Inmathematics, thecommutator gives an indication of the extent to which a certainbinary operation fails to becommutative. There are different definitions used ingroup theory andring theory.

Thecommutator of two elements,g andh, of agroupG, is the element

[g,h] =g⁻¹h⁻¹gh.^[1]

This element is equal to the group's identity if and only ifg andh commute (that is, if and only ifgh =hg).

The set of all commutators of a group is not in general closed under the group operation, but thesubgroup ofGgenerated by all commutators is closed and is called thederived group or thecommutator subgroup ofG. Commutators are used to definenilpotent andsolvable groups and the largestabelianquotient group.

The definition of the commutator above is used throughout this article, but many group theorists define the commutator as

[g,h] =ghg⁻¹h⁻¹.^[2]

Using the first definition, this can be expressed as[g⁻¹,h⁻¹].

Commutator identities are an important tool ingroup theory.^[3] The expressiona^x denotes theconjugate ofa byx, defined asx⁻¹ax.

1. ${\displaystyle x^y=x[x,y].}$
2. ${\displaystyle [y,x]=[x,y{]}^{-1}.}$
3. ${\displaystyle [x,zy]=[x,y]\cdot [x,z{]}^y}$ and${\displaystyle [xz,y]=[x,y{]}^z\cdot [z,y].}$
4. ${\displaystyle \left[x,y^{-1}\right]=[y,x{]}^{y^{-1}}}$ and${\displaystyle \left[x^{-1},y\right]=[y,x{]}^{x^{-1}}.}$
5. ${\displaystyle {\left[\left[x,y^{-1}\right],z\right]}^y\cdot {\left[\left[y,z^{-1}\right],x\right]}^z\cdot {\left[\left[z,x^{-1}\right],y\right]}^x=1}$ and${\displaystyle \left[\left[x,y\right],z^x\right]\cdot \left[[z,x],y^z\right]\cdot \left[[y,z],x^y\right]=1.}$

Identity (5) is also known as theHall–Witt identity, afterPhilip Hall andErnst Witt. It is a group-theoretic analogue of theJacobi identity for the ring-theoretic commutator (see next section).

N.B., the above definition of the conjugate ofa byx is used by some group theorists.^[4] Many other group theorists define the conjugate ofa byx asxax⁻¹.^[5] This is often written${\displaystyle {}^{x}a}$. 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 ofsolvable groups andnilpotent groups. For instance, in any group, second powers behave well:

${\displaystyle (xy{)}^2=x^2{y}^2[y,x][[y,x],y].}$

If thederived subgroup is central, then

${\displaystyle (xy{)}^n=x^n{y}^n[y,x{]}^{(\genfrac{}{}{0ex}{}{n}{2})}.}$

Rings often do not support division. Thus, thecommutator of two elementsa andb of a ring (or anyassociative algebra) is defined differently by

${\displaystyle [a,b]=ab-ba.}$

The commutator is zero if and only ifa andb commute. Inlinear algebra, if twoendomorphisms 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 aLie bracket, every associative algebra can be turned into aLie algebra.

Theanticommutator of two elementsa andb of a ring or associative algebra is defined by

${\displaystyle \{a,b\}=ab+ba.}$

Sometimes${\displaystyle [a,b{]}_{+}}$ is used to denote anticommutator, while${\displaystyle [a,b{]}_{-}}$ is then used for commutator.^[6] The anticommutator is used less often, but can be used to defineClifford algebras andJordan algebras and in the derivation of theDirac equation inparticle physics.

The commutator of two operators acting on aHilbert space is a central concept inquantum mechanics, since it quantifies how well the twoobservables described by these operators can be measured simultaneously. Theuncertainty principle is ultimately a theorem about such commutators, by virtue of theRobertson–Schrödinger relation.^[7] Inphase space, equivalent commutators of functionstar-products are calledMoyal brackets and are completely isomorphic to the Hilbert space commutator structures mentioned.

The commutator has the following properties:

1. ${\displaystyle [A+B,C]=[A,C]+[B,C]}$
2. ${\displaystyle [A,A]=0}$
3. ${\displaystyle [A,B]=-[B,A]}$
4. ${\displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}$

Relation (3) is calledanticommutativity, while (4) is theJacobi identity.

