Weak Lie Symmetry and extended Lie algebra

TL;DR

This paper introduces the concept of weak Lie symmetry, explores its applications in symmetry reduction, and develops a new class of extended Lie algebras that relate to tangent Lie algebroids and Lorentz geometries.

Contribution

It proposes weak Lie symmetry as a generalization of Lie symmetry and introduces extended Lie algebras linked to tangent Lie algebroids, with applications in geometry and physics.

Findings

Weak Lie symmetry reduces traditional symmetry groups.

Extended Lie algebras form involutive distributions and relate to tangent Lie algebroids.

Lorentz geometries can be constructed on these algebroids using an extended Cartan-Killing form.

Abstract

The concept of weak Lie motion (weak Lie symmetry) is introduced through ${\cal{L}}_{\xi}{\cal{L}}_{\xi}g_{ab}=0,$ (${\cal{L}}_{\xi}{\cal{L}}_{\xi}f=0$). 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).