Editing Commutator

{{Short description|Operation measuring the failure of two entities to commute}}
{{About|the mathematical concept|the electrical component|Commutator (electric)|the relation between [[conjugate variables|canonical conjugate entities]]|Canonical commutation relation|other uses|Commutation (disambiguation){{!}}Commutation}}
{{Use shortened footnotes|date=November 2022}}
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]].

== Group theory ==
The '''commutator''' of two elements, {{mvar|g}} and {{mvar|h}}, of a [[group (mathematics)|group]] {{mvar|G}}, is the element
: {{math|1=[''g'', ''h''] = ''g''<sup>−1</sup>''h''<sup>−1</sup>''gh''}}.<ref>{{harvtxt|Herstein|1975|p=252}}</ref>

This element is equal to the group's identity if and only if {{mvar|g}} and {{mvar|h}} commute (that is, if and only if {{math|1=''gh'' = ''hg''}}).

The set of all commutators of a group is not in general closed under the group operation, but the [[subgroup]] of ''G'' [[Generating set of a group|generated]] by all commutators is closed and is called the ''derived group'' or the ''[[commutator subgroup]]'' of ''G''. Commutators are used to define [[nilpotent group|nilpotent]] and [[solvable group|solvable]] groups and the largest [[Abelian group|abelian]] [[quotient group]].

The definition of the commutator above is used throughout this article, but many group theorists define the commutator as
: {{math|1=[''g'', ''h''] = ''ghg''<sup>−1</sup>''h''<sup>−1</sup>}}.<ref>{{harvtxt|Fraleigh|1976|p=108}}</ref>

Using the first definition, this can be expressed as {{math|1=[''g''<sup>−1</sup>, ''h''<sup>−1</sup>]}}.

=== Identities (group theory) ===
Commutator identities are an important tool in [[group theory]].<ref>{{harvtxt|McKay|2000|p=4}}</ref> The expression {{math|''a<sup>x</sup>''}} denotes the [[conjugate (group theory)#Definition|conjugate]] of {{mvar|a}} by {{mvar|x}}, defined as {{math|''x''<sup>−1</sup>''ax''}}.
# <math>x^y = x^{-1}[x, y].</math>
# <math>[y, x] = [x,y]^{-1}.</math>
# <math>[x, zy] = [x, y]\cdot [x, z]^y</math> and <math>[x z, y] = [x, y]^z \cdot [z, y].</math>
# <math>\left[x, y^{-1}\right] = [y, x]^{y^{-1}}</math> and <math>\left[x^{-1}, y\right] = [y, x]^{x^{-1}}.</math>
# <math>\left[\left[x, y^{-1}\right], z\right]^y \cdot \left[\left[y, z^{-1}\right], x\right]^z \cdot \left[\left[z, x^{-1}\right], y\right]^x = 1</math> and <math>\left[\left[x, y\right], z^x\right] \cdot  \left[[z ,x], y^z\right] \cdot \left[[y, z], x^y\right] = 1.</math>

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 {{mvar|a}} by {{mvar|x}} is used by some group theorists.<ref>{{harvtxt|Herstein|1975|p=83}}</ref>  Many other group theorists define the conjugate of {{mvar|a}} by {{mvar|x}} as {{math|''xax''<sup>−1</sup>}}.<ref>{{harvtxt|Fraleigh|1976|p=128}}</ref>  This is often written <math>{}^x a</math>.  Similar identities hold for these conventions.

Many identities that are true modulo certain subgroups are also used.  These can be particularly useful in the study of [[solvable group]]s and [[nilpotent group]]s.  For instance, in any group, second powers behave well:
: <math>(xy)^2 = x^2 y^2 [y, x][[y, x], y].</math>

If the [[derived subgroup]] is central, then
: <math>(xy)^n = x^n y^n [y, x]^\binom{n}{2}.</math>

== Ring theory ==
<!-- This section is linked from [[Lie algebra]] -->
[[ring (algebra)|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
: <math>[a, b] = ab - ba.</math>

The commutator is zero if and only if ''a'' and ''b'' commute. In [[linear algebra]], if two [[endomorphism]]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 algebra|Lie bracket]], every associative algebra can be turned into a [[Lie algebra]].

