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Complex Lie algebra

Given a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$, a real Lie algebra ${\displaystyle {\mathfrak {g}}_{0}}$ is said to be a real formof${\displaystyle {\mathfrak {g}}}$ if the complexification ${\displaystyle {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} }$ is isomorphic to ${\displaystyle {\mathfrak {g}}}$.

A real form ${\displaystyle {\mathfrak {g}}_{0}}$ is abelian (resp. nilpotent, solvable, semisimple) if and only if ${\displaystyle {\mathfrak {g}}}$ is abelian (resp. nilpotent, solvable, semisimple).^[1] On the other hand, a real form ${\displaystyle {\mathfrak {g}}_{0}}$ is simple if and only if either ${\displaystyle {\mathfrak {g}}}$ is simple or ${\displaystyle {\mathfrak {g}}}$ is of the form ${\displaystyle {\mathfrak {s}}\times {\overline {\mathfrak {s}}}}$ where ${\displaystyle {\mathfrak {s}},{\overline {\mathfrak {s}}}}$ are simple and are complex conjugates of each other.^[1]

Complex Lie algebra of a complex Lie group

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

Suppose ${\displaystyle {\mathfrak {g}}}$ is semisimple and there is the decomposition ${\displaystyle {\mathfrak {g}}={\mathfrak {n}}^{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$ given by a choice of a root system and positive roots. Then the exponential map defines an isomorphism from ${\displaystyle {\mathfrak {n}}}$ to a closed subgroup ${\displaystyle U\subset G}$.^[2] The Lie subgroup ${\displaystyle B\subset G}$ corresponding to ${\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}}$ is closed and is the semidirect product of ${\displaystyle H}$ and ${\displaystyle U}$.^[3]; the conjugates of ${\displaystyle B}$ are called Borel subgroups.

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.
• Jean-Pierre Serre: Complex Semisimple Lie Algebras, Springer, Berlin, 2001. ISBN 3-5406-7827-1

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