Commutator

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.

This element is equal to the group's identity if and only if g and h commute (from the definition gh = hg [g, h], being [g, h] equal to the identity if and only if gh = hg).

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 other group theorists define the commutator as

Commutator identities are an important tool in group theory.[3] The expression ax denotes the conjugate of a by x, defined as x−1ax.

Identity (5) is also known as the Hall–Witt identity, after Philip Hall and Ernst Witt. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).

Many 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:

The commutator of two elements a and b of a ring (including any associative algebra) is defined by

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 in terms of one basis, then they are so represented in terms of 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

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.

Relation (3) is called anticommutativity, while (4) is the Jacobi identity.

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.[8] For example:

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as

By the Jacobi identity, it is also a derivation over the commutation operation:

The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation: