Complex Lie algebra

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

Given a complex Lie algebra $𝔤$ , its conjugate $\overline{𝔤}$ is a complex Lie algebra with the same underlying real vector space but with $i = \sqrt{- 1}$ acting as $- i$ instead.^[1] As a real Lie algebra, a complex Lie algebra $𝔤$ 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).

Real form

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Given a complex Lie algebra $𝔤$ , a real Lie algebra $𝔤_{0}$ is said to be a real form of $𝔤$ if the complexification $𝔤_{0} \otimes_{ℝ} ℂ$ is isomorphic to $𝔤$ .

A real form $𝔤_{0}$ is abelian (resp. nilpotent, solvable, semisimple) if and only if $𝔤$ is abelian (resp. nilpotent, solvable, semisimple).^[2] On the other hand, a real form $𝔤_{0}$ is simple if and only if either $𝔤$ is simple or $𝔤$ is of the form $𝔰 \times \overline{𝔰}$ where $𝔰, \overline{𝔰}$ are simple and are the conjugates of each other.^[2]

The existence of a real form in a complex Lie algebra $𝔤$ implies that $𝔤$ is isomorphic to its conjugate;^[1] indeed, if $𝔤 = 𝔤_{0} \otimes_{ℝ} ℂ = 𝔤_{0} \oplus i 𝔤_{0}$ , then let $τ : 𝔤 \to \overline{𝔤}$ denote the $ℝ$ -linear isomorphism induced by complex conjugate and then

τ (i (x + i y)) = τ (i x - y) = - i x - y = - i τ (x + i y)

,

which is to say $τ$ is in fact a $ℂ$ -linear isomorphism.

Conversely,Template:Clarify suppose there is a $ℂ$ -linear isomorphism $τ : 𝔤 \tilde{\to} \overline{𝔤}$ ; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define $𝔤_{0} = {z \in 𝔤 | τ (z) = z}$ , which is clearly a real Lie algebra. Each element $z$ in $𝔤$ can be written uniquely as $z = 2^{- 1} (z + τ (z)) + i 2^{- 1} (i τ (z) - i z)$ . Here, $τ (i τ (z) - i z) = - i z + i τ (z)$ and similarly $τ$ fixes $z + τ (z)$ . Hence, $𝔤 = 𝔤_{0} \oplus i 𝔤_{0}$ ; i.e., $𝔤_{0}$ is a real form.