Complex Lie algebra

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

Real form

Given a complex Lie algebra $\mathfrak{𝔤}$, a real Lie algebra ${\mathfrak{𝔤}}_0$ is said to be a real form of $\mathfrak{𝔤}$ if the complexification ${\mathfrak{𝔤}}_0{\otimes }_{\mathbb{ℝ}}\mathbb{ℂ}$ is isomorphic to $\mathfrak{𝔤}$.

A real form ${\mathfrak{𝔤}}_0$ is abelian (resp. nilpotent, solvable, semisimple) if and only if $\mathfrak{𝔤}$ is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form ${\mathfrak{𝔤}}_0$ is simple if and only if either $\mathfrak{𝔤}$ is simple or $\mathfrak{𝔤}$ is of the form $\mathfrak{𝔰}\times \overline{\mathfrak{𝔰}}$ where $\mathfrak{𝔰},\overline{\mathfrak{𝔰}}$ are simple and are the conjugates of each other.

The existence of a real form in a complex Lie algebra $\mathfrak{𝔤}$ implies that $\mathfrak{𝔤}$ is isomorphic to its conjugate; indeed, if $\mathfrak{𝔤}={\mathfrak{𝔤}}_0{\otimes }_{\mathbb{ℝ}}\mathbb{ℂ}={\mathfrak{𝔤}}_0\oplus i{\mathfrak{𝔤}}_0$, then let $\tau :\mathfrak{𝔤}\to \overline{\mathfrak{𝔤}}$ denote the $\mathbb{ℝ}$-linear isomorphism induced by complex conjugate and then

$\tau (i(x+iy))=\tau (ix-y)=-ix-y=-i\tau (x+iy)$,

which is to say $\tau$ is in fact a $\mathbb{ℂ}$-linear isomorphism.

Conversely, suppose there is a $\mathbb{ℂ}$-linear isomorphism $\tau :\mathfrak{𝔤}\stackrel{\sim }{\to }\overline{\mathfrak{𝔤}}$; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define ${\mathfrak{𝔤}}_0=\{z\in \mathfrak{𝔤}|\tau (z)=z\}$, which is clearly a real Lie algebra. Each element $z$ in $\mathfrak{𝔤}$ can be written uniquely as $z=2^{-1}(z+\tau (z))+i{2}^{-1}(i\tau (z)-iz)$. Here, $\tau (i\tau (z)-iz)=-iz+i\tau (z)$ and similarly $\tau$ fixes $z+\tau (z)$. Hence, $\mathfrak{𝔤}={\mathfrak{𝔤}}_0\oplus i{\mathfrak{𝔤}}_0$; i.e., ${\mathfrak{𝔤}}_0$ is a real form.

Complex Lie algebra of a complex Lie group

Let $\mathfrak{𝔤}$ be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group $G$. Let $\mathfrak{𝔥}$ be a Cartan subalgebra of $\mathfrak{𝔤}$ and $H$ the Lie subgroup corresponding to $\mathfrak{𝔥}$; the conjugates of $H$ are called Cartan subgroups.

Suppose there is the decomposition $\mathfrak{𝔤}={\mathfrak{𝔫}}^{-}\oplus \mathfrak{𝔥}\oplus {\mathfrak{𝔫}}^{+}$ given by a choice of positive roots. Then the exponential map defines an isomorphism from ${\mathfrak{𝔫}}^{+}$ to a closed subgroup $U\subset G$. The Lie subgroup $B\subset G$ corresponding to the Borel subalgebra $\mathfrak{𝔟}=\mathfrak{𝔥}\oplus {\mathfrak{𝔫}}^{+}$ is closed and is the semidirect product of $H$ and $U$; the conjugates of $B$ are called Borel subgroups.

References

• Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9. 
• Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5. .
• Serre, Jean-Pierre (2001). Complex Semisimple Lie Algebras. Berlin: Springer. ISBN 3-5406-7827-1.