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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 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. There are different definitions used in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
and
ring theory In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their re ...
.


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 subgroup ...
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 s ...
'' 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 class ...
and solvable groups and the largest abelian
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
. 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 group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. The expression denotes the conjugate of by , defined as . # x^y = x
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
# , x= ,y. # , zy=
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
cdot , zy and z, y=
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
z \cdot
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
# \left , y^\right= , x and \left ^, y\right= , x. # \left left[x,_y^\right_z\right.html" ;"title=",_y^\right.html" ;"title="left[x, y^\right">left[x, y^\right z\right">,_y^\right.html" ;"title="left[x, y^\right">left[x, y^\right z\righty \cdot \left[\left[y, z^\right], x\right]^z \cdot \left[\left[z, x^\right], y\right]^x = 1 and \left[\left[x, y\right], z^x\right] \cdot \leftz ,x], y^z\right] \cdot \lefty, z], x^y\right] = 1. Identity (5) is also known as the ''Hall–Witt identity'', after
Philip Hall Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thomps ...
and
Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...
. It is a group-theoretic analogue of the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
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 ^x a. 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 group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...
s and
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with . Intui ...
s. For instance, in any group, second powers behave well: :(xy)^2 = x^2 y^2 , x , x y]. 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 :(xy)^n = x^n y^n , x\binom.


Ring theory

Rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
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 multiplic ...
) is defined differently by : , b= ab - ba. 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 matrices. ...
, if two
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
, every associative algebra can be turned into a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. The anticommutator of two elements and of a ring or associative algebra is defined by : \ = ab + ba. Sometimes ,b+ is used to denote anticommutator, while ,b- 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 hyperc ...
s and
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan alg ...
s 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 par ...
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
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
s described by these operators can be measured simultaneously. The
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
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 usually ...
, 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 le ...
s and are completely isomorphic to the Hilbert space commutator structures mentioned.


Identities (ring theory)

The commutator has the following properties:


Lie-algebra identities

# + B, C= , C+ , C/math> #
, A The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= 0 # , B= -
, A The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
/math> # ,_[B,_C_+_[B,_[C,_A.html"_;"title=",_C.html"_;"title=",_[B,_C">,_[B,_C_+_[B,_[C,_A">,_C.html"_;"title=",_[B,_C">,_[B,_C_+_[B,_[C,_A_+_[C,_[A,_B.html" ;"title=",_C">,_[B,_C_+_[B,_[C,_A.html" ;"title=",_C.html" ;"title=", [B, C">, [B, C + [B, [C, A">,_C.html" ;"title=", [B, C">, [B, C + [B, [C, A + [C, [A, B">,_C">,_[B,_C_+_[B,_[C,_A.html" ;"title=",_C.html" ;"title=", [B, C">, [B, C + [B, [C, A">,_C.html" ;"title=", [B, C">, [B, C + [B, [C, A + [C, [A, B = 0 Relation (3) is called anticommutativity, while (4) is the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
.


Additional identities

#
, BC The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= , B + B , C/math> # , BCD= , BD + B , C + BC , D/math> #
, BCDE The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= , BDE + B , CE + BC , D + BCD , E/math> # B, C= A , C+ , C # BC, D= AB , D+ A , D + , DC # BCD, E= ABC , E+ AB , E + A , ED + , ECD # , B + C= , B+ , C/math> # + B, C + D= , C+ , D+ , C+ , D/math> # B, CD= A , C + , CD + CA , D+ C , D =A , C + AC ,D+ ,CB + C , D # A, C , D = [A,_B_C.html"_;"title=",_B.html"_;"title="[A,_B">[A,_B_C">,_B.html"_;"title="[A,_B">[A,_B_C_D.html" ;"title=",_B">[A,_B_C.html" ;"title=",_B.html" ;"title="[A, B">[A, B C">,_B.html" ;"title="[A, B">[A, B C D">,_B">[A,_B_C.html" ;"title=",_B.html" ;"title="[A, B">[A, B C">,_B.html" ;"title="[A, B">[A, B C D+ [B, C], D], A] + [C, D], A], B] + [D, A], B], C] If is a fixed element of a ring ''R'', identity (1) can be interpreted as a product rule, Leibniz rule for the map \operatorname_A: R \rightarrow R given by \operatorname_A(B) = , B/math>. In other words, the map ad''A'' defines a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
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. Identities (7), (8) express Z-
bilinearity In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
. Some of the above identities can be extended to the anticommutator using the above ± subscript notation. For example: # B, C\pm = A , C- + , C\pm B # B, CD\pm = A , C- D + AC , D- + , C- DB + C , D\pm B # A,B ,D= [B,C+,A.html"_;"title=",C.html"_;"title="[B,C">[B,C+,A">,C.html"_;"title="[B,C">[B,C+,A+,D.html" ;"title=",C">[B,C+,A.html" ;"title=",C.html" ;"title="[B,C">[B,C+,A">,C.html" ;"title="[B,C">[B,C+,A+,D">,C">[B,C+,A.html" ;"title=",C.html" ;"title="[B,C">[B,C+,A">,C.html" ;"title="[B,C">[B,C+,A+,D[B,D]_+,A]_+,C]+[A,D]_+,B]_+,C]- ,C+,B]_+,D] #\left[A, , C\pm\right] + \left[B, [C, A]_\pm\right] + \left[C, [A, B]_\pm\right] = 0 # ,BC\pm = ,B- C + B ,C\pm # ,BC= ,B\pm C \mp B ,C\pm


