In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.
Given a complex Lie algebra
is a complex Lie algebra with the same underlying real vector space but with
[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).
Given a complex Lie algebra
, a real Lie algebra
is said to be a real form of
is abelian (resp.
nilpotent, solvable, semisimple) if and only if
is abelian (resp.
nilpotent, solvable, semisimple).
[2] On the other hand, a real form
are simple and are the conjugates of each other.
[2] The existence of a real form in a complex Lie algebra
is isomorphic to its conjugate;[1] indeed, if
-linear isomorphism induced by complex conjugate and then which is to say
Conversely,[clarification needed] suppose there is a
; without loss of generality, we can assume it is the identity function on the underlying real vector space.
, which is clearly a real Lie algebra.
be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group
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
is closed and is the semidirect product of
are called Borel subgroups.
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