Definition:Conjugation (Abstract Algebra)
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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 $a$.
Also see