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

1.2 Lie groups and Lie algebras 9
We are now in the position of giving a more precise definition of a Lie
group.
Lie group - A Lie group G of dimension n is a set of elements which satisfy
the following conditions:
1. they form a group;
2. they form an analytic manifold of dimension n;
3. for any two elements a and b of G, the mapping φ(a, b) = a ◦ b of the
Cartesian product G × G onto G is analytic;
4. for any element a of G, the mapping φ(a) = a −1 of G onto G is analytic.
1.2.1 Linear Lie groups
The Lie groups that are important for physical applications are of the type
known as linear Lie groups, for which a simpler definition can be given.
Let us consider a n-dimensional vector space V over the field F (such
as the field R of real numbers and the field C of complex numbers) and the
general linear group GL(N, F) of N × N matrices. A Lie group G is said to
be a linear Lie group if it is isomorphic to a subgroup G ′ of GL(N, F). In
particular, a real linear Lie group is isomorphic to a subgroup of the linear
group GL(N, R) of N × N real matrices.
A linear Lie group G of dimension n satisfies the following conditions:
1. G possesses a faithful finite-dimensional representation D. Suppose that
this representation has dimension m; then the distance between two elements
g and g ′ of G is given, according to Eq. (1.16), by
{ ∑
m
d(g, g ′ ) = | D(g) ij − D(g ′ ) ij | 2} 1/2
, (1.17)
i,j=1
and the set of matrices D(g) satisfies the requirement of a metric space.
2. There exists a real number δ > 0 such that every element g of G lying in
the open set V δ , centered on the identity e and defined by d(g, e) < δ, can
be parametrized by n independent real parameters (x 1 , x 2 , ..., x n ), with
e corresponding to x 1 = x 2 = ... = x n = 0. Then every element of V δ
corresponds to one and only one point in a n-dimensional real Euclidean
space R n . The number n is the dimension of the linear Lie group.
3. There exists a real number ǫ > 0 such that every point in R n for which
n∑
x 2 i < ǫ 2 (1.18)
i=1
corresponds to some element g in the open set V δ defined above and the
correspondence is one-to-one.