Commutator
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.
Contents
1 Group theory
1.1 Identities (group theory)
2 Ring theory
2.1 Identities (ring theory)
2.1.1 Lie-algebra identities
2.1.2 Additional identities
3 Graded rings and algebras
4 Derivations
4.1 General Leibniz rule
5 See also
6 Notes
7 References
8 Further reading
9 External links
Group theory
The commutator of two elements, g and h, of a group G, is the element
- [g, h] = g−1h−1gh
and is equal to the group's identity if and only if g and h commute (that is, if and only if gh = hg). 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 and solvable groups and the largest abelian quotient group.
The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as
[g, h] = ghg−1h−1.[1][2]
Identities (group theory)
Commutator identities are an important tool in group theory.[3] The expression ax denotes the conjugate of a by x, defined as x−1ax.
- xy=x[x,y].{displaystyle x^{y}=x[x,y].}
- [y,x]=[x,y]−1.{displaystyle [y,x]=[x,y]^{-1}.}
[x,zy]=[x,y]⋅[x,z]y{displaystyle [x,zy]=[x,y]cdot [x,z]^{y}}and [xz,y]=[x,y]z⋅[z,y].{displaystyle [xz,y]=[x,y]^{z}cdot [z,y].}
[x,y−1]=[y,x]y−1{displaystyle left[x,y^{-1}right]=[y,x]^{y^{-1}}}and [x−1,y]=[y,x]x−1.{displaystyle left[x^{-1},yright]=[y,x]^{x^{-1}}.}
[[x,y−1],z]y⋅[[y,z−1],x]z⋅[[z,x−1],y]x=1{displaystyle left[left[x,y^{-1}right],zright]^{y}cdot left[left[y,z^{-1}right],xright]^{z}cdot left[left[z,x^{-1}right],yright]^{x}=1}and [[x,y],zx]⋅[[z,x],yz]⋅[[y,z],xy]=1.{displaystyle left[left[x,yright],z^{x}right]cdot left[[z,x],y^{z}right]cdot left[[y,z],x^{y}right]=1.}
![{displaystyle x^{y}=x[x,y].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbfda5e3214a078a56860ef1d958532efd6ecc6c)
![{displaystyle [y,x]=[x,y]^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c8c7dabf4a3f20a3f9a04a295468426eb22009)
![{displaystyle [xz,y]=[x,y]^{z}cdot [z,y].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51d505bd20b4dfad61942e780f9019f2342a28df)
![{displaystyle left[x^{-1},yright]=[y,x]^{x^{-1}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec4802d72362fa5f8b6f440bee17b73b8f9ff48)
![{displaystyle left[left[x,yright],z^{x}right]cdot left[[z,x],y^{z}right]cdot left[[y,z],x^{y}right]=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/969627eb5e4b01b893d68ba8cb5aefb6fb9dcb4e)
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 a by x is used by some group theorists.[4] Many other group theorists define the conjugate of a by x as xax−1.[5] This is often written xa{displaystyle {}^{x}a}
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:
- (xy)2=x2y2[y,x][[y,x],y].{displaystyle (xy)^{2}=x^{2}y^{2}[y,x][[y,x],y].}
![{displaystyle (xy)^{2}=x^{2}y^{2}[y,x][[y,x],y].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c75b2bcc9c723ec68c6e482e9e1e002b02082d3)
If the derived subgroup is central, then
- (xy)n=xnyn[y,x](n2).{displaystyle (xy)^{n}=x^{n}y^{n}[y,x]^{binom {n}{2}}.}
![{displaystyle (xy)^{n}=x^{n}y^{n}[y,x]^{binom {n}{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/128ae0855aada1e122d118d63c2bfa18a08eb603)
Ring theory
The commutator of two elements a and b of a ring or an associative algebra is defined by
- [a,b]=ab−ba.{displaystyle [a,b]=ab-ba.}
![{displaystyle [a,b]=ab-ba.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c73ba7fbd6260acd540051cc2c6f9131ff0d7f8e)
It is zero if and only if a and b commute. In linear algebra, 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 a and b of a ring or an associative algebra is defined by
- {a,b}=ab+ba.{displaystyle {a,b}=ab+ba.}
Sometimes the brackets [ ]+ are also used to denote anticommutators, while [ ]− is then used for commutators.[6] The anticommutator is used less often than the commutator, but can be used, for example, to define Clifford algebras, Jordan algebras and is utilized to derive 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 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.[7] In phase space, equivalent commutators of function star-products are called Moyal brackets, and are completely isomorphic to the Hilbert-space commutator structures mentioned.
