Finitely Generated over Orbits Lie Algebras and Algebraic Representation of the Gradient System

Abstract

We have seen that a finite-dimensional Lie algebra Λ \( \subseteq \) F n (or Der (Rn)) can be associated with a full rank Lie algebra of analytical vector fields. Generally, the new Lie algebra is no longer finite-dimensional but it can be characterized using a global system of generators provided the space R of the coefficients is replaced by analytical functions C ω (R M ; R). That is to say, the new Lie algebra L(q1,·,q m ) can be determined by a system q1,·q m ,Qm+1,·,Q M \( \subseteq \) L(q1,·,q m ) in involution over the ring of analytical functions. It is the task of the finite generation over orbits of a Lie algebra to preserve both the infinite dimensionality property and the analysis of the associated gradient systems.