Symmetries and Group Theory in Particle Physics: An Introduction to ...

1.2 Lie groups and Lie algebras 15
Examples
1. The real Lie algebra of SU(N). Let A(t) = e at be a one-parameter subgroup of
SU(N). Since A is a N × N matrix, which satisfies the conditions A † A = AA † = I
and detA = 1, one gets: a † = −a and tr(a) = 0. Then the real Lie algebra of SU(N)
is the set of all traceless and anti-hermitian N × N matrices.
2. The real Lie algebra of SL(N, R). The elements of the one-parameter subgroup
are real N × N matrices A with det A = 1. Then the real Lie algebra of SL(N, R)
is the set of traceless real N × N matrices.
Adjoint representation of a Lie algebra - Given a real Lie algebra L of
dimension n and a basis a 1 , a 2 , ...a n for L, we define for any a ∈ L the n × n
matrix ad(a) by the relation
[a, a s ] =
n∑
ad(a) ps a p . (1.31)
p=1
The quantities ad(a) ps are the entries of the set of matrices ad(a) which form
a n-dimensional representation, called the adjoint representation of L. This
representation plays a key role in the analysis of semi-simple Lie algebras, as
it will be shown in the next Section.
Structure constants - Let us consider the real Lie algebra L of dimension
n and a basis a 1 , a 2 , ...a n . Then, since [a r , a s ] ∈ L, one can write in general
[a r , a s ] =
Eqs. (1.31) and (1.32) together imply
n∑
c p rsa p . (1.32)
p=1
{ad(a r )} ps = c p rs . (1.33)
The n 3 real number c p rs are called structure constants of L with respect to the
basis a 1 , a 2 , ...a n . The structure constants are not independent. In fact, from
the relations which define the real Lie algebra it follows:
c p rs = −cp sr
c s pqc t rs + c s qrc t ps + c s rpc t qs = 0 .
(1.34)
It is useful to define the n × n matrix g whose entries are expressed in terms
of the structure constants:
g ij = ∑ l,k
c l ikc k jl . (1.35)
Casimir operators - Let us consider the real vector space V of dimension n
of a semi-simple Lie algebra with basis a 1 , a 2 , ...a n and composition law given