Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group

Abstract

We present a general method for reducing the problem of finding all continuous subgroups of a given Lie group G with a nontrivial invariant subgroup N, to that of classifying the subgroups of N and the subgroups of the factor group G/N. The method is applied to classify all continuous subgroups of the Poincaré group (PG) and of the Lorentz group extended by dilatations [the homogeneous similitude group (HSG)]. Lists of representatives of each conjugacy class of subalgebras of the Lie algebras of the groups PG and HSG are given in the form of tables.