Complex Lie algebra

In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.

Given a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$, its conjugate ${\displaystyle {\overline {\mathfrak {g}}}}$ is a complex Lie algebra with the same underlying real vector space but with ${\displaystyle i={\sqrt {-1}}}$ acting as ${\displaystyle -i}$ instead.^[1] As a real Lie algebra, a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$ is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers).