# nLab commutant

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

In an associative algebra $A$, the commutant of a set $B \subset A$ of elements of $A$ is the set

$B' = \{a \in A | \forall b \in B: a b = b a \}$

of elements in $A$ that commute with all elements in $B$.

## Properties

The operation of taking a commutant is a contravariant map $P(A) \to P(A)$ that is adjoint to itself in the sense of Galois connections. In other words, we have for any two subsets $B, C \subseteq A$ the equivalence

$B \subseteq C' \qquad iff \qquad C \subseteq B'.$

Hence $B \subseteq B''$ and also $B' = B'''$.

Revised on July 27, 2011 15:57:10 by Urs Schreiber (89.204.137.107)