complex conjugation in nLab

Contents

Contents

Idea

Complex conjugation is the operation on complex numbers which reverses the sign of the imaginary part, hence the function

ℂ ⟶(−) * ℂ a+ib ↦ a−ibAAAAAAfora,b∈ℝ. \array{ \mathbb{C} & \overset{ \;\;\; (-)^\ast \;\;\; }{ \longrightarrow } & \mathbb{C} \\ a + \mathrm{i} b &\mapsto& a - \mathrm{i} b } \phantom{AAAAAA} for\;\; a,b \in \mathbb{R} \,.

More generally, the anti-involution on any star-algebra may be referred to as conjugation. For instance one speaks of quaternionic conjugation for the analogous operation on quaternions:

ℍ ⟶(−) * ℍ a+ib+jc+kd ↦ a−ib−jc−kdAAAAAAfora,b,c,d∈ℝ. \array{ \mathbb{H} & \overset{ \;\;\; (-)^\ast \;\;\; }{ \longrightarrow } & \mathbb{H} \\ a + \mathrm{i} b + \mathrm{j} c + \mathrm{k} d &\mapsto& a - \mathrm{i} b - \mathrm{j} c - \mathrm{k} d } \phantom{AAAAAA} for\;\; a, b, c, d \in \mathbb{R} \,.

• real part, imaginary part

• anti-involution

• anti-linear map, anti-dual linear space

• automorphism of the complex numbers

For an unrelated (or vaguely related) notion with a similar name see at conjugacy class and adjoint action.