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 formEdit

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 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  .[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.

NotesEdit

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

ReferencesEdit

  • 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