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.

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 if and only if is abelian. 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 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. The Lie subgroup corresponding to the Borel subalgebra is closed and is the semidirect product of and ; the conjugates of are called Borel subgroups.