# Dictionary Definition

commutator n : switch for reversing the direction
of an electric current

# User Contributed Dictionary

## English

### Noun

- 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#### See also

# Extensive Definition

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

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

### Identities

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.

## Ring theory

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

- [a, b] = ab − ba

### Identities

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 \,\!

Additional relations:

- [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:

- e^Be^=B+[A,B]+\frac[A,[A,B]]+\frac[A,[A,[A,B]]]+...

## Graded Rings and Algebras

When 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## Derivations

Especially 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.

Examples:

- (x) (x)(y) = [x,[x,y]\,]
- (x) (a+b)(y) = [x,[a+b,y]\,]

## See also

## References

- 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