Complex Lie algebra

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

Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate g ¯ {\displaystyle {\overline {\mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i = − 1 {\displaystyle i={\sqrt {-1}}} acting as − i {\displaystyle -i} instead.^[1] As a real Lie algebra, a complex Lie algebra g {\displaystyle {\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).

Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , a real Lie algebra g 0 {\displaystyle {\mathfrak {g}}_{0}} is said to be a real form of g {\displaystyle {\mathfrak {g}}} if the complexification g 0 ⊗ R C {\displaystyle {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} } is isomorphic to g {\displaystyle {\mathfrak {g}}} .

A real form g 0 {\displaystyle {\mathfrak {g}}_{0}} is abelian (resp. nilpotent, solvable, semisimple) if and only if g {\displaystyle {\mathfrak {g}}} is abelian (resp. nilpotent, solvable, semisimple).^[2] On the other hand, a real form g 0 {\displaystyle {\mathfrak {g}}_{0}} is simple if and only if either g {\displaystyle {\mathfrak {g}}} is simple or g {\displaystyle {\mathfrak {g}}} is of the form s × s ¯ {\displaystyle {\mathfrak {s}}\times {\overline {\mathfrak {s}}}} where s , 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 g {\displaystyle {\mathfrak {g}}} implies that g {\displaystyle {\mathfrak {g}}} is isomorphic to its conjugate;^[1] indeed, if g = g 0 ⊗ R C = g 0 ⊕ i g 0 {\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} ={\mathfrak {g}}_{0}\oplus i{\mathfrak {g}}_{0}} , then let τ : g → g ¯ {\displaystyle \tau :{\mathfrak {g}}\to {\overline {\mathfrak {g}}}} denote the R {\displaystyle \mathbb {R} } -linear isomorphism induced by complex conjugate and then

τ ( i ( x + i y ) ) = τ ( i x − y ) = − i x − y = − i τ ( x + i y ) {\displaystyle \tau (i(x+iy))=\tau (ix-y)=-ix-y=-i\tau (x+iy)} ,

which is to say τ {\displaystyle \tau } is in fact a C {\displaystyle \mathbb {C} } -linear isomorphism.

Conversely,^{[clarification needed]} suppose there is a C {\displaystyle \mathbb {C} } -linear isomorphism τ : g → ∼ 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 g 0 = { z ∈ g | τ ( z ) = z } {\displaystyle {\mathfrak {g}}_{0}=\{z\in {\mathfrak {g}}|\tau (z)=z\}} , which is clearly a real Lie algebra. Each element z {\displaystyle z} in g {\displaystyle {\mathfrak {g}}} can be written uniquely as z = 2 − 1 ( z + τ ( z ) ) + i 2 − 1 ( i τ ( z ) − i z ) {\displaystyle z=2^{-1}(z+\tau (z))+i2^{-1}(i\tau (z)-iz)} . Here, τ ( i τ ( z ) − i z ) = − i z + i τ ( z ) {\displaystyle \tau (i\tau (z)-iz)=-iz+i\tau (z)} and similarly τ {\displaystyle \tau } fixes z + τ ( z ) {\displaystyle z+\tau (z)} . Hence, g = g 0 ⊕ i g 0 {\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}\oplus i{\mathfrak {g}}_{0}} ; i.e., g 0 {\displaystyle {\mathfrak {g}}_{0}} is a real form.

Let g {\displaystyle {\mathfrak {g}}} be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group G {\displaystyle G} . Let h {\displaystyle {\mathfrak {h}}} be a Cartan subalgebra of g {\displaystyle {\mathfrak {g}}} and H {\displaystyle H} the Lie subgroup corresponding to h {\displaystyle {\mathfrak {h}}} ; the conjugates of H {\displaystyle H} are called Cartan subgroups.

Suppose there is the decomposition g = n − ⊕ h ⊕ 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 n + {\displaystyle {\mathfrak {n}}^{+}} to a closed subgroup U ⊂ G {\displaystyle U\subset G} .^[3] The Lie subgroup B ⊂ G {\displaystyle B\subset G} corresponding to the Borel subalgebra b = h ⊕ n + {\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}} is closed and is the semidirect product of H {\displaystyle H} and U {\displaystyle U} ;^[4] the conjugates of B {\displaystyle B} are called Borel subgroups.

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• Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. Vol. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5..
• Serre, Jean-Pierre (2001). Complex Semisimple Lie Algebras. Berlin: Springer. ISBN 3-5406-7827-1.

• Tanguy Marsault, Laurent Schoeffel, Complex Lies, Real Physics: The Role of Algebra Complexification