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-1B-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


 * $$\left.\frac{\partial\alpha}{\partial s}\right|_{s=0}=A$$


 * $$\left.\frac{\partial\beta}{\partial t}\right|_{t=0}=B$$

then
 * $$\left.\frac{\partial}{\partial s}\frac{\partial}{\partial s}\alpha(s)\beta(t)\alpha^{-1}(s)\beta^{-1}(t)\right|_{s=0,t=0}=[A,B].$$