1. ${\displaystyle [A,BC]=[A,B]C+B[A,C]}$
2. ${\displaystyle [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]}$
3. ${\displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}$
4. ${\displaystyle [AB,C]=A[B,C]+[A,C]B}$
5. ${\displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}$
6. ${\displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}$
7. ${\displaystyle [A,B+C]=[A,B]+[A,C]}$
8. ${\displaystyle [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]}$
9. ${\displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B=A[B,C]D+AC[B,D]+[A,C]DB+C[A,D]B}$
10. ${\displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}$

IfA is a fixed element of a ringR, identity (1) can be interpreted as aLeibniz rule for the map${\displaystyle {\mathrm{ad}}_A:R\to R}$ given by${\displaystyle {\mathrm{ad}}_A(B)=[A,B]}$. In other words, the map ad_A defines aderivation on the ringR. 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) expressZ-bilinearity.

From identity (9), one finds that the commutator of integer powers of ring elements is:

${\displaystyle [A^N,B^M]=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}{A}^n{B}^m[A,B]B^{M-m-1}{A}^{N-n-1}=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}{B}^m{A}^n[A,B]A^{N-n-1}{B}^{M-m-1}}$

Some of the above identities can be extended to the anticommutator using the above ± subscript notation.^[8]For example:

1. ${\displaystyle [AB,C{]}_{\pm }=A[B,C{]}_{-}+[A,C{]}_{\pm }B}$
2. ${\displaystyle [AB,CD{]}_{\pm }=A[B,C{]}_{-}D+AC[B,D{]}_{-}+[A,C{]}_{-}DB+C[A,D{]}_{\pm }B}$
3. ${\displaystyle [[A,B],[C,D]]=[[[B,C{]}_{+},A{]}_{+},D]-[[[B,D{]}_{+},A{]}_{+},C]+[[[A,D{]}_{+},B{]}_{+},C]-[[[A,C{]}_{+},B{]}_{+},D]}$
4. ${\displaystyle \left[A,[B,C{]}_{\pm }\right]+\left[B,[C,A{]}_{\pm }\right]+\left[C,[A,B{]}_{\pm }\right]=0}$
5. ${\displaystyle [A,BC{]}_{\pm }=[A,B{]}_{-}C+B[A,C{]}_{\pm }=[A,B{]}_{\pm }C\mp B[A,C{]}_{-}}$
6. ${\displaystyle [A,BC]=[A,B{]}_{\pm }C\mp B[A,C{]}_{\pm }}$

Consider a ring or algebra in which theexponential${\displaystyle e^A=\exp (A)=1+A+\frac{1}{2!}{A}^2+\cdots }$ can be meaningfully defined, such as aBanach algebra or a ring offormal power series.

In such a ring,Hadamard's lemma applied to nested commutators gives:$e^{A}B{e}^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+\cdots =e^{{\mathrm{ad}}_A}(B).$ (For the last expression, seeAdjoint derivation below.) This formula underlies theBaker–Campbell–Hausdorff expansion of log(exp(A) exp(B)).

A similar expansion expresses the group commutator of expressions${\displaystyle e^A}$ (analogous to elements of aLie group) in terms of a series of nested commutators (Lie brackets),$${\displaystyle e^A{e}^B{e}^{-A}{e}^{-B}=\exp \left([A,B]+\frac{1}{2!}[A+B,[A,B]]+\frac{1}{3!}\left(\frac{1}{2}[A,[B,[B,A]]]+[A+B,[A+B,[A,B]]]\right)+\cdots \right).}$$

When dealing withgraded algebras, the commutator is usually replaced by thegraded commutator, defined in homogeneous components as

${\displaystyle [\omega ,\eta {]}_{gr}:=\omega \eta -(-1{)}^{\mathrm{deg}\omega \mathrm{deg}\eta }\eta \omega .}$

Especially if one deals with multiple commutators in a ringR, another notation turns out to be useful. For an element${\displaystyle x\in R}$, we define theadjoint mapping${\displaystyle {\mathrm{a}\mathrm{d}}_x:R\to R}$ by:

