# nLab commutator

Contents

### Context

#### Algebra

higher algebra

universal algebra

# 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

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

### 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 $(A,[-,-])$.

Last revised on May 5, 2020 at 11:30:21. See the history of this page for a list of all contributions to it.