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== Complex Lie algebra of a complex Lie group == |
== Complex Lie algebra of a complex Lie group == |
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Let <math>\mathfrak{g}</math> be a complex Lie algebra that is the Lie algebra of a [[complex Lie group]] <math>G</math>. |
Let <math>\mathfrak{g}</math> be a complex Lie algebra that is the Lie algebra of a [[complex Lie group]] <math>G</math>. Let <math>\mathfrak{h}</math> a [[Cartan subalgebra]] of <math>\mathfrak{g}</math> and <math>H</math> the Lie group corresponding to <math>\mathfrak{h}</math>; the conjugates of <math>H</math> are called [[Cartan subgroup]]s. |
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Suppose <math>\mathfrak{g}</math> is semisimple and there is the decomposition <math>\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+</math> given by a choice of a root system and positive roots. Then the [[exponential map (Lie theory)|exponential map]] defines an isomorphism from <math>\mathfrak{n}</math> to a |
Suppose <math>\mathfrak{g}</math> is semisimple and there is the decomposition <math>\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+</math> given by a choice of a root system and positive roots. Then the [[exponential map (Lie theory)|exponential map]] defines an isomorphism from <math>\mathfrak{n}</math> to a subclosed group <math>U</math> of <math>G</math>.<ref>{{harvnb|Serre|Theorem 6 (a)}}</ref> The Lie group <math>B</math> corresponding to <math>\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}</math> is closed and is the semidirect product of <math>H</math> and <math>U</math>. |
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== References == |
== References == |
In mathematics, a complex Lie algebra is a Lie algebra over the complex 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).[1] 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 complex conjugates of each other.[1]
Let be a complex Lie algebra that is the Lie algebra of a complex Lie group
. Let
aCartan 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 a root system and positive roots. Then the exponential map defines an isomorphism from
to a subclosed group
of
.[2] The Lie group
corresponding to
is closed and is the semidirect product of
and
.
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