Complex Lie algebra

In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math], its conjugate [math]\displaystyle{ \overline{\mathfrak g} }[/math] is a complex Lie algebra with the same underlying real vector space but with [math]\displaystyle{ i = \sqrt{-1} }[/math] acting as [math]\displaystyle{ -i }[/math] instead. As a real Lie algebra, a complex Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] 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 [math]\displaystyle{ \mathfrak{g} }[/math], a real Lie algebra [math]\displaystyle{ \mathfrak{g}_0 }[/math] is said to be a real form of [math]\displaystyle{ \mathfrak{g} }[/math] if the complexification [math]\displaystyle{ \mathfrak{g}_0 \otimes_{\mathbb{R}}\mathbb{C} }[/math] is isomorphic to [math]\displaystyle{ \mathfrak{g} }[/math].

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

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

[math]\displaystyle{ \tau(i(x + iy)) = \tau(ix - y) = -ix- y = -i\tau(x + iy) }[/math],

which is to say [math]\displaystyle{ \tau }[/math] is in fact a [math]\displaystyle{ \mathbb{C} }[/math]-linear isomorphism.

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

Complex Lie algebra of a complex Lie group

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

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