nLab
commutant

Contents

Definition

In an associative algebra AA, the commutant of a set BAB \subset A of elements of AA is the set

B={aA|bB:ab=ba} B' = \{a \in A | \forall b \in B: a b = b a \}

of elements in AA that commute with all elements in BB.

Properties

The operation of taking a commutant is a contravariant map P(A)P(A)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,CAB, C \subseteq A the equivalence

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

Hence BBB \subseteq B'' and also B=BB' = B'''.

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