Chapter 18

Semi-direct Products

The theory of a free particle is largely determined by its group of symmetries,
the group of symmetries of three dimensional space, a group which includes a
subgroupR^3 of spatial translations, and a subgroupSO(3) of rotations. The
second subgroup acts non-trivially on the first, since the direction of a transla-
tion is rotated by an element ofSO(3). In later chapters dealing with special
relativity, these groups get enlarged to include a fourth dimension, time, and the
theory of a free particle will again be determined by the action of these groups,
now on space-time, not just space. In chapters 15 and 16 we studied two groups
acting on phase space: the Heisenberg groupH 2 d+1and the symplectic group
Sp(2d,R). In this situation also, the second group acts non-trivially on the first
by automorphisms (see 16.19).
This situation of two groups, with one acting on the other by automorphisms,
allows one to construct a new sort of product of the two groups, called the semi-
direct product, and this will be the topic for this chapter. The general theory
of such a construction will be given, but our interest will be in certain specific
examples: the semi-direct product ofR^3 andSO(3), the semi-direct product of
H 2 d+1andSp(2d,R), and the Poincar ́e group (which will be discussed later, in
chapter 42). This chapter will just be concerned with the groups and their Lie
algebras, with their representations the topics of later chapters (19, 20 and 42).

18.1 An example: the Euclidean group

Given two groupsG′andG′′, a product group is formed by taking pairs of
elements (g′,g′′)∈G′×G′′. However, when the two groups act on the same
space, but elements ofG′andG′′don’t commute, a different sort of product
group is needed to describe the group action. As an example, consider the
case of pairs (a 2 ,R 2 ) of elementsa 2 ∈R^3 andR 2 ∈SO(3), acting onR^3 by
translation and rotation

v→(a 2 ,R 2 )·v=a 2 +R 2 v