Commutator

For an electrical switch that periodically reverses the current see commutator (electric)

In mathematics, the commutator gives an indication of how poorly 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

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 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 groups.

N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as

h = ghg⁻¹h⁻¹

Identities

In the sequel the expression a^x denotes the conjugated (by x) element x⁻¹a x.

= [x,y ⁻¹
⁻¹" target="_blank" >^*,z]" target="_blank" >^y ^* ^z ^*,y]^x = 1
= ^*
= ^*^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).

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.

Ring theory

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

b = ab − ba

It is zero if and only if a and b commute. In linear algebra, if two matrices commute in one basis they will commute in any basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about these commutators.

Likewise, the anticommutator is defined as ab + ba, often written { a, b }. See also Poisson algebra.

Identities

The commutator has the following properties:

Lie-algebra relations:

= − [B,A
^* = 0
^*]" target="_blank" >+ ^* + ^*] = 0 (the Jacobi identity)

Additional relations:

= ^*
= A^*B
= ^*
= AB^*C" target="_blank" >+ [A,DBC

If A is a fixed element of a ring R, the first additional relation can also be interpreted as a Leibniz rule for the map $D_A: R \rightarrow R$ given by $B \mapsto$ ^* \ . In other words: the map D_A defines a derivation on the ring R.

Alternate notation

Especially if one deals with multiple Commutators another notation turns out to be useful:

$\mathcal C_{x}\;y =$ ^*

Examples:

$\mathcal C_{x}^3\;y =$ ^*\,]\,]
$\mathcal C_{x}^2\;\mathcal C_{a+b}\;y =$ ^*\,]\,]

References

Abstract algebra | Group theory | Binary operations

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Group theory

Identities

Ring theory

Identities

Alternate notation

See also

References