Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

Abstract

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot–Guldberg–Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot–Guldberg–Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.