First - Order Differential Invariants of the Splitting Subgroups of the Poincaré Group P ( 1 , 4 )

The functional bases of the first-order differential invariants for the splitting subgroups of the Poincaré group P (1, 4) are constructed. Some of the results obtained are presented. The differential invariants of Lie groups of point transformations play an important role in geometry (see, for example, [14]), group analysis of differential equations (see, for example, [12, 14, 15]), etc. In particular, with the help of these invariants we can construct differential equations with non-trivial symmetry groups. Differential invariants have been studied in many works (see, for example, [8, 9, 11–15, 17–19]). The present paper is devoted to the construction of functional bases of the first-order differential invariants for the splitting subgroups of the generalized Poincaré group P (1, 4). The group P (1, 4) is the group of rotations and translations of the five-dimensional Minkowski space M(1, 4). This group has many applications in theoretical and mathematical physics (see, for example, [5, 7, 10]). In order to present some of the results obtained, we consider the Lie algebra of the group P (1, 4). 1. The Lie algebra of the group P (1, 4) and its non-conjugate subalgebras. The Lie algebra of the group P (1, 4) is given by the 15 basis elements Mμν = −Mνμ (μ, ν = 0, 1, 2, 3, 4) and P ′ μ (μ = 0, 1, 2, 3, 4), satisfying the commutation relations [ P ′ μ, P ′ ν ] = 0, [ M ′ μν , P ′ σ ] = gμσP ′ ν − gνσP ′ μ, [ M ′ μν ,M ′ ρσ ] = gμρM ′ νσ + gνσM ′ μρ − gνρM ′ μσ − gμσM ′ νρ,