# 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

$B\prime =\left\{a\in A\mid \forall b\in B:ab=ba\right\}$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 $P\left(A\right)\to P\left(A\right)$ 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

$B\subseteq C\prime \phantom{\rule{2em}{0ex}}\mathrm{iff}\phantom{\rule{2em}{0ex}}C\subseteq B\prime .$B \subseteq C' \qquad iff \qquad C \subseteq B'.

Hence $B\subseteq B″$ and also $B\prime =B‴$.

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