If the original representation was on a complex vector spaceV, the extended
one will act on the same space. If the original representation was on a real
vector spaceV, the extended one will act on the complexificationVC. Some of
the examples of these phenomena that we will encounter are the following:
- The adjoint representation
ad:g→gl(dimg,R) =M(dimg,R)extends to a complex representationad:gC→gl(dimg,C) =M(dimg,C)- Complexndimensional representations
π′:su(2)→M(n,C)ofsu(2) extend to representationsπ′:su(2)C=sl(2,C)→M(n,C)Doing this allows one to classify the finite dimensional irreducible repre-
sentations ofsu(2) by studyingsl(2,C) representations (see section 8.1.2).- We will see that complex representations of a real Lie algebra called the
Heisenberg Lie algebra play a central role in quantum theory and in quan-
tum field theory. An important technique for constructing such represen-
tations (using so-called “annihilation” and “creation” operators) does so
by extending the representation to the complexification of the Heisenberg
Lie algebra (see section 22.4). - Quantum field theories based on complex fields start with a Heisenberg
Lie algebra that is already complex (see chapter 37 for the case of non-
relativistic fields, section 44.1.2 for relativistic fields). The use of annihila-
tion and creation operators for such theories thus involves complexifying
a Lie algebra that is already complex, requiring the use of the general
notion of complexification discussed in this section.
5.6 For further reading
The material of this section is quite conventional mathematics, with many good
expositions, although most aimed at a higher level than ours. Examples at a
similar level to this one are [85] and [93], which cover basics of Lie groups and
Lie algebras, but without representations. The notes [40] and book [42] of Brian
Hall are a good source for the subject at a somewhat more sophisticated level
than adopted here. Some parts of the proofs given in this chapter are drawn
from those two sources.