Commutator

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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, g and h, of a group G, is the element

[g, h] = g⁻¹h⁻¹gh.

It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups.

The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as

[g, h] = ghg⁻¹h⁻¹.

Identities (group theory)

Commutator identities are an important tool in group theory. The expression a^x denotes the conjugate of a by x, defined as x⁻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 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. Many other group theorists define the conjugate of a by x as xax⁻¹. This is often written ${\displaystyle {}^{x}a}$. 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 groups and nilpotent groups. For instance, in any group, second powers behave well:

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

If the derived subgroup is central, then

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

Ring theory

The commutator of two elements a and b of a ring or an associative algebra is defined by

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

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

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

Sometimes the brackets [ ]₊ are also used to denote anticommutators, while [ ]₋ is then used for commutators. The anticommutator is used less often than the commutator, but can be used for example to define Clifford algebras, Jordan algebras and is utilised 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. 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

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

The third relation is called anticommutativity, while the fourth is the Jacobi identity.

Additional identities

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. ${\displaystyle [AB,CD]=A[B,CD]+[A,CD]B}$

An additional identity may be found for this last expression, in the form:

1. ${\displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B}$
2. ${\displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}$

If A is a fixed element of a ring R, the first additional identity can be interpreted as a Leibniz rule for the map ${\displaystyle \operatorname {ad} _{A}:R\rightarrow R}$ given by ${\displaystyle \operatorname {ad} _{A}(B)=[A,B]}$. In other words, the map ad_A defines a derivation on the ring R. The second and third identities represent Leibniz rules for more than two factors that are valid for any derivation. Identities 4–6 can also be interpreted as Leibniz rules for a certain derivation.

Hadamard's lemma, applied on nested commutators holds, and underlies the Baker–Campbell–Hausdorff expansion of log(exp(A) exp(B)):

• ${\displaystyle e^{A}Be^{-A}\equiv B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \equiv e^{\operatorname {ad} (A)}B.}$

This formula is valid in any ring or algebra where the exponential function can be meaningfully defined, for instance 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,

• ${\displaystyle \ln \left(e^{A}e^{B}e^{-A}e^{-B}\right)=[A,B]+{\frac {1}{2!}}[(A+B),[A,B]]+{\frac {1}{3!}}\left({\frac {1}{2}}[A,[B,[B,A]]]+[(A+B),[(A+B),[A,B]]]\right)+\cdots .}$

These identities can be written more generally using the subscript convention to include the anticommutator:^[8] (defined above), for instance

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

Graded rings and algebras

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as

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

Derivations

Especially if one deals with multiple commutators, another notation turns out to be useful, the adjoint representation:

${\displaystyle \operatorname {ad} (x)(y)=[x,y].}$

Then ad(x) is a linear derivation:

${\displaystyle \operatorname {ad} (x+y)=\operatorname {ad} (x)+\operatorname {ad} (y)}$ and ${\displaystyle \operatorname {ad} (\lambda x)=\lambda \operatorname {ad} (x)}$

and, crucially, it is a Lie algebra homomorphism:

${\displaystyle \operatorname {ad} ([x,y])=[\operatorname {ad} (x),\operatorname {ad} (y)]~.}$

By contrast, it is not always an algebra homomorphism; it does not hold in general:

${\displaystyle \operatorname {ad} (xy)\,{\stackrel {?}{=}}\,\operatorname {ad} (x)\operatorname {ad} (y)}$

Examples

${\displaystyle {\begin{aligned}\operatorname {ad} (x)\operatorname {ad} (x)(y)&=[x,[x,y]\,]\\\operatorname {ad} (x)\operatorname {ad} (a+b)(y)&=[x,[a+b,y]\,]\end{aligned}}}$

General Leibniz rule

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

${\displaystyle x^{n}y=\sum _{k=0}^{n}{\binom {n}{k}}\left(\operatorname {ad} (x)\right)^{k}(y)\,x^{n-k}}$

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