Complex Lie Algebra

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

Given a complex Lie algebra $\mathfrak$, its conjugate $\overline$ is a complex Lie algebra with the same underlying real vector space but with $i = \sqrt$ acting as $-i$ instead. As a real Lie algebra, a complex Lie algebra $\mathfrak$ 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 $\mathfrak$, a real Lie algebra $\mathfrak_0$ is said to be a real form of $\mathfrak$ if the complexification $\mathfrak_0 \otimes_\mathbb$ is isomorphic to $\mathfrak$.

A real form $\mathfrak_0$ is abelian (resp. nilpotent, solvable, semisimple) if and only if $\mathfrak$ is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form $\mathfrak_0$ is simple if and only if either $\mathfrak$ is simple or $\mathfrak$ is of the form $\mathfrak \times \overline$ where $\mathfrak, \overline$ 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 $\mathfrak = \mathfrak_0 \otimes_ \mathbb = \mathfrak_0 \oplus i\mathfrak_0$, then let $\tau : \mathfrak \to \overline$ denote the $\mathbb$-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 $\mathbb$-linear isomorphism.

Conversely, suppose there is a $\mathbb$-linear isomorphism $\tau: \mathfrak \overset\to \overline$; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define $\mathfrak_0 = \$, which is clearly a real Lie algebra. Each element $z$ in $\mathfrak$ can be written uniquely as $z = 2^(z + \tau(z)) + i 2^(i\tau(z) - iz)$. Here, $\tau(i\tau(z) - iz) = -iz + i\tau(z)$ and similarly $\tau$ fixes $z + \tau(z)$. Hence, $\mathfrak = \mathfrak_0 \oplus i \mathfrak_0$; i.e., $\mathfrak_0$ is a real form.

Complex Lie algebra of a complex Lie group

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

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

Notes

References

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Lie algebras