commutator n : switch for reversing the direction of an electric current
- an electrical switch, in a generator or motor, that periodically reverses the direction of an electric current
- (of a group) an element of the form ghg−1h−1 where g and h are elements of the group; it is equal to the group's identity if and only if g and h commute
- (of a ring) an element of the form ab-ba, where a and b are elements of the ring, it is identical to the ring's zero element if and only if a and b commute
Translations*Swedish: kommutator *Hungarian: kommutátor*Swedish: kommutator
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.
The commutator of two elements g and h of a group G is the element
- [g, h] = g−1h−1gh
N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as
- [g, h] = ghg−1h−1
In the sequel the expression ax denotes the conjugated (by x) element x−1a x.
- [y,x] = [x,y]^\,.
- [[x, y^], z]^y\cdot[[y, z^], x]^z\cdot[[z, x^], y]^x = 1.
- [x y, z] = [x, z]^y\cdot [y, z].
- [x, y z] = [x, z]\cdot [x, y]^z.
The second identity is also known under the name Hall-Witt identity. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section). The fourth identity follows from the first and third.
N.B. The above definition of the conjugate of a by x is used by group theorists. Many other mathematicians define the conjugate of a by x as xax−1. This is usually written ^x a.
The commutator of two elements a and b of a ring or an associative algebra is defined by
- [a, b] = ab − ba
IdentitiesThe commutator has the following properties:
- [A,A] = 0 \,\!
- [A,B] = - [B,A] \,\!
- [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 \,\!
- [A,BC] = [A,B]C + B[A,C] \,\!
- [AB,C] = A[B,C] + [A,C]B \,\!
- [A,BC] = [AB,C] + [CA,B] \,\!
- [ABC,D] = AB[C,D] + A[B,D]C + [A,D]BC \,\!
- [A,B], C], D] + [[[B,C], D], A] + [[[C, D], A], B] + [[[D, A], B], C] = [[A, C], [B, D \,\!
If A is a fixed element of a ring \scriptstyle\mathfrak , the first additional relation can also be interpreted as a Leibniz rule for the map \scriptstyle D_A: R \rightarrow R given by \scriptstyle B \mapsto [A,B]. In other words: the map D_A defines a derivation on the ring \scriptstyle\mathfrak .
The following identity involving commutators, a special case of the Baker-Campbell-Hausdorff formula, is also useful:
Graded Rings and AlgebrasWhen dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as \ [\omega,\eta]_ := \omega\eta - (-1)^ \eta\omega
DerivationsEspecially if one deals with multiple commutators, another notation turns out to be useful involving the adjoint representation:
\operatorname (x)(y) = [x, y] .
Then (x) is a derivation and is linear, i.e., (x+y)= (x)+ (y) and (\lambda x)=\lambda\,\operatorname (x), and a Lie algebra homomorphism, i.e, ([x, y])=[ (x), (y)], but it is not always an algebra homomorphism, i.e the identity \operatorname(xy) = \operatorname(x)\operatorname(y) does not hold in general.
- (x) (x)(y) = [x,[x,y]\,]
- (x) (a+b)(y) = [x,[a+b,y]\,]
- Introduction to Quantum Mechanics
- Introductory Quantum Mechanics
commutator in Czech: Komutátor (algebra)
commutator in Danish: Kommutator (matematik)
commutator in German: Kommutator (Mathematik)
commutator in Italian: Commutatore
commutator in Hebrew: קומוטטור
commutator in Dutch: Commutator (wiskunde)
commutator in Polish: Komutator (operatorów)
commutator in Portuguese: Comutador (matemática)
commutator in Finnish: Kommutaattori
commutator in Ukrainian: Комутатор (математика)
commutator in Chinese: 交換子