# Commutator

Let A, B, ... be operators. Then the commutator of A and B is defined as [A,B]=AB-BA.

Let a, b, ... be constants, then identities include [f(x),x] = 0

[A,A] = 0

[A,B] = -[B,A]

[A,BC] = [A,B]C+B[A,C]

[AB,C] = [A,C]B+A[B,C]

[a+A,b+B] = [A,B]

[A+B,C+D] = [A,C]+[A,D]+[B,C]+[B,D].

Let A and B be tensors. Then
[A,B]=delAB-delBA.

There is a related notion of commutator in the theory of groups. The commutator of two group elements A and B is ABA^{-1}B^{-1}, and two elements A and B are said to commute when their commutator is the identity element. When the group is a Lie group, the Lie bracket in its Lie algebra is an infinitesimal version of the group commutator. For instance, let A and B be square matrices, and let α(s) and β(t) be paths in the Lie group of nonsingular matrices which satisfy
α(0)=β(0) = I

(δα)/(δs)|(s=0) = A

(δβ)/(δs)|(s=0) = B,

then
δ/(δs)δ/(δt)α(s)β(t)α^{-1}(s)β^{-1}(t)|((s=0,t=0))=2[A,B].