Identities (ring theory)
The commutator has the following properties:
Lie-algebra identities
- [A+B,C]=[A,C]+[B,C]{displaystyle [A+B,C]=[A,C]+[B,C]}
- [A,A]=0{displaystyle [A,A]=0}
- [A,B]=−[B,A]{displaystyle [A,B]=-[B,A]}
- [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0{displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}
![{displaystyle [A+B,C]=[A,C]+[B,C]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f1c95436e1c9c40871bc7a3704072ae067dc10)
![{displaystyle [A,A]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c816c8f8631ef069ad0b1aee64ed996ad8309ed)
![{displaystyle [A,B]=-[B,A]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37b04a066443e96c0b27d4b14f7f5d51c7b2b0c3)
![{displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046dbc0d4387302c8d321afb443e35bc1890fcac)
The third relation is called anticommutativity, while the fourth is the Jacobi identity.
Additional identities
- [A,BC]=[A,B]C+B[A,C]{displaystyle [A,BC]=[A,B]C+B[A,C]}
- [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]{displaystyle [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]}
- [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]{displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}
- [AB,C]=A[B,C]+[A,C]B{displaystyle [AB,C]=A[B,C]+[A,C]B}
- [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC{displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}
- [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD{displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}
- [A,B+C]=[A,B]+[A,C]{displaystyle [A,B+C]=[A,B]+[A,C]}
- [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]{displaystyle [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]}
![{displaystyle [A,BC]=[A,B]C+B[A,C]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7c96785faa7d6e5ecc7d8ada53830b24338c60)
![{displaystyle [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa03949d714777910e992c14aeed34adf42d9766)
![{displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b79c370dc9984ff5e2720b3e06c6153af7ba572)
![{displaystyle [AB,C]=A[B,C]+[A,C]B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ba3177011b65b292d8dc39dd2e3391f29b0a07)
![{displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9e755fd1fe3c26c40e007256c6976b9612e5d9b)
![{displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e00b8c22f5a63ff3dbbcf8a74f0d01f43994afa)
![{displaystyle [A,B+C]=[A,B]+[A,C]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f50e382d516600cfe9f7a34376ebe694bb62177)
![{displaystyle [A+B,C+D]=[A,C]+[A,D]+[B,C]+[B,D]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e47836a02ce95e65488370410a80446b1ae636c)
An additional identity may be found for this last expression, in the form:
- [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B{displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B}
- [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]{displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}
![{displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309878de035824af6f921ace77d4640671b41e15)
![{displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33e847e641c8d2810846e666be23d2512b08eb2f)
If A is a fixed element of a ring R, the first additional identity can be interpreted as a Leibniz rule for the map adA:R→R{displaystyle operatorname {ad} _{A}:Rrightarrow R}
![{displaystyle operatorname {ad} _{A}(B)=[A,B]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a40684049019f333e05818bcd1ce5b48f839561)
Hadamard's lemma, applied on nested commutators holds, and underlies the Baker–Campbell–Hausdorff expansion of log(exp(A) exp(B)):
- eABe−A≡B+[A,B]+12![A,[A,B]]+13![A,[A,[A,B]]]+⋯≡ead(A)B.{displaystyle e^{A}Be^{-A}equiv B+[A,B]+{frac {1}{2!}}[A,[A,B]]+{frac {1}{3!}}[A,[A,[A,B]]]+cdots equiv e^{operatorname {ad} (A)}B.}
![{displaystyle e^{A}Be^{-A}equiv B+[A,B]+{frac {1}{2!}}[A,[A,B]]+{frac {1}{3!}}[A,[A,[A,B]]]+cdots equiv e^{operatorname {ad} (A)}B.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f69985124bc6724a34fe3b3dfde62991063937)
This formula is valid in any ring or algebra in which the exponential function can be meaningfully defined, for example, in a Banach algebra or in a ring of formal power series.
