FIRST–ORDER DIFFERENTIAL INVARIANTS OF THE SPLITTING SUBGROUPS OF THE POINCARÉ GROUP P (1, 4)

V. M. FEDORCHUK and V. I. FEDORCHUK

Abstract

The differential invariants of the Lie groups of the point transformations play an important role in geometry (see, for example, [1]), group analysis of differential equations (see, for example, [1–3]), etc.

Our communication is devoted to the construction of the differential invariants for nonconjugate subgroups of the generalized Poincaré group P (1, 4). The group P (1, 4) is a group of rotations and translations of the five–dimensional Minkowski space M(1, 4). For all splitting subgroups of the group P (1, 4), the first–order differential invariants have been found.

The results obtained have been used for the construction of the first-order differential equations in the space M(1, 4) × R(u), which are invariant under splitting subgroups of the group P (1, 4).

Since the Lie algebra of the group P (1, 4) contains as subalgebras the Lie algebra of the Poincaré group P (1, 3) and the Lie algebra of the extended Galilei group G̃(1, 3) [ 4 ], the obtained differential equations can be used in relativistic and non–relativistic physics.

1. Olver P.J., Applications of the Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.

2. Lie S., Transformationsgruppen, In: 3 Bd, Leipzig, 1893, Bd 3.

3. Ovsiannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982.

4. Fushchych W.I. and Nikitin A.G., Symmetries of Equations of Quantum Mechanics, Allerton Press Inc., New York, 1994.

V. M. FEDORCHUK Institute of Mathematics, Pedagogical Academy PL – 30–084 Kraków, ul. Podchora̧żych 2 POLAND and Pidstryhach Institute of Applied Problems of Mechanics and Mathematics National Ukrainian Academy of Sciences 3b Naukowa St, 79–053, Lviv UKRAINE e-mail: vas fedorchuk@yahoo.com

V. I. FEDORCHUK Ivan Franko Lviv National University 1 Universytetska St, Lviv, 7900 UKRAINE