nLab commutator

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Contents

Definition

In rings and algebras

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

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

In groups

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

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

Properties

References

On commutators in operator algebra:

Last revised on December 7, 2023 at 15:54:52. See the history of this page for a list of all contributions to it.