Lorentzian Algebraic Ricci Solitons on Four-Dimensional Nilpotent Lie Groups

There exist seven nonequivalent Lorentzian metrics on the nilpotent four-dimensional Lie group G₄ . We show that three of these metrics do not yield algebraic Ricci solitons, while two are confirmed to be algebraic Ricci solitons. Among the latter, one metric is Ricci-flat under a certain condition but not flat. The remaining two metrics admit an algebraic Ricci soliton structure only in specific cases, depending on particular parameter relations. In contrast, for the nilpotent Lie group H₃ × R , five Lorentzian metrics are algebraic Ricci solitons and one is flat.