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.[1] 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).

Real form

Given a complex Lie algebra , a real Lie algebra is said to be a real form of if the complexification is isomorphic to .

A real form is abelian (resp. nilpotent, solvable, semisimple) if and only if is abelian (resp. nilpotent, solvable, semisimple).[2] 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.[2]

The existence of a real form in a complex Lie algebra implies that is isomorphic to its conjugate;[1] 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 group

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 .[3] The Lie subgroup corresponding to the Borel subalgebra is closed and is the semidirect product of and ;[4] the conjugates of are called Borel subgroups.

Notes

  1. Knapp 2002, Ch. VI, § 9.
  2. Serre 2001, Ch. II, § 8, Theorem 9.
  3. Serre 2001, Ch. VIII, § 4, Theorem 6 (a).
  4. 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.


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