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commutator, device used in an electric generator to convert the alternating current produced in the generator into direct current before the current is sent into an external circuit; it is basically a rotary switching device synchronized with the frequency of the alternating current. Commutators are also used in electric motors to switch currents in order to maintain magnetic polarities necessary to keep the shafts of the motors turning.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

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

### Identities

## Ring theory

### Identities

The commutator has the following properties:## 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:## See also

## References

The commutator of two elements g and h of a group G is the element

- [g, h] = g
^{−1}h^{−1}gh

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}

Commutator identities are an important tool in group theory, . The expression a^{x} denotes x^{−1}a x.

- $x^y\; =\; x[x,y].$
- $[y,x]\; =\; [x,y]^\{-1\}.,$
- $[x\; y,\; z]\; =\; [x,\; z]^ycdot\; [y,\; z]$ and $[x,\; y\; z]\; =\; [x,\; z]cdot\; [x,\; y]^z.$
- $[x,\; y^\{-1\}]\; =\; [y,\; x]^\{y^\{-1\}\}$ and $[x^\{-1\},\; y]\; =\; [y,\; x]^\{x^\{-1\}\}.$
- $[[x,\; y^\{-1\}],\; z]^ycdot[[y,\; z^\{-1\}],\; x]^zcdot[[z,\; x^\{-1\}],\; y]^x\; =\; 1$ and $[[x,y],z^x][[z,x],y^z][[y,z],x^y]=1.$

The last identities are 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}. This is usually 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

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

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

- [a, b] = ab − ba

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:

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

If $A$ is a fixed element of a ring $scriptstylemathfrak\{R\}$, 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 $scriptstylemathfrak\{R\}$.

The following identity involving commutators, a special case of the Baker-Campbell-Hausdorff formula, is also useful:

- $e^\{A\}Be^\{-A\}=B+[A,B]+frac\{1\}\{2!\}[A,[A,B]]+frac\{1\}\{3!\}[A,[A,[A,B]]]+...$

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

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:

- $\{rm\; ad\}\; (x)\{rm\; ad\}\; (x)(y)\; =\; [x,[x,y],]$
- $\{rm\; ad\}\; (x)\{rm\; ad\}\; (a+b)(y)\; =\; [x,[a+b,y],]$

- Anticommutativity
- Derivation (abstract algebra)
- Pincherle derivative
- Poisson bracket
- Canonical commutation relation

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