Definition:Conjugation on Algebra

Definition

Let $A = \struct {A_F, \oplus}$ be an algebra over a field $F$.

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

$\forall a \in A: \map C {\map C a} = a$

$\forall a, b \in A: \map C {a \oplus b} = \map C b \oplus \map C a$

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

Let $a \in A$.

Then $\map C a$ is called the conjugate of $a$.

$\map C a$ is usually written $a^*$ in the general context of algebras.

When $A$ is the set of complex numbers, $\map C a$ is usually written $\overline a$ and is referred to as the complex conjugate of $a$.