Complex Lie algebra
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.
Given a complex Lie algebra , its conjugate is a complex Lie algebra with the same underlying real vector space but with acting as instead. As a real Lie algebra, a complex Lie algebra 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).
A real form is abelian (resp. nilpotent, solvable, semisimple) if and only if is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form is simple if and only if either is simple or is of the form where are simple and are the conjugates of each other.
The existence of a real form in a complex Lie algebra implies that is isomorphic to its conjugate; indeed, if , then let denote the -linear isomorphism induced by complex conjugate and then
which is to say is in fact a -linear isomorphism.
Conversely, suppose there is a -linear isomorphism ; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define , which is clearly a real Lie algebra. Each element in can be written uniquely as . Here, and similarly fixes . Hence, ; i.e., is a real form.
Complex Lie algebra of a complex Lie groupEdit
Let be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group . Let be a Cartan subalgebra of and the Lie subgroup corresponding to ; the conjugates of are called Cartan subgroups.
Suppose there is the decomposition given by a choice of positive roots. Then the exponential map defines an isomorphism from to a closed subgroup . The Lie subgroup corresponding to the Borel subalgebra is closed and is the semidirect product of and ; the conjugates of are called Borel subgroups.
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- Serre, Ch. II, § 8, Theorem 9. harvnb error: no target: CITEREFSerre (help)
- Serre, Ch. VIII, § 4, Theorem 6 (a). harvnb error: no target: CITEREFSerre (help)
- Serre, Ch. VIII, § 4, Theorem 6 (b). harvnb error: no target: CITEREFSerre (help)
- 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
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