nLab
complex conjugation

Contents

Context

Algebra

Complex geometry

Contents

Idea

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

() * a+ib aibAAAAAAfora,b. \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 quaternionc conjugation for the analogous operation on quaternions:

() * a+ib+jc+kd aibjckdAAAAAAfora,b,c,d. \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} \,.

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

Created on January 26, 2021 at 07:17:53. See the history of this page for a list of all contributions to it.