Use of the same expansion expresses the above Lie group commutator in terms of a series of nested Lie bracket (algebra) commutators,
- ln(eAeBe−Ae−B)=[A,B]+12![(A+B),[A,B]]+13!(12[A,[B,[B,A]]]+[(A+B),[(A+B),[A,B]]])+⋯.{displaystyle ln left(e^{A}e^{B}e^{-A}e^{-B}right)=[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 .}
![{displaystyle ln left(e^{A}e^{B}e^{-A}e^{-B}right)=[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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a041c0b240e7930f91b0690d6f42670cc88c21)
These identities can be written more generally using the subscript convention to include the anticommutator defined above.[8]
For example,
- [AB,C]−=A[B,C]∓±[A,C]∓B{displaystyle [AB,C]_{-}=A[B,C]_{mp }pm [A,C]_{mp }B}
- [AB,CD]−=A[B,C]∓D±AC[B,D]∓+[A,C]∓DB±C[A,D]∓B{displaystyle [AB,CD]_{-}=A[B,C]_{mp }Dpm AC[B,D]_{mp }+[A,C]_{mp }DBpm C[A,D]_{mp }B}
- [A,[B,C]±]+[B,[C,A]±]+[C,[A,B]±]=0{displaystyle left[A,[B,C]_{pm }right]+left[B,[C,A]_{pm }right]+left[C,[A,B]_{pm }right]=0}
![{displaystyle [AB,C]_{-}=A[B,C]_{mp }pm [A,C]_{mp }B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e511049bfa3c8609faaff6d9442c84cc6d150ec1)
![{displaystyle [AB,CD]_{-}=A[B,C]_{mp }Dpm AC[B,D]_{mp }+[A,C]_{mp }DBpm C[A,D]_{mp }B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9de6c7299f77fbc8951c9c4c51e741a3febcd0f)
![{displaystyle left[A,[B,C]_{pm }right]+left[B,[C,A]_{pm }right]+left[C,[A,B]_{pm }right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa38c6ec51fd44c85b6a7dd7820d8ebaee4f2cf)
Graded rings and algebras
When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as
- [ω,η]gr:=ωη−(−1)degωdegηηω.{displaystyle [omega ,eta ]_{gr}:=omega eta -(-1)^{deg omega deg eta }eta omega .}
![{displaystyle [omega ,eta ]_{gr}:=omega eta -(-1)^{deg omega deg eta }eta omega .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a31537974b529c2386b07f032b0452d7ab70e81)
Derivations
Especially if one deals with multiple commutators, another notation turns out to be useful, the adjoint representation:
- ad(x)(y)=[x,y].{displaystyle operatorname {ad} (x)(y)=[x,y].}
![{displaystyle operatorname {ad} (x)(y)=[x,y].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dde7131b403b5370e2b1953a25c135c22baaa670)
Then ad(x) is a linear derivation:
ad(x+y)=ad(x)+ad(y){displaystyle operatorname {ad} (x+y)=operatorname {ad} (x)+operatorname {ad} (y)}and ad(λx)=λad(x){displaystyle operatorname {ad} (lambda x)=lambda operatorname {ad} (x)}
and, crucially, it is a Lie algebra homomorphism:
- ad([x,y])=[ad(x),ad(y)] .{displaystyle operatorname {ad} ([x,y])=[operatorname {ad} (x),operatorname {ad} (y)]~.}
![{displaystyle operatorname {ad} ([x,y])=[operatorname {ad} (x),operatorname {ad} (y)]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee397f615a77a3daa1d8ff2c2a0c0c819a966b0e)
By contrast, it is not always an algebra homomorphism; it does not hold in general:
- ad(xy)=?ad(x)ad(y){displaystyle operatorname {ad} (xy),{stackrel {?}{=}},operatorname {ad} (x)operatorname {ad} (y)}
- Examples
- ad(x)ad(x)(y)=[x,[x,y]]ad(x)ad(a+b)(y)=[x,[a+b,y]]{displaystyle {begin{aligned}operatorname {ad} (x)operatorname {ad} (x)(y)&=[x,[x,y],]\operatorname {ad} (x)operatorname {ad} (a+b)(y)&=[x,[a+b,y],]end{aligned}}}
![{displaystyle {begin{aligned}operatorname {ad} (x)operatorname {ad} (x)(y)&=[x,[x,y],]\operatorname {ad} (x)operatorname {ad} (a+b)(y)&=[x,[a+b,y],]end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cee80704369858ec1d45a85aeda878f0325be05c)
General Leibniz rule
The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:
- xny=∑k=0n(nk)(ad(x))k(y)xn−k{displaystyle x^{n}y=sum _{k=0}^{n}{binom {n}{k}}left(operatorname {ad} (x)right)^{k}(y),x^{n-k}}
Replacing x by the differentiation operator ∂{displaystyle partial }
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
^ Fraleigh (1976, p. 108)
^ Herstein (1975, p. 65)
^ McKay (2000, p. 4)
^ Herstein (1975, p. 83)
^ Fraleigh (1976, p. 128)
^ McMahon (2008)
^ Liboff (2003, pp. 140–142)
^
Lavrov, P.M. "Jacobi -type identities in algebras and superalgebras" (PDF)..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
References
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.), John Wiley & Sons
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, 18, University of London, ISBN 978-0-902480-17-9, MR 1802994
McMahon, D. (2008), Quantum Field Theory, USA: McGraw Hill, ISBN 978-0-07-154382-8
Further reading
McKenzie, R.; Snow, J. (2005), "Congruence modular varieties: commutator theory", in Kudryavtsev, V. B.; Rosenberg, I. G., Structural Theory of Automata, Semigroups, and Universal Algebra, Springer, pp. 273–329
External links
Hazewinkel, Michiel, ed. (2001) [1994], "Commutator", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
