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Weak Lie symmetry and extended Lie algebra
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The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of 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).
Title: Weak Lie symmetry and extended Lie algebra
Description:
The concept of weak Lie motion (weak Lie symmetry) is introduced.
Applications given exhibit a reduction of the usual symmetry, e.
g.
, in the case of 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).
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