Weak Lie Symmetry and extended Lie algebra

The concept of weak Lie motion (weak Lie symmetry) is introduced through ${\mathcal{L}}_{\xi }{\mathcal{L}}_{\xi }{g}_{ab}=0,$ (${\mathcal{L}}_{\xi }{\mathcal{L}}_{\xi }f=0$). Applications are given which exhibit a reduction of the usual symmetry, e.g., in the case of the the rotation group. In this context, a particular generalization of Lie algebras is found (“extended Lie algebras”) which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).