# Definition:Conjugation (Abstract Algebra)

## Definition

Let $A = \left({A_F, \oplus}\right)$ be an algebra over a field $F$.

Let $C: A_F \to A_F$ be a mapping such that:

$\forall a \in A: C \left({C \left({a}\right)}\right) = a$
$\forall a, b \in A: C \left({a \oplus b}\right) = C \left({b}\right) \oplus C \left({a}\right)$

Then $C$ is called a conjugation on $A$.

### Conjugate

Let $a \in A$.

Then $C \left({a}\right)$ is called the conjugate of $a$.

## Notation

$C \left({a}\right)$ is usually written $a^*$ in the general context of algebras.

When $A$ is the set of complex numbers, $C \left({a}\right)$ is usually written $\overline a$ and is referred to as the complex conjugate of $a$.