Lie groups with conformal vector fields induced by derivations

Abstract

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 \u2061 ( p , q ) ≥ 3 . Finally, we give the classification for the Riemannian and Lorentzian cases.