The '''anticommutator''' of two elements {{mvar|a}} and {{mvar|b}} of a ring or associative algebra is defined by
: <math>\{a, b\} = ab + ba.</math>

Sometimes <math>[a,b]_+</math> is used to denote anticommutator, while <math>[a,b]_-</math> is then used for commutator.<ref>{{harvtxt|McMahon|2008}}</ref> The anticommutator is used less often, but can be used to define [[Clifford algebra]]s and [[Jordan algebra]]s and in the derivation of the [[Dirac equation]] 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 [[observable]]s described by these operators can be measured simultaneously. The [[uncertainty principle]] is ultimately a theorem about such commutators, by virtue of the [[Uncertainty relation|Robertson–Schrödinger relation]].<ref>{{harvtxt|Liboff|2003|pp=140–142}}</ref> In [[phase space]], equivalent commutators of function [[Moyal product|star-products]] are called [[Moyal bracket]]s and are completely isomorphic to the Hilbert space commutator structures mentioned.

=== Identities (ring theory) ===
The commutator has the following properties:

==== Lie-algebra identities ====
# <math>[A + B, C] = [A, C] + [B, C]</math>
# <math>[A, A] = 0</math>
# <math>[A, B] = -[B, A]</math>
# <math>[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0</math>

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

==== Additional identities ====
# <math>[A, BC] = [A, B]C + B[A, C]</math>
# <math>[A, BCD] = [A, B]CD + B[A, C]D + BC[A, D]</math>
# <math>[A, BCDE] = [A, B]CDE + B[A, C]DE + BC[A, D]E + BCD[A, E]</math>
# <math>[AB, C] = A[B, C] + [A, C]B</math>
# <math>[ABC, D] = AB[C, D] + A[B, D]C + [A, D]BC</math>
# <math>[ABCD, E] = ABC[D, E] + AB[C, E]D + A[B, E]CD + [A, E]BCD</math>
# <math>[A, B + C] = [A, B] + [A, C]</math>
# <math>[A + B, C + D] = [A, C] + [A, D] + [B, C] + [B, D]</math>
# <math>[AB, CD] = A[B, C]D + [A, C]BD + CA[B, D] + C[A, D]B =A[B, C]D + AC[B,D] + [A,C]DB + C[A, D]B</math>
# <math>[[A, C], [B, D]] = [[[A, B], C], D] + [[[B, C], D], A] + [[[C, D], A], B] + [[[D, A], B], C]</math>

If {{mvar|A}} is a fixed element of a ring ''R'', identity (1) can be interpreted as a [[product rule|Leibniz rule]] for the map <math>\operatorname{ad}_A: R \rightarrow R</math> given by <math>\operatorname{ad}_A(B) = [A, B]</math>.  In other words, the map ad<sub>''A''</sub> defines a [[derivation (abstract algebra)|derivation]] 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'''-[[Bilinear map|bilinearity]].

From identity (9), one finds that the commutator of integer powers of ring elements is:
: <math>[A^N, B^M] = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} A^{n}B^{m} [A,B] B^{N-n-1}A^{M-m-1} = \sum_{n=0}^{N-1}\sum_{m=0}^{M-1} B^{n}A^{m} [A,B] A^{N-n-1}B^{M-m-1}</math>

