Complex Lie algebra

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

Given a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$, its conjugate ${\displaystyle {\overline {\mathfrak {g}}}}$ is a complex Lie algebra with the same underlying real vector space but with ${\displaystyle i={\sqrt {-1}}}$ acting as ${\displaystyle -i}$ instead.[1] As a real Lie algebra, a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle {\mathfrak {g}}}$, a real Lie algebra ${\displaystyle {\mathfrak {g}}_{0}}$ is said to be a real form of ${\displaystyle {\mathfrak {g}}}$ 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).[2] 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 the conjugates of each other.[2]

The existence of a real form in a complex Lie algebra ${\displaystyle {\mathfrak {g}}}$ implies that ${\displaystyle {\mathfrak {g}}}$ is isomorphic to its conjugate;[1] indeed, if ${\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} ={\mathfrak {g}}_{0}\oplus i{\mathfrak {g}}_{0}}$, then let ${\displaystyle \tau$ :{\mathfrak {g}}\to {\overline {\mathfrak {g}}}} denote the ${\displaystyle \mathbb {R} }$-linear isomorphism induced by complex conjugate and then

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

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

Conversely, suppose there is a ${\displaystyle \mathbb {C} }$-linear isomorphism ${\displaystyle \tau$ :{\mathfrak {g}}{\overset {\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 ${\displaystyle {\mathfrak {g}}_{0}=\{z\in {\mathfrak {g}}|\tau (z)=z\}}$, which is clearly a real Lie algebra. Each element ${\displaystyle z}$ in ${\displaystyle {\mathfrak {g}}}$ can be written uniquely as ${\displaystyle z=2^{-1}(z+\tau (z))+i2^{-1}(i\tau (z)-iz)}$. Here, ${\displaystyle \tau (i\tau (z)-iz)=-iz+i\tau (z)}$ and similarly ${\displaystyle \tau }$ fixes ${\displaystyle z+\tau (z)}$. Hence, ${\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\oplus i{\mathfrak {g}}_{0}}$; i.e., ${\displaystyle {\mathfrak {g}}_{0}}$ is a real form.

Complex Lie algebra of a complex Lie group

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

Suppose there is the decomposition ${\displaystyle {\mathfrak {g}}={\mathfrak {n}}^{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$ given by a choice of positive roots. Then the exponential map defines an isomorphism from ${\displaystyle {\mathfrak {n}}^{+}}$ to a closed subgroup ${\displaystyle U\subset G}$.[3] The Lie subgroup ${\displaystyle B\subset G}$ corresponding to the Borel subalgebra ${\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}}$ is closed and is the semidirect product of ${\displaystyle H}$ and ${\displaystyle U}$;[4] the conjugates of ${\displaystyle B}$ are called Borel subgroups.

Notes

1. Knapp, Ch. VI, § 9. harvnb error: no target: CITEREFKnapp (help)
2. Serre, Ch. II, § 8, Theorem 9. harvnb error: no target: CITEREFSerre (help)
3. Serre, Ch. VIII, § 4, Theorem 6 (a). harvnb error: no target: CITEREFSerre (help)
4. Serre, Ch. VIII, § 4, Theorem 6 (b). harvnb error: no target: CITEREFSerre (help)

References

• Fulton, William; Harris, Joe (1991). 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. MR 1153249. OCLC 246650103.
• Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5.CS1 maint: ref=harv (link).
• Jean-Pierre Serre: Complex Semisimple Lie Algebras, Springer, Berlin, 2001. ISBN 3-5406-7827-1