(Almost) Ricci Solitons in Lorentzian–Sasakian Hom-Lie Groups

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional sl(2,R) and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentzian–Sasakian Hom-Lie algebras are investigated. If v is a contact 1-form, conditions under which the Ricci curvature tensor is v-parallel are given. Ricci solitons for Lorentzian–Sasakian Hom-Lie algebras are also studied. It is shown that a Ricci soliton vector field ζ is conformal whenever the Lorentzian–Sasakian Hom-Lie algebra is Ricci semisymmetric. To illustrate the use of the theory, a two-parameter family of three-dimensional Lorentzian–Sasakian Hom-Lie algebras which are not Lie algebras is given and their Ricci solitons are computed.