# Contents

## Definition

### In rings and algebras

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

$[x,y] := x y - y x \,.$

### In groups

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

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

## Properties

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

Revised on January 4, 2017 07:36:11 by Urs Schreiber (46.183.103.8)