TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the commutator gives an indication of the extent to which a certain
binary operation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
fails to be
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. There are different definitions used in
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
and
ring theory In algebra, ring theory is the study of ring (mathematics), 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 ...
.

# Group theory

The commutator of two elements, and , of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, 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 (mathematics), 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 ...
of ''G'' generated by all commutators is closed and is called the ''derived group'' or the ''
commutator subgroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
'' of ''G''. Commutators are used to define
nilpotent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and solvable groups and the largest abelian
quotient group A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...
. 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 mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
. The expression denotes the
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...

of by , defined as . # $x^y = x$
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
# 
, x The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
= ,y. # 
, zy 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 (ty ...
=
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
cdot
, z The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
y and
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
z \cdot
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
#
, x 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 (t ...
and
, x 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 (t ...
. # and $\left\left[\left\left[x, y\right\right], z^x\right\right] \cdot \leftz ,x\right], y^z\right\right] \cdot \lefty, z\right], x^y\right\right] = 1.$ Identity (5) is also known as the ''Hall–Witt identity'', after
Philip Hall Philip Hall FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resources Su ...

and
Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...
. It is a group-theoretic analogue of the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s and
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a Group (mathematics), group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series termina ...
s. For instance, in any group, second powers behave well: :$\left(xy\right)^2 = x^2 y^2$
, x 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 (t ...
, x 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 (t ...
y]. If the
derived subgroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
is central, then :$\left(xy\right)^n = x^n y^n$
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\binom.

# Ring theory

The commutator of two elements ''a'' and ''b'' of a ring (including any
associative algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
) is defined by : 
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
= ab - ba. It 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 matrix (mat ...
, 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 group ...
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 "Lee") 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 \rightar ...
, every associative algebra can be turned into a
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. The anticommutator of two elements and of a ring or an associative algebra is defined by : $\ = ab + ba.$ 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 over a field, algebra generated by a vector space with a quadratic form, and is a Unital algebra, unital associative algebra. As algebra over a field, ''K''-algebras, they generalize the real nu ...
s and
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra algebra over a field, over a field whose product (mathematics), multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (Jordan identity). ...
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 Dirac equation#Covariant form and relativistic invariance, free form, or including Dirac equation#Comparison with t ...
in particle physics. The commutator of two operators acting on a
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is a central concept in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, since it quantifies how well the two
observable In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
s described by these operators can be measured simultaneously. The
uncertainty principle In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking ...

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 Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called param ...

, equivalent commutators of function star-products are called
Moyal bracket In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
s, and are completely isomorphic to the Hilbert space commutator structures mentioned.

## Identities (ring theory)

The commutator has the following properties:

### Lie-algebra identities

#
, A The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
= 0 # 
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
= -
, A The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> # Relation (3) is called anticommutativity, while (4) is the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

#
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
+ B , C/math> # 
, BCD 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 (t ...
=
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
D + B , C + BC , D/math> #
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
DE + B , CE + BC , D + BCD , E/math> # # # #
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
+ , C/math> #
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math>. In other words, the map ad''A'' defines a
derivation Derivation may refer to: * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Derivation (linguistics) * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
. Some of the above identities can be extended to the anticommutator using the above ± subscript notation. For example: # # # #$\left[A,BC\right]_\pm = \left[A,B\right]_- C + B\left[A,C\right]_\pm$ #$\left[A,BC\right] = \left[A,B\right]_\pm C \mp B\left[A,C\right]_\pm$

### Exponential identities

Consider a ring or algebra in which the [
exponential Exponential may refer to any of several mathematical topics related to exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and ...
$e^A = \exp\left(A\right) = 1 + A + \tfracA^2 + \cdots$ can be meaningfully defined, such as a
Banach algebra Banach is a Polish-language surname of several possible origins."Banach"
at genezanazwisk.pl (the webpage cites the sources)
or a ring of
formal power series In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. In such a ring, Hadamard's lemma applied to nested commutators gives: $e^A Be^ \ =\ B +$
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
+ \frac ,_[A,_B_+_\frac[A,__,_[A,_B.html"_;"title=",_B.html"_;"title=",_[A,_B">,_[A,_B_+_\frac[A,__,_[A,_B">,_B.html"_;"title=",_[A,_B">,_[A,_B_+_\frac[A,__,_[A,_B+_\cdots __\_=\__e^(B). _(For_the_last_expression,_see_''Adjoint_derivation''_below.)_This_formula_underlies_the_Baker–Campbell–Hausdorff_formula#An_important_lemma.html" ;"title=",_B">,_[A,_B.html" ;"title=",_B.html" ;"title=", [A, B">, [A, B + \frac[A, , [A, B">,_B.html" ;"title=", [A, B">, [A, B + \frac[A, , [A, B+ \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), :$\begin &e^A e^B e^ e^ \\=& \exp\!\left\left($
, B The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
+ \frac[AB, [A, B + \frac \left(\frac [A, [B, [B, A] + [AB, [AB, [A, B]\right) + \cdots\right). \end

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. For an element $x\in R$, we define the adjoint mapping $\mathrm_x:R\to R$ by: :$\operatorname_x\left(y\right) =$
, y The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
= xy-yx. This mapping is a
derivation Derivation may refer to: * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Derivation (linguistics) * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...
on the ring ''R'': :$\mathrm_x\!\left(yz\right) \ =\ \mathrm_x\!\left(y\right) \,z \,+\, y\,\mathrm_x\!\left(z\right).$ By the
Jacobi identity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, it is also a derivation over the commutation operation: : Composing such mappings, we get for example and $\operatorname_x^2\!(z) \ =\ \operatorname_x\!(\operatorname_x\!(z)) \ =\ [x, [x, z]\,].$ We may consider $\mathrm$ itself as a mapping, $\mathrm: R \to \mathrm\left(R\right)$, where $\mathrm\left(R\right)$ is the ring of mappings from ''R'' to itself with composition as the multiplication operation. Then $\mathrm$ is a
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
homomorphism, preserving the commutator: : 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-tim ...
, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation: :$x^n y = \sum_^n \binom \operatorname_x^k\!\left(y\right)\, x^.$ Replacing ''x'' by the differentiation operator $\partial$, and ''y'' by the multiplication operator $m_f : g \mapsto fg$, we get $\operatorname\left(\partial\right)\left(m_f\right) = m_$, and applying both sides to a function ''g'', the identity becomes the usual Leibniz rule for the ''n''-th derivative $\partial^\!\left(fg\right)$.

*
Anticommutativity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
AssociatorIn abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...
*
Baker–Campbell–Hausdorff formula In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
*
Canonical commutation relation In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking i ...
*
Centralizer In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
a.k.a. commutant *
Derivation (abstract algebra)In mathematics, a derivation is a function on an algebra over a field, algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring (mathematics), ring or a field (mathematics), field ''K'', ...
*
Moyal bracket In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
*
Pincherle derivative In mathematics, the Pincherle derivative ''T’'' of a linear operator ''T'':K 'x''→ K 'x''on the vector space of polynomials in the variable ''x'' over a field (mathematics), field K is the commutator of ''T'' with the multiplication by ''x ...
*
Poisson bracket In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
* Ternary commutator *
Three subgroups lemmaIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* * * * * *