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.

Real form

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). 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.

The existence of a real form in a complex Lie algebra $𝔤$ implies that $𝔤$ is isomorphic to its conjugate; 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. On the other hand, a Lie algebra can be isomorphic to its conjugate but admit no real form; see and references therein.

Complex Lie algebra of a complex Lie group

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

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

Notes

1 2 Knapp 2002, Ch. VI, § 9.
1 2 Serre 2001, Ch. II, § 8, Theorem 9.
↑ "Algebra isomorphic to its complex conjugate". MathOverflow. Retrieved 2026-10-12.
↑ Serre 2001, Ch. VIII, § 4, Theorem 6 (a).
↑ Serre 2001, Ch. VIII, § 4, Theorem 6 (b).

References

Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. Vol. 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.

Complex Lie algebra

Real form

Complex Lie algebra of a complex Lie group

Notes

References

Further reading