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.<ref>{{harvtxt|Lavrov|2014}}</ref>
For example:
# <math>[AB, C]_\pm = A[B, C]_- + [A, C]_\pm B</math>
# <math>[AB, CD]_\pm = A[B, C]_- D + AC[B, D]_- + [A, C]_- DB + C[A, D]_\pm B</math>
# <math>[[A,B],[C,D]]=[[[B,C]_+,A]_+,D]-[[[B,D]_+,A]_+,C]+[[[A,D]_+,B]_+,C]-[[[A,C]_+,B]_+,D]</math>
# <math>\left[A, [B, C]_\pm\right] + \left[B, [C, A]_\pm\right] + \left[C, [A, B]_\pm\right] = 0</math>
# <math>[A,BC]_\pm = [A,B]_- C + B[A,C]_\pm = [A,B]_\pm C \mp B[A,C]_-</math>
# <math>[A,BC] = [A,B]_\pm C \mp B[A,C]_\pm</math>

==== Exponential identities ====
Consider a ring or algebra in which the [[exponential function|exponential]] <math>e^A = \exp(A) = 1 + A + \tfrac{1}{2!}A^2 + \cdots</math> can be meaningfully defined, such as a [[Banach algebra]] or a ring of [[formal power series]].

In such a ring, [[Hadamard's lemma]] applied to nested commutators gives: <math display="inline">e^A Be^{-A}
  \ =\  B + [A, B] + \frac{1}{2!}[A, [A, B]] + \frac{1}{3!}[A, [A, [A, B]]] + \cdots
  \ =\  e^{\operatorname{ad}_A}(B).
</math> (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 <math>e^A</math> (analogous to elements of a [[Lie group]]) in terms of a series of nested commutators (Lie brackets),
<math display="block">e^A e^B e^{-A} e^{-B} =
\exp\!\left(  [A, B] + \frac{1}{2!}[A{+}B, [A, B]] + \frac{1}{3!} \left(\frac{1}{2} [A, [B, [B, A]]] + [A{+}B, [A{+}B, [A, B]]]\right) + \cdots\right). </math>

== Graded rings and algebras ==
When dealing with [[graded algebra]]s, the commutator is usually replaced by the '''graded commutator''', defined in homogeneous components as 
: <math>[\omega, \eta]_{gr} := \omega\eta - (-1)^{\deg \omega \deg \eta} \eta\omega.</math>

== Adjoint derivation ==
Especially if one deals with multiple commutators in a ring ''R'', another notation turns out to be useful. For an element <math>x\in R</math>, we define the [[adjoint representation of a Lie algebra|adjoint]] mapping <math>\mathrm{ad}_x:R\to R</math> by:
: <math>\operatorname{ad}_x(y) = [x, y] = xy-yx.</math>

This mapping is a [[Derivation (differential algebra)|derivation]] on the ring ''R'': 
: <math>\mathrm{ad}_x\!(yz) \ =\ \mathrm{ad}_x\!(y) \,z \,+\, y\,\mathrm{ad}_x\!(z).</math>
By the [[Jacobi identity]], it is also a derivation over the commutation operation: 
: <math>\mathrm{ad}_x[y,z] \ =\ [\mathrm{ad}_x\!(y),z] \,+\, [y,\mathrm{ad}_x\!(z)] .</math>

Composing such mappings, we get for example <math>\operatorname{ad}_x\operatorname{ad}_y(z) = [x, [y, z]\,] </math> and <math display="block">\operatorname{ad}_x^2\!(z) \ =\ 
\operatorname{ad}_x\!(\operatorname{ad}_x\!(z)) \ =\ 
[x, [x, z]\,].</math> We may consider <math>\mathrm{ad}</math> itself as a mapping, <math>\mathrm{ad}: R \to \mathrm{End}(R) </math>, where <math>\mathrm{End}(R)</math> is the ring of mappings from ''R'' to itself with composition as the multiplication operation. Then <math>\mathrm{ad}</math> is a [[Lie algebra]] homomorphism, preserving the commutator:
: <math>\operatorname{ad}_{[x, y]} = \left[ \operatorname{ad}_x, \operatorname{ad}_y \right].</math>

By contrast, it is '''not''' always a ring homomorphism: usually <math>\operatorname{ad}_{xy} \,\neq\, \operatorname{ad}_x\operatorname{ad}_y </math>.

