2014/09/26 by José F. Cariñena, J. F. Cariñena, F. Falceto +5 · 15 citations
Mathematics · Physics and Astronomy · #Adjoint representation #Adjoint representation of a Lie algebra #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebra over a field #Class (philosophy) #Computer science #Geometry #Graded Lie algebra #Integrable system #Lie algebra #Lie bracket of vector fields #Lie conformal algebra #Mathematics #Nilpotent #Nonlinear Waves and Solitons #Pure mathematics #Vector field #math-ph #math.CA #math.MP #msc:34A34 #msc:34C15 #msc:37J35 #msc:70H06 #nlin.SI
paper · pdf · doi:10.1088/1751-8113/48/21/215206
published in Journal of Physics A Mathematical and Theoretical 48(21), 215206 (Institute of Physics) · 18 pages
arxiv created 2014/09/26 · openalex publication_date 2015/05/08 · arxiv updated 2019/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, we extend the Lie theory of integration by quadratures of systems of ordinary differential equations in two different ways. First, we consider a finite-dimensional Lie algebra of vector fields and discuss the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a prescribed way. It turns out that the conditions can be expressed in a purely algebraic way. In the second step, we generalize the construction to the case in which we substitute the Lie algebra of vector fields by a module (generalized distribution). We obtain a much larger class of explicitly integrable systems, replacing standard concepts of solvable (or nilpotent) Lie algebra with distributional solvability (nilpotency).