements is: Some of the above identities can be extended to the anticommutator using the above ± subscript notation.
can be meaningfully defined, such as a Banach algebra or a ring of formal power series.
In such a ring, Hadamard's lemma applied to nested commutators gives:
(analogous to elements of a Lie group) in terms of a series of nested commutators (Lie brackets),
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 in a ring R, another notation turns out to be useful.
by: This mapping is a derivation on the ring R: By the Jacobi identity, it is also a derivation over the commutation operation: Composing such mappings, we get for example
is the ring of mappings from R to itself with composition as the multiplication operation.
is a Lie algebra homomorphism, preserving the commutator: By contrast, it is not always a ring homomorphism: usually
The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation: Replacing
, and applying both sides to a function g, the identity becomes the usual Leibniz rule for the nth derivative