Complex Lie algebra explained

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

Given a complex Lie algebra

ak{g} , its conjugate \overline{ ak g} is a complex Lie algebra with the same underlying real vector space but with i = \sqrt{ -1 } acting as - i instead. As a real Lie algebra, a complex Lie algebra ak{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

See main article: Real form.

Given a complex Lie algebra

ak{g} , a real Lie algebra ak{g} _{0} is said to be a real form of ak{g} if the complexification ak{g} _{0} ⊗ _{R} C is isomorphic to ak{g} .

A real form

ak{g} _{0} is abelian (resp. nilpotent, solvable, semisimple) if and only if ak{g} is abelian (resp. nilpotent, solvable, semisimple). On the other hand, a real form ak{g} _{0} is simple if and only if either ak{g} is simple or ak{g} is of the form ak{s} x \overline{ ak{s}} where ak{s}, \overline{ ak{s}} are simple and are the conjugates of each other.

The existence of a real form in a complex Lie algebra

ak g implies that ak g is isomorphic to its conjugate; indeed, if ak{g} = ak{g} _{0} ⊗ _{R} C = ak{g} _{0} ⊕ i ak{g} _{0} , then let \tau : ak{g} \to \overline{ ak{g}} denote the R -linear isomorphism induced by complex conjugate and then\tau(i(x + iy)) = \tau(ix - y) = - ix - y = - i\tau(x + iy) ,which is to say \tau is in fact a C -linear isomorphism.

Conversely, suppose there is a

C -linear isomorphism \tau: ak{g} \overset{\sim}\to \overline{ ak{g}} ; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define ak{g} _{0} = \{ z \in ak{g} | \tau(z) = z \} , which is clearly a real Lie algebra. Each element z in ak{g} can be written uniquely as z = 2 ^{-1} (z + \tau(z)) + i 2 ^{-1} (i\tau(z) - iz) . Here, \tau(i\tau(z) - iz) = - iz + i\tau(z) and similarly \tau fixes z + \tau(z) . Hence, ak{g} = ak{g} _{0} ⊕ i ak{g} _{0} ; i.e., ak{g} _{0} is a real form.

Complex Lie algebra of a complex Lie group

Let

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

Suppose there is the decomposition

ak{g} = ak{n} ^{-} ⊕ ak{h} ⊕ ak{n} ^{+} given by a choice of positive roots. Then the exponential map defines an isomorphism from ak{n} ^{+} to a closed subgroup U \subset G . The Lie subgroup B \subset G corresponding to the Borel subalgebra ak{b} = ak{h} ⊕ ak{n} ^{+} is closed and is the semidirect product of H and U ; the conjugates of B are called Borel subgroups.

References

• • Book: Knapp, A. W.. A. W. Knapp. Lie groups beyond an introduction. 0-8176-4259-5. Birkhäuser. Progress in Mathematics. 120. 2nd. 2002. Boston·Basel·Berlin. .
• Book: Serre, Jean-Pierre . Complex Semisimple Lie Algebras . Springer . Berlin . 2001 . 3-5406-7827-1.