Quantum numbers for particles in de Sitter space

All subalgebras of the Lie algebra of the de Sitter group O(4,1) are classified with respect to conjugacy under the group itself. The maximal continuous subgroups are shown to be O(4), O(3,1), D⧠E(3) (the Euclidean group extended by dilatations), and O(2) ⊗O(2,1). Representatives of each conjugacy class are shown in the figures, also demonstrating all mutual inclusions. For each subalgebra we either derive all invariants (both polynomial and nonpolynomial ones) or prove that they have none. The mathematical results are used to discuss different possible sets of quantum numbers, characterizing elementary particle states in de Sitter space (or the states of any physical system, described by this de Sitter group).