Ricci solitons on four-dimensional Lorentzian Lie groups

Abstract

We determine all non-Einstein Ricci solitons on four-dimensional Lorentzian Lie groups whose soliton vector field is left-invariant. In addition to pp-wave and plane wave Lie groups, there are four families of Lorentzian metrics on semi-direct extensions mathbb {R}^3rtimes mathbb {R} and E(1,1)rtimes mathbb {R}. We show that some of these Ricci solitons are conformally Einstein and they may be expanding, steady or shrinking.