5.5 Complexification
Conventional physics discussion of Lie algebra representations proceed by as-
suming complex coefficients are allowed in all calculations, since we are inter-
ested in complex representations. An important subtlety is that the Lie algebra
is a real vector space, often in a confusing way, as a subspace of complex ma-
trices. To properly keep track of what is going on one needs to understand the
notion of “complexification” of a vector space or Lie algebra. In some cases this
is easily understood as just going from real to complex coefficients, but in other
cases a more complicated construction is necessary. The reader is advised that
it might be a good idea to just skim this section at first reading, coming back
to it later only as needed to make sense of exactly how things work when these
subtleties make an appearance in a concrete problem.
The way we have defined a Lie algebrag, it is a real vector space, not a
complex vector space. Even ifGis a group of complex matrices, its tangent
space at the identity will not necessarily be a complex vector space. Consider
for example the casesG=U(1) andG=SU(2), whereu(1) =Randsu(2) =
R^3. While the tangent space to the groupGL(n,C) of all invertible complex
matrices is a complex vector space (M(n,C), allnbynmatrices), imposing
some condition such as unitarity picks out a subspace ofM(n,C) which generally
is just a real vector space, not a complex one. So the adjoint representation
(Ad,g) is in general not a complex representation, but a real representation,
with
Ad(g)∈GL(g) =GL(dimg,R)
The derivative of this is the Lie algebra representation
ad:X∈g→ad(X)∈gl(dimg,R)and once we pick a basis ofg, we can identifygl(dimg,R) =M(dimg,R). So,
for eachX∈gwe get a real linear operator on a real vector space.
We most often would like to work with not real representations, but complex
representations, since it is for these that Schur’s lemma applies (the proof of
2.1 also applies to the Lie algebra case), and representation operators can be
diagonalized. To get from a real Lie algebra representation to a complex one,
we can “complexify”, extending the action of real scalars to complex scalars.
If we are working with real matrices, complexification is nothing but allowing
complex entries and using the same rules for multiplying matrices as before.
More generally, for any real vector space we can define:
Definition.The complexificationVC of a real vector spaceV is the space of
pairs(v 1 ,v 2 )of elements ofV with multiplication bya+bi∈Cgiven by
(a+ib)(v 1 ,v 2 ) = (av 1 −bv 2 ,av 2 +bv 1 )One should think of the complexification ofV asVC=V+iV