Exponential identities

Consider a ring or algebra in which the
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
e^A = \exp(A) = 1 + A + \tfracA^2 + \cdots can be meaningfully defined, such as a
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
or a ring of
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
. In such a ring,
Hadamard's lemma In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polyn ...
applied to nested commutators gives: e^A Be^ \ =\ B + , B+ \frac
, [A, B The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
+ \frac[A,
, [A, B The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
] + \cdots \ =\ e^(B). (For the last expression, see ''Adjoint derivation'' below.) This formula underlies the Baker–Campbell–Hausdorff formula#An important lemma, Baker–Campbell–Hausdorff expansion of log(exp(''A'') exp(''B'')). A similar expansion expresses the group commutator of expressions e^A (analogous to elements of a Lie group) in terms of a series of nested commutators (Lie brackets), e^A e^B e^ e^ = \exp\!\left( , B+ \frac B,_[A,_B_+_\frac_\left(\frac_ B,_[A,_B_+_\frac_\left(\frac_[A,_[B,_[B,_A">,_B.html"_;"title="B,_[A,_B">B,_[A,_B_+_\frac_\left(\frac_[A,_[B,_[B,_A.html" ;"title=",_[B,_[B,_A.html" ;"title=",_B.html" ;"title="B, [A, B">B, [A, B + \frac \left(\frac [A, [B, [B, A">,_B.html" ;"title="B, [A, B">B, [A, B + \frac \left(\frac [A, [B, [B, A">,_[B,_[B,_A.html" ;"title=",_B.html" ;"title="B, [A, B">B, [A, B + \frac \left(\frac [A, [B, [B, A">,_B.html" ;"title="B, [A, B">B, [A, B + \frac \left(\frac [A, [B, [B, A+ [AB, B, [A, B]\right) + \cdots\right).


Graded rings and algebras

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as : omega, \eta := \omega\eta - (-1)^ \eta\omega.


Adjoint derivation

Especially if one deals with multiple commutators in a ring ''R'', another notation turns out to be useful. For an element x\in R, we define the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
mapping \mathrm_x:R\to R by: :\operatorname_x(y) =
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= xy-yx. This mapping is a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a proc ...
on the ring ''R'': :\mathrm_x\!(yz) \ =\ \mathrm_x\!(y) \,z \,+\, y\,\mathrm_x\!(z). By the
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
, it is also a derivation over the commutation operation: :\mathrm_x ,z\ =\ mathrm_x\!(y),z\,+\, ,\mathrm_x\!(z). Composing such mappings, we get for example \operatorname_x\operatorname_y(z) = ,_[y,_z,.html" ;"title=",_z.html" ;"title=", [y, z">, [y, z,">,_z.html" ;"title=", [y, z">, [y, z, and \operatorname_x^2\!(z) \ =\ \operatorname_x\!(\operatorname_x\!(z)) \ =\ [x, [x, z]\,]. We may consider \mathrm itself as a mapping, \mathrm: R \to \mathrm(R) , where \mathrm(R) is the ring of mappings from ''R'' to itself with composition as the multiplication operation. Then \mathrm is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
homomorphism, preserving the commutator: :\operatorname_ = \left \operatorname_x, \operatorname_y \right By contrast, it is not always a ring homomorphism: usually \operatorname_ \,\neq\, \operatorname_x\operatorname_y .


General Leibniz rule

The
general Leibniz rule In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if f and g are n-times differentiable functions, then the product fg is also n-t ...
, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation: :x^n y = \sum_^n \binom \operatorname_x^k\!(y)\, x^. Replacing ''x'' by the differentiation operator \partial, and ''y'' by the multiplication operator m_f : g \mapsto fg, we get \operatorname(\partial)(m_f) = m_, and applying both sides to a function ''g'', the identity becomes the usual Leibniz rule for the ''n''-th derivative \partial^\!(fg).


See also

*
Anticommutativity In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
*
Associator In abstract algebra, the term associator is used in different ways as a measure of the non-associativity of an algebraic structure. Associators are commonly studied as triple systems. Ring theory For a non-associative ring or algebra R, the associ ...
* Baker–Campbell–Hausdorff formula *
Canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
*
Centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
a.k.a. commutant *
Derivation (abstract algebra) In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies L ...
*
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 le ...
*
Pincherle derivative In mathematics, the Pincherle derivative T' of a linear operator T: \mathbb \to \mathbb /math> on the vector space of polynomials in the variable ''x'' over a field \mathbb is the commutator of T with the multiplication by ''x'' in the algebra of ...
*
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
* Ternary commutator *
Three subgroups lemma In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity. Notation In what follows, the following notation will be employe ...


Notes


References

* * * * * * *


Further reading

*


External links

* {{Authority control Abstract algebra Group theory Binary operations Mathematical identities