Chapter 5

Lie Algebras and Lie

Algebra Representations

In this chapter we will introduce Lie algebras and Lie algebra representations,
which provide a tractable linear construction that captures much of the behavior
of Lie groups and Lie group representations. We have so far seen the case of
U(1), for which the Lie algebra is trivial, and a little bit about theSU(2) case,
where the first non-trivial Lie algebra appears. Chapters 6 and 8 will provide
details showing how the general theory works out for the basic examples of
SU(2),SO(3) and their representations. The very general nature of the material
in this chapter may make it hard to understand until one has some experience
with examples that only appear in later chapters. The reader is thus advised
that it may be a good idea to first skim the material of this chapter, returning
for a deeper understanding and better insight into these structures after first
seeing them in action later on in more concrete contexts.
For a groupGwe have defined unitary representations (π,V) for finite di-
mensional vector spacesVof complex dimensionnas homomorphisms

π:G→U(n)

Recall that in the case ofG=U(1) (see the proof of theorem 2.3) we could use
the homomorphism property ofπto determineπin terms of its derivative at the
identity. This turns out to be a general phenomenon for Lie groupsG: we can
study their representations by considering the derivative ofπat the identity,
which we will callπ′. Because of the homomorphism property, knowingπ′is
often sufficient to characterize the representationπit comes from.π′is a linear
map from the tangent space toGat the identity to the tangent space ofU(n)
at the identity. The tangent space toGat the identity will carry some extra
structure coming from the group multiplication, and this vector space with this
structure will be called the Lie algebra ofG. The linear mapπ′will be an
example of a Lie algebra representation.
The subject of differential geometry gives many equivalent ways of defining