withv 1 in the first copy ofV,v 2 in the second copy. Then the rule for mul-
tiplication by a complex number comes from the standard rules for complex
multiplication.
Given a real Lie algebrag, the complexificationgCis pairs of elements (X,Y)
ofg, with the above rule for multiplication by complex scalars, which can be
thought of as
gC=g+ig
The Lie bracket ongextends to a Lie bracket ongCby the rule
[(X 1 ,Y 1 ),(X 2 ,Y 2 )] = ([X 1 ,X 2 ]−[Y 1 ,Y 2 ],[X 1 ,Y 2 ] + [Y 1 ,X 2 ])which can be understood by the calculation
[X 1 +iY 1 ,X 2 +iY 2 ] = [X 1 ,X 2 ]−[Y 1 ,Y 2 ] +i([X 1 ,Y 2 ] + [Y 1 ,X 2 ])With this Lie bracketgCis a Lie algebra over the complex numbers.
For many of the cases we will be interested in, this level of abstraction is not
really needed, since they have the property thatVwill be given as a subspace
of a complex vector space, with the property thatV∩iV= 0, in which caseVC
will just be the larger subspace you get by taking complex linear combinations
of elements ofV. For example,gl(n,R), the Lie algebra of realnbynmatrices,
is a subspace ofgl(n,C), the complex matrices, and one can see that
gl(n,R)C=gl(n,C)Recalling our discussion from section 5.2.2 ofu(n), a real Lie algebra, with
elements certain complex matrices (the skew-Hermitian ones), multiplication
byigives the Hermitian ones, and complexifying will give all complex matrices
so
u(n)C=gl(n,C)
This example shows that two different real Lie algebras (u(n) andgl(n,R))
may have the same complexification. For yet another example,so(n) is the
Lie algebra of all real antisymmetric matrices,so(n)Cis the Lie algebra of all
complex antisymmetric matrices.
For an example where the general definition is needed and the situation
becomes easily confusing, consider the case ofgl(n,C), thinking of it as a Lie
algebra and thus a real vector space. The complexification of this real vector
space will have twice the (real) dimension, so
gl(n,C)C=gl(n,C) +igl(n,C)will not be what you get by just allowing complex coefficients (gl(n,C)), but
something built out of two copies of this.
Given a representationπ′of a real Lie algebrag, it can be extended to a
representation ofgCby complex linearity, defining
π′(X+iY) =π′(X) +iπ′(Y)