Commutator

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This article is about the mathematical concept. For the relation between canonical conjugate entities, see Canonical commutation relation. For the type of electrical switch, see Commutator (electric).

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[edit]

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.[1][2]

Identities (group theory)[edit]

Commutator identities are an important tool in group theory.[3] The expression ax denotes the conjugate of a by x, defined as x−1a x.

  1. x^y = x[x,y].\,
  2. [y,x] = [x,y]^{-1}.\,
  3. [x, z y] = [x, y]\cdot [x, z]^y and [x z, y] = [x, y]^z\cdot [z, y].
  4. [x, y^{-1}] = [y, x]^{y^{-1}} and [x^{-1}, y] = [y, x]^{x^{-1}}.
  5. [[x, y^{-1}], z]^y\cdot[[y, z^{-1}], x]^z\cdot[[z, x^{-1}], y]^x = 1 and [[x,y],z^x]\cdot [[z,x],y^z]\cdot [[y,z],x^y]=1.

Identity 5 is also known as the Hall–Witt identity.  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 {}^x a.  Similar identities hold for these conventions.

A wide range of 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 = x^2y^2[y,x][[y,x],y].\,

If the derived subgroup is central, then

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

Ring theory[edit]

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

[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 with respect to one basis, then they are so represented with respect to 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 .

Sometimes the brackets [ ]+ are also used.[6] 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.

In physics, this is an important overarching principle in quantum mechanics. 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)[edit]

The commutator has the following properties:

Lie-algebra relations:

  • [A,A] = 0
  • [A,B] = -[B,A]
  • [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0

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

Additional relations:

  •  [A+B,C] = [A,C]+[B,C]
  •  [A,BC] = [A,B]C + B[A,C]
  •  [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]
  •  [AB,C] = A[B,C] + [A,C]B
  •  [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
  •  [AB,CD] = A[B,CD] +[A,CD]B = A[B,C]D + AC[B,D] +[A,C]DB + C[A,D]B
  •  [[[A,B], C], D] + [[[B,C], D], A] + [[[C, D], A], B] + [[[D, A], B], C] = [[A, C], [B, D]]
  •  [AB, C]=A\{B, C\}-\{A, C\}B,    where \{A, B\} = AB + BA is the anticommutator defined above.

If A is a fixed element of a ring ℜ, the second additional relation can also be interpreted as a Leibniz rule for the map  \scriptstyle D_A: R \rightarrow R given by B ↦ [A,B]. In other words, the map DA defines a derivation on the ring ℜ.

The following identity ("Hadamard Lemma") involving nested commutators, underlying the Campbell–Baker–Hausdorff expansion of log (exp A exp B), is also useful:

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

Use of the same expansion expresses the above Lie group commutator in terms of a series of nested Lie bracket (algebra) commutators,

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

These identities differ slightly for the anticommutator (defined above)

  •  \{A,BC\} = \{A,B\}C - B[A,C]

Graded rings and algebras[edit]

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as \ [\omega,\eta]_{gr} := \omega\eta - (-1)^{\deg \omega \deg \eta} \eta\omega.

Derivations[edit]

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

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

Then  {\rm ad} (x) is a derivation and  {\rm ad} is linear, i.e., {\rm ad} (x+y)={\rm ad} (x)+{\rm ad} (y) and {\rm ad} (\lambda x)=\lambda\,\operatorname{ad} (x), and a Lie algebra homomorphism, i.e., {\rm ad} ([x, y])=[{\rm ad} (x), {\rm ad}(y)], but it is not always an algebra homomorphism, i.e. the identity \operatorname{ad}(xy) = \operatorname{ad}(x)\operatorname{ad}(y) does not hold in general.

Examples:

  • {\rm ad} (x){\rm ad} (x)(y) = [x,[x,y]\,]
  • {\rm ad} (x){\rm ad} (a+b)(y) = [x,[a+b,y]\,].

See also[edit]

Notes[edit]

  1. ^ Fraleigh (1976, p. 108)
  2. ^ Herstein (1964, p. 55)
  3. ^ McKay (2000, p. 4)
  4. ^ Herstein (1964, p. 70)
  5. ^ Fraleigh (1976, p. 128)
  6. ^ McMahon (2008)
  7. ^ Liboff (2003, pp. 140–142)

References[edit]

External links[edit]