We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we obtain new solitons on G2, G5, and G6, and we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons.Remark 1.2. Let G be a semi-simple Lie group, g a left-invariant Riemannian metric. If g is an algebraic Ricci soliton, then g is Einstein ([16]).Next we introduce Ricci solitons. Let g 0 be a pseudo-Riemannian metric on a manifold M n . If g 0 satisfieswhere ̺ denotes the Ricci tensor, X is a vector field and c is a real constant, then (M n , g 0 , X, c) is called a Ricci soliton structure and g 0 is the Ricci soliton.Moreover we say that the Ricci soliton g 0 is a gradient Ricci soliton if the vector field X satisfies X = ∇f , where f is a function. The Ricci soliton g 0 is said to be a non-gradient Ricci soliton if the vector field X satisfies X = ∇f for any function f . If c is positive, zero, or negative, then g 0 is called a shrinking, steady, or expanding Ricci soliton, respectively. According to [10], we check that a Ricci soliton is a Ricci flow solution.Proposition 1.3 ([10]). A pseudo-Riemannian metric g 0 is Ricci soliton if and only if g 0 is the initial metric of the Ricci flow equation,and the solution is expressed as g(t) = c(t)(ϕ t ) * g 0 , where c(t) is a scaling parameter, and ϕ t is a diffeomorphism.An interesting example of Ricci solitons is (R 2 , g st , X, c), where the metric g st is the Euclidean metric on R 2 , the vector field X is X = ∇f, f = |x| 2 2 and c is a real number. This is a gradient Ricci soliton structure and so, g st is the gradient Ricci soliton, named Gaussian soliton. In the closed Riemannian case, Perelman [23] proved that any Ricci soliton is a gradient Ricci soliton, and any steady or expanding Ricci soliton is an Einstein metric with the Einstein constant zero or negative, respectively. However in the non-compact Riemannian case, a Ricci soliton is not necessarily gradient and a steady or expanding Ricci soliton is not necessarily Einstein. In fact, any left-invariant Riemannian metric on the three-dimensional Heisenberg group is an expanding non-gradient Ricci soliton which is not an Einstein metric (see [1], [14], [17]). In the Riemannian case, all homogeneous non-trivial Ricci solitons are expanding Ricci solitons. In the pseudo-Riemannian case, there are shrinking homogeneous nontrivial Ricci solitons discovered in [20], the vectors fields of these Ricci solitons are not left-invariant. In [16], Lauret studied the relation between algebraic Ricci solitons and Ricci solitons on Riemannian manifolds. More precisely, he proved that any left-invariant Riemannian algebraic Ricci soliton metric is a Ricci soliton. This was extended by the second author to the pseudo-Riemannian case :

Abstract. We show that Lorentzian manifolds whose isometry group is of dimension at least 1 2 n(n − 1) + 1 are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally conformally flat and symmetric Lorentzian Ricci solitons which are not rigid.

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Homogeneous pseudo-Riemannian structures of linear type are reviewed and studied. In the Riemannian case, they furnish characterisations of the real, complex and quaternionic hyperbolic spaces. In the Lorentzian case, a related class gives characterisations of singular homogeneous plane waves.

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