In
mathematics, the commutator gives an indication of the extent to which a certain
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
fails to be
commutative. There are different definitions used in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
and
ring theory.
Group theory
The commutator of two elements, and , of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, is the element
: .
This element is equal to the group's identity if and only if and commute (from the definition , being equal to the identity if and only if ).
The set of all commutators of a group is not in general closed under the group operation, but the
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of ''G''
generated by all commutators is closed and is called the ''derived group'' or the ''
commutator subgroup
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.
The commutator subgroup is important because it is the smallest normal ...
'' of ''G''. Commutators are used to define
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
and
solvable groups and the largest
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
quotient group.
The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as
:.
Identities (group theory)
Commutator identities are an important tool in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
. The expression denotes the
conjugate of by , defined as .
#
#
#
and
#
and
#
and
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).
N.B., the above definition of the conjugate of by is used by some group theorists. Many other group theorists define the conjugate of by as . This is often written
. Similar identities hold for these conventions.
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:
:
If the
derived subgroup
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.
The commutator subgroup is important because it is the smallest normal ...
is central, then
:
Ring theory
Rings often do not support division. Thus, the commutator of two elements ''a'' and ''b'' of a ring (or any
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...
) is defined differently by
:
The commutator 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 and of a ring or associative algebra is defined by
:
Sometimes
is used to denote anticommutator, while
is then used for commutator. The anticommutator is used less often, but can be used to define
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
s and
Jordan algebras and in the derivation of the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac p ...
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
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
, equivalent commutators of function
star-products are called
Moyal bracket
In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.
The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a len ...
s and are completely isomorphic to the Hilbert space commutator structures mentioned.
Identities (ring theory)
The commutator has the following properties:
Lie-algebra identities
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