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 :

The three-dimensional Heisenberg group H 3 has three left-invariant Lorentzian metrics g 1 , g 2 , and g 3 as in Rahmani (J. Geom. Phys. 9(3), 295-302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g 1 as a Lorentz Ricci Soliton. This Ricci Soliton g 1 is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.

This paper provides a study of algebraic Ricci solitons in the pseudo-Riemannian case. In the Riemannian case, all nontrivial homogeneous algebraic Ricci solitons are expanding algebraic Ricci solitons. In this paper, we obtain a steady algebraic Ricci soliton and a shrinking algebraic Ricci soliton in the Lorentzian setting.of the central problems in differential geometry. For instance, Perelman [19] was able to prove Thurston's geometrization conjecture.From the definition of the Ricci flow, a fixed point of the Ricci flow is a Ricci-flat metric, and a fixed point of the normalized Ricci flow is an Einstein metric. From Proposition 2.3, Ricci solitons change only by diffeomorphism and rescaling, and are regarded as generalized fixed points. In other words, let M(M n ) be the space of Riemannian metrics on M n , and D(M n ) the diffeomorphism group of M n . Considering the dynamical system of the Ricci flow on the moduli space M(M n )/D(M n ) × R + , we regard Ricci solitons as fixed points.In this paper, we study left-invariant pseudo-Riemannian metrics and algebraic Ricci solitons on Loretzian Lie groups. The concept of an algebraic Ricci soliton was first introduced by Lauret in the Riemannian case (see [13]). Lauret proved that algebraic Ricci solitons on homogeneous Riemannian manifolds are Ricci solitons. In general, problems for Ricci solitons are second-order differential equations. However, problems for algebraic Ricci solitons are algebraic equations. Therefore, algebraic Ricci solitons allow us to construct Ricci solitons in an algebraic way, i.e., using algebraic Ricci soliton theory, the study of Ricci solitons on homogeneous manifolds becomes algebraic. So far, Ricci solitons have been studied in the Riemannian case. Recently, the study of Ricci solitons in the pseudo-Riemannian setting has started with special attention to the Lorentzian case. In the Riemannian case, all homogeneous nontrivial Ricci solitons are expanding Ricci solitons. In the pseudo-Riemannian case, there are shrinking homogeneous non-trivial Ricci solitons discovered in [18], while all vector fields of these Ricci solitons are not left-invariant. And 3-dimensional homogeneous Lorentzian Ricci solitons with left-invariant vector fields are classified in [3]. Other results about Lorentzian Ricci solitons are found in [2], [4], [6], [7].In this paper, we study algebraic Ricci solitons on Lorentzian Lie groups in the Lorentzian case. By using algebraic Ricci soliton theory, we can construct homogeneous Lorentzian Ricci solitons in an algebraic way. For example, we can construct the Ricci soliton in [18] by using algebraic Ricci solitons and Theorem 2.5. Recall that all homogeneous non-trivial solvsolitons are expanding in the Riemannian setting. In this paper, we construct Lorentzian algebraic Ricci solitons on the Heisenberg group H N and the oscillator groups G m (λ) and on three-dimensional Lorentzian Lie groups. In particular, we obtain new Lorentzian Ricci solitons on H N and G m (λ). This paper is organized as follows. In Sec...

Abstract. We classify, up to isometry, non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons. It turns out that those of Cerny-Kowalski's types A, C and D are algebraic Ricci solitons, whereas those of type B are not. Thus, we give new examples of algebraic Ricci solitons. Mathematics Subject Classification (2000). 53C50, 53C21, 53C25.

In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. As one of applications, we show that any left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.