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 it admits a real form (and is said to be defined over the real numbers).
Given a complex Lie algebra , a real Lie algebra
is said to be a real formof
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
issimple if and only if either
is simple or
is of the form
where
are simple and are the complex-conjugates of each other.[2]
Let be a complex Lie algebra that is the Lie algebra of a complex Lie group
. Let
be a Cartan subalgebraof
and
the Lie group corresponding to
; the conjugates of
are called Cartan subgroups.
Suppose is semisimple and 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.
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