Lie groups with conformal vector fields induced by derivations

Abstract

A pseudo-Riemannian Lie group (G,〈⋅,⋅〉) is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of type (p,q). This paper is to study pseudo-Riemannian Lie groups with non-Killing conformal vector fields induced by derivations which is an extension from non-Killing left-invariant conformal vector fields. First we prove that a Riemannian (i.e. type (n,0)), Lorentzian (i.e. type (n−1,1)) or trans-Lorentzian (i.e. type (n−2,2)) Lie group with such a vector field is solvable. Then we construct non-solvable unimodular pseudo-Riemannian Lie groups with such vector fields for any min⁡(p,q)≥3. Finally, we give the classification for the Riemannian and Lorentzian cases.