COMMUTATOR

:''For an electrical switch that periodically reverses the current 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.

Contents

Group theory

Identities

Ring theory

Identities

Graded Rings and Algebras

Derivations

See also

References

Group theory

The 'commutator' of two elements ''g'' and ''h'' of a group ''G'' is the element
:[''g'', ''h''] = ''g''⁻¹''h''⁻¹''gh''
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.
N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as
:[''g'', ''h''] = ''ghg''⁻¹''h''⁻¹

Identities

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

★ $[y,x]\; =\; [x,y]^\{-1\},$.

★ $+\; [B,[C,A]]\; +\; [C,[A,B]]\; =\; 0\; ,!$
The second relation is called anticommutativity, while the third is the Jacobi identity.
''Additional relations:''

★ $[A,BC]\; =\; [A,B]C\; +\; B[A,C]\; ,!$

★ $[AB,C]\; =\; A[B,C]\; +\; [A,C]B\; ,!$

★ $[A,BC]\; =\; [AB,C]\; +\; [CA,B]\; ,!$

★ $[ABC,D]\; =\; AB[C,D]\; +\; A[B,D]C\; +\; [A,D]BC\; ,!$

★ $,!$
If ''$A$'' is a fixed element of a ring $scriptstylemathfrak\{R\}$, the first additional relation can also be interpreted as a Leibniz rule for the map $scriptstyle\; D\_A:\; R$
ightarrow R given by $scriptstyle\; B\; mapsto\; [A,B]$. In other words: the map ''$D\_A$'' defines a derivation on the ring $scriptstylemathfrak\{R\}$.

Graded Rings and Algebras

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

Derivations

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 $\{$
m ad} (x) is a derivation and $\{$
m ad} is linear, ''i.e.'', $\{$
m ad} (x+y)={
m ad} (x)+{
m ad} (y) and $\{$
m ad} (lambda x)=lambda,operatorname{ad} (x), and a Lie algebra homomorphism, ''i.e'', $\{$
m ad} ([x, y])=[{
m ad} (x), {
m 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:

★ $\{$
m ad} (x){
m ad} (x)(y) = [x,[x,y],]

★ $\{$
m ad} (x){
m ad} (a+b)(y) = [x,[a+b,y],]

See also

★ Anticommutativity

★ Derivation (abstract algebra)

★ Lie algebra

★ Pincherle derivative

★ Poisson bracket

★ Canonical commutation relation

References

★ Introduction to Quantum Mechanics, , David J., Griffiths, Prentice Hall, 2004, ISBN 0-13-805326-X

★ Introductory Quantum Mechanics, Liboff, Richard L., , , Addison-Wesley, 2002, ISBN 0-8053-8714-5