# Involution

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An endomorphism of order two, that is, a mapping of an object onto itself whose square is the identity morphism (see also Category with involution). A periodic mapping, that is, a morphism some non-zero power of which is the identity morphism, is also sometimes called an involution. The minimum of such powers is called the period of the involution.

Often, by the involutions of a group $G$ one means the elements of order two in $G$.

An involution in an algebra $E$ over the field of real or complex numbers is a mapping $x \rightarrow x ^ {*}$ of $E$ onto itself satisfying the following involution axioms: 1) $x ^ {**} = x$ for all $x \in E$; 2) $( x + y ) ^ {*} = x ^ {*} + y ^ {*}$ for all $x , y \in E$; 3) $( \lambda x ) ^ {*} = \overline \lambda \; x ^ {*}$ for all $x \in E$ and for all $\lambda$ in the corresponding field; and 4) $( x y ) ^ {*} = y ^ {*} x ^ {*}$ for all $x , y \in E$. An algebra $E$ over the complex field endowed with an involution is called a symmetric algebra or involution algebra.

How to Cite This Entry:
Involution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Involution&oldid=47428