nLab commutator

Definition

In rings and algebras

For $A$ a ring or associative algebra, the commutator of two elements $x,y \in A$ is the element

$$[x,y] \coloneqq x y - y x \,.$$

In groups

For $G$ a group, the group commutator of two elements $a,b \in G$ is the element

$$[a,b] \coloneqq a^{-1} b^{-1} a b \,.$$

Properties

• For $A$ an associative algebra, the underlying vector space of $A$ equipped with the commutator bracket is a Lie algebra $(A,[-,-])$.

Related concepts

• Lie bracket

• group commutator. commutator subgroup

• anti-commutator

• graded commutator

• associator

• commutator function

• canonical commutation relation, Weyl algebra

References

On commutators in operator algebra:

• Calvin R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Ergebnisse der Mathematik und ihrer Grenzgebiete 36, Springer (1967) [doi:10.1007/978-3-642-85938-0]