Chapter 15: Some Important Lie Groups and Their Lie Algebras

As examples of physically important Lie groups, we describe in this chapter the following groups:

• the group T₃ of translations in three-dimensional Euclidean space;

• the group R(3) or SO(3) of proper orthogonal real rotations in threedimensional space;

• the Euclidean group E(3) consisting of a combination of T₃ and R(3);

• the Galilei group, which is the group of space-time transformations describing Newtonian relativity; and

• the proper homogeneous and inhomogeneous Lorentz groups describing Poincaŕe relativity