In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
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 ...
and
ring theory.
Group theory
The commutator of two elements, and , of a
group , 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 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
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 cl ...
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 ...
. The expression denotes the
conjugate of by , defined as .
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#
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and
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and
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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) is defined differently by
:
The commutator is zero if and only if ''a'' and ''b'' commute. In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, 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 algebras and
Jordan algebras and in the derivation of the
Dirac equation in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
.
The commutator of two operators acting on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is a central concept in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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 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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