=== General Leibniz rule ===
The [[general Leibniz rule]], expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:
: <math>x^n y = \sum_{k = 0}^n \binom{n}{k} \operatorname{ad}_x^k\!(y)\, x^{n - k}.</math>

Replacing <math>x</math> by the differentiation operator <math>\partial</math>, and <math>y</math> by the multiplication operator <math>m_f : g \mapsto fg</math>, we get <math>\operatorname{ad}(\partial)(m_f) = m_{\partial(f)}</math>, and applying both sides to a function ''g'', the identity becomes the usual Leibniz rule for the ''n''th derivative <math>\partial^{n}\!(fg)</math>.

== See also ==
* [[Anticommutativity]]
* [[Associator]]
* [[Baker–Campbell–Hausdorff formula]]
* [[Canonical commutation relation]]
* [[Centralizer]] a.k.a. commutant
* [[Derivation (abstract algebra)]]
* [[Moyal bracket]]
* [[Pincherle derivative]]
* [[Poisson bracket]]
* [[Ternary commutator]]
* [[Three subgroups lemma]]

== Notes ==
{{reflist}}

== References ==
* {{citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading |url=https://books.google.com/books?id=hyHvAAAAMAAJ&q=commutator}}
* {{citation | last1 = Griffiths | first1 = David J. | author1-link = David J. Griffiths | title = Introduction to Quantum Mechanics | edition = 2nd | publisher = [[Prentice Hall]] | year = 2004 | isbn = 0-13-805326-X | url-access = registration | url = https://archive.org/details/introductiontoel00grif_0 }}
* {{citation | first = I. N. | last = Herstein | author-link= Israel Nathan Herstein | year = 1975 | title = Topics In Algebra | edition= 2nd | publisher = Wiley |isbn=0471010901}}
* {{citation|last=Lavrov|first=P.M.|title=Jacobi -type identities in algebras and superalgebras|journal=Theoretical and Mathematical Physics|year=2014|volume=179|issue=2|pages=550–558|doi=10.1007/s11232-014-0161-2|arxiv=1304.5050|bibcode=2014TMP...179..550L|s2cid=119175276}}
* {{citation | last1=Liboff | first1=Richard L. | author1-link = Richard L. Liboff | title=Introductory Quantum Mechanics | edition = 4th | publisher = [[Addison-Wesley]] | year=2003 | isbn=0-8053-8714-5}}
* {{citation | last1=McKay | first1=Susan | title=Finite p-groups | publisher = [[University of London]] | series=Queen Mary Maths Notes | isbn=978-0-902480-17-9 | mr=1802994 | year=2000 | volume=18}}
* {{citation | first1 = D. | last1 = McMahon | year = 2008 | isbn = 978-0-07-154382-8 | title = Quantum Field Theory | publisher = [[McGraw Hill]] }}

== Further reading ==
* {{citation | author1-first= R.| author1-last= McKenzie | author2-first= J. | author2-last= Snow | author1-link= Ralph McKenzie | contribution= Congruence modular varieties: commutator theory | title= Structural Theory of Automata, Semigroups, and Universal Algebra | editor1-first= V. B. | editor1-last= Kudryavtsev | editor2-first= I. G. | editor2-last= Rosenberg| pages= 273–329 | year= 2005 | publisher= Springer|chapter-url=https://www.researchgate.net/publication/226377308 |doi=10.1007/1-4020-3817-8_11 |isbn=9781402038174 |volume=207 |series=NATO Science Series II}}

== External links ==
* {{springer|title=Commutator|id=p/c023430}}

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[[Category:Abstract algebra]]
[[Category:Group theory]]
[[Category:Binary operations]]
[[Category:Mathematical identities]]