${\displaystyle {\mathrm{ad}}_x(y)=[x,y]=xy-yx.}$

This mapping is aderivation on the ringR:

${\displaystyle {\mathrm{a}\mathrm{d}}_x(yz)={\mathrm{a}\mathrm{d}}_x(y)z+y{\mathrm{a}\mathrm{d}}_x(z).}$

By theJacobi identity, it is also a derivation over the commutation operation:

${\displaystyle {\mathrm{a}\mathrm{d}}_x[y,z]=[{\mathrm{a}\mathrm{d}}_x(y),z]+[y,{\mathrm{a}\mathrm{d}}_x(z)].}$

Composing such mappings, we get for example${\displaystyle {\mathrm{ad}}_x{\mathrm{ad}}_y(z)=[x,[y,z]]}$ and$${\displaystyle {\mathrm{ad}}_x^2(z)={\mathrm{ad}}_x({\mathrm{ad}}_x(z))=[x,[x,z]].}$$ We may consider${\displaystyle \mathrm{a}\mathrm{d}}$ itself as a mapping,${\displaystyle \mathrm{a}\mathrm{d}:R\to \mathrm{E}\mathrm{n}\mathrm{d}(R)}$, where${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}(R)}$ is the ring of mappings fromR to itself with composition as the multiplication operation. Then${\displaystyle \mathrm{a}\mathrm{d}}$ is aLie algebra homomorphism, preserving the commutator:

${\displaystyle {\mathrm{ad}}_{[x,y]}=\left[{\mathrm{ad}}_x,{\mathrm{ad}}_y\right].}$

By contrast, it isnot always a ring homomorphism: usually${\displaystyle {\mathrm{ad}}_{xy}\ne {\mathrm{ad}}_x{\mathrm{ad}}_y}$.

Thegeneral Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:

${\displaystyle x^{n}y=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k}){\mathrm{ad}}_x^k(y)x^{n-k}.}$

Replacing${\displaystyle x}$ by the differentiation operator${\displaystyle \mathrm{\partial }}$, and${\displaystyle y}$ by the multiplication operator${\displaystyle m_f:g\mapsto fg}$, we get${\displaystyle \mathrm{ad}(\mathrm{\partial })(m_f)=m_{\mathrm{\partial }(f)}}$, and applying both sides to a functiong, the identity becomes the usual Leibniz rule for thenth derivative${\displaystyle {\mathrm{\partial }}^n(fg)}$.

• 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

• Fraleigh, John B. (1976),A First Course In Abstract Algebra (2nd ed.), Reading:Addison-Wesley,ISBN 0-201-01984-1
• Griffiths, David J. (2004),Introduction to Quantum Mechanics (2nd ed.),Prentice Hall,ISBN 0-13-805326-X
• Herstein, I. N. (1975),Topics In Algebra (2nd ed.), Wiley,ISBN 0471010901
• Lavrov, P.M. (2014), "Jacobi -type identities in algebras and superalgebras",Theoretical and Mathematical Physics,179 (2):550–558,arXiv:1304.5050,Bibcode:2014TMP...179..550L,doi:10.1007/s11232-014-0161-2,S2CID 119175276
• Liboff, Richard L. (2003),Introductory Quantum Mechanics (4th ed.),Addison-Wesley,ISBN 0-8053-8714-5
• McKay, Susan (2000),Finite p-groups, Queen Mary Maths Notes, vol. 18,University of London,ISBN 978-0-902480-17-9,MR 1802994
• McMahon, D. (2008),Quantum Field Theory,McGraw Hill,ISBN 978-0-07-154382-8

• McKenzie, R.; Snow, J. (2005),"Congruence modular varieties: commutator theory", in Kudryavtsev, V. B.; Rosenberg, I. G. (eds.),Structural Theory of Automata, Semigroups, and Universal Algebra, NATO Science Series II, vol. 207, Springer, pp. 273–329,doi:10.1007/1-4020-3817-8_11,ISBN 9781402038174

• "Commutator",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Abstract algebra
• Group theory
• Binary operations
• Mathematical identities

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