Complex Lie algebra – Wikipedia

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

Given a complex Lie algebra

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ , its conjugate

{displaystyle {overline {mathfrak {g}}}}

${displaystyle {overline {mathfrak {g}}}}$ is a complex Lie algebra with the same underlying real vector space but with

{displaystyle i={sqrt {-1}}}

$i={sqrt {-1}}$ acting as

{displaystyle -i}

$-i$ instead.^[1] As a real Lie algebra, a complex Lie algebra

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{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ 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[edit]

Given a complex Lie algebra

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ , a real Lie algebra

{displaystyle {mathfrak {g}}_{0}}

$mathfrak{g}_0$ is said to be a real form of

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ if the complexification

{displaystyle {mathfrak {g}}_{0}otimes _{mathbb {R} }mathbb {C} }

${displaystyle {mathfrak {g}}_{0}otimes _{mathbb {R} }mathbb {C} }$ is isomorphic to

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ .

A real form

{displaystyle {mathfrak {g}}_{0}}

$mathfrak{g}_0$ is abelian (resp. nilpotent, solvable, semisimple) if and only if

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ is abelian (resp. nilpotent, solvable, semisimple).^[2] On the other hand, a real form

{displaystyle {mathfrak {g}}_{0}}

$mathfrak{g}_0$ is simple if and only if either

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ is simple or

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ is of the form

{displaystyle {mathfrak {s}}times {overline {mathfrak {s}}}}

${displaystyle {mathfrak {s}}times {overline {mathfrak {s}}}}$ where

{displaystyle {mathfrak {s}},{overline {mathfrak {s}}}}

${displaystyle {mathfrak {s}},{overline {mathfrak {s}}}}$ are simple and are the conjugates of each other.^[2]

The existence of a real form in a complex Lie algebra

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ implies that

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ is isomorphic to its conjugate;^[1] indeed, if

{displaystyle {mathfrak {g}}={mathfrak {g}}_{0}otimes _{mathbb {R} }mathbb {C} ={mathfrak {g}}_{0}oplus i{mathfrak {g}}_{0}}

${displaystyle {mathfrak {g}}={mathfrak {g}}_{0}otimes _{mathbb {R} }mathbb {C} ={mathfrak {g}}_{0}oplus i{mathfrak {g}}_{0}}$ , then let

{displaystyle tau :{mathfrak {g}}to {overline {mathfrak {g}}}}

${displaystyle tau :{mathfrak {g}}to {overline {mathfrak {g}}}}$ denote the

{displaystyle mathbb {R} }

$mathbb {R}$ -linear isomorphism induced by complex conjugate and then

{displaystyle tau (i(x+iy))=tau (ix-y)=-ix-y=-itau (x+iy)}

which is to say

{displaystyle tau }

$tau$ is in fact a

{displaystyle mathbb {C} }

$mathbb {C}$ -linear isomorphism.

Conversely, suppose there is a

{displaystyle mathbb {C} }

$mathbb {C}$ -linear isomorphism

{displaystyle tau :{mathfrak {g}}{overset {sim }{to }}{overline {mathfrak {g}}}}

${displaystyle tau :{mathfrak {g}}{overset {sim }{to }}{overline {mathfrak {g}}}}$ ; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define

{displaystyle {mathfrak {g}}_{0}={zin {mathfrak {g}}|tau (z)=z}}

${displaystyle {mathfrak {g}}_{0}={zin {mathfrak {g}}|tau (z)=z}}$ , which is clearly a real Lie algebra. Each element

{displaystyle z}

$z$ in

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ can be written uniquely as

{displaystyle z=2^{-1}(z+tau (z))+i2^{-1}(itau (z)-iz)}

${displaystyle z=2^{-1}(z+tau (z))+i2^{-1}(itau (z)-iz)}$ . Here,

{displaystyle tau (itau (z)-iz)=-iz+itau (z)}

${displaystyle tau (itau (z)-iz)=-iz+itau (z)}$ and similarly

{displaystyle tau }

$tau$ fixes

{displaystyle z+tau (z)}

${displaystyle z+tau (z)}$ . Hence,

{displaystyle {mathfrak {g}}={mathfrak {g}}_{0}oplus i{mathfrak {g}}_{0}}

${displaystyle {mathfrak {g}}={mathfrak {g}}_{0}oplus i{mathfrak {g}}_{0}}$ ; i.e.,

{displaystyle {mathfrak {g}}_{0}}

$mathfrak{g}_0$ is a real form.

Complex Lie algebra of a complex Lie group[edit]

Let

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group

{displaystyle G}

$G$ . Let

{displaystyle {mathfrak {h}}}

${mathfrak {h}}$ be a Cartan subalgebra of

{displaystyle {mathfrak {g}}}

${mathfrak {g}}$ and

{displaystyle H}

$H$ the Lie subgroup corresponding to

{displaystyle {mathfrak {h}}}

${mathfrak {h}}$ ; the conjugates of

{displaystyle H}

$H$ are called Cartan subgroups.

Suppose there is the decomposition

{displaystyle {mathfrak {g}}={mathfrak {n}}^{-}oplus {mathfrak {h}}oplus {mathfrak {n}}^{+}}

${displaystyle {mathfrak {g}}={mathfrak {n}}^{-}oplus {mathfrak {h}}oplus {mathfrak {n}}^{+}}$ given by a choice of positive roots. Then the exponential map defines an isomorphism from

{displaystyle {mathfrak {n}}^{+}}

${displaystyle {mathfrak {n}}^{+}}$ to a closed subgroup

{displaystyle Usubset G}

${displaystyle Usubset G}$ .^[3] The Lie subgroup

{displaystyle Bsubset G}

${displaystyle Bsubset G}$ corresponding to the Borel subalgebra

{displaystyle {mathfrak {b}}={mathfrak {h}}oplus {mathfrak {n}}^{+}}

${displaystyle {mathfrak {b}}={mathfrak {h}}oplus {mathfrak {n}}^{+}}$ is closed and is the semidirect product of

{displaystyle H}

$H$ and

{displaystyle U}

$U$ ;^[4] the conjugates of

{displaystyle B}

$B$ are called Borel subgroups.

^ ^a ^b Knapp 2002, Ch. VI, § 9.
^ ^a ^b Serre 2001, Ch. II, § 8, Theorem 9.
^ Serre 2001, Ch. VIII, § 4, Theorem 6 (a).
^ Serre 2001, Ch. VIII, § 4, Theorem 6 (b).

References[edit]

[Knapp-1] Knapp 2002, Ch. VI, § 9.

[Theorem_9-2] Serre 2001, Ch. II, § 8, Theorem 9.

[3] Serre 2001, Ch. VIII, § 4, Theorem 6 (a).

[4] Serre 2001, Ch. VIII, § 4, Theorem 6 (b).

Complex Lie algebra – Wikipedia

Real form[edit]

Complex Lie algebra of a complex Lie group[edit]

References[edit]

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