The Ricci operator of completely solvable metric lie algebras

We study the Ricci curvature of completely solvablemetric Lie algebras. In particular,we prove that the Ricci operator of every completely solvable nonunimodular or every noncommutative nilpotent metric Lie algebra has at least two negative eigenvalues.     

On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension

The Ricci curvature of solvable metric Lie algebras is studied. In particular, we prove that the Ricci operator of any metric nonunimodular solvable Lie algebra of dimension not exceeding 6 has at least two negative eigenvalues, that generalizes the known results.     

Moduli of Einstein and non-Einstein nilradicals

A nilpotent Lie algebra is called an Einstein nilradical if the corresponding Lie group admits a left-invariant Ricci soliton metric. While these metrics are of independent interest, their existence is intimately related to the existence of Einstein metrics on solvable Lie groups. In this note we are concerned with the following question: How are the Einstein and non-Einstein nilradicals distributed among nilpotent Lie algebras? A full answer to this question is not known and we restrict to the class of 2-step nilpotent Lie groups. Within this class, it is known that a generic group admits a Ricci soliton metric. Using techniques from Geometric Invariant Theory, we study the set of non-generic algebras to learn more about the distribution of non-Einstein nilradicals. Many new (continuous) families of non-isomorphic, non-Einstein nilradicals are constructed. Moreover, the dimension of these families can be arbitrarily large (depending on the dimension of the underlying Lie group). To show such large classes of Lie groups are pairwise non-isomorphic, a new technique is developed to distinguish between Lie algebras.     

Curvatures of homogeneous Randers spaces

We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2)$\text{SU}\left(2\right)$.     

Ricci soliton homogeneous nilmanifolds

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric for some and some derivationD of the Lie algebra ofN, where Ric denotes the Ricci operator of . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set. Received August 24, 1999 / Revised October 2, 2000 / Published online February 5, 2001     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Negative eigenvalues of the Ricci operator of solvable metric Lie algebras

In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra has this property.     

Two-step solvable Lie algebras and weight graphs

By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras \mathfrakg\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product \mathfrakg?^? T_i\mathfrak{g}\overrightarrow{\oplus } T_{i} to be two-step solvable, where $T_{i}TT over \mathfrakg\mathfrak{g} which induces a decomposition of \mathfrakg\mathfrak{g} into one-dimensional weight spaces without zero weights. In particular we show that the semidirect product of such a Lie algebra with a maximal torus of derivations cannot be itself two-step solvable. We also obtain some applications to rigid Lie algebras, as a geometrical proof of the nonexistence of two-step nonsplit solvable rigid Lie algebras in dimensions n\geqslant 3n\geqslant 3.     

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.     

Sub-semi-Riemannian geometry of general H-type groups

We study a special class of nilpotent Lie groups of step 2, that generalizes the class of the so-called H(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of an inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms. We present the geodesic equation for sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general H-type groups. We also present some results on sectional curvature and the Ricci tensor of semi-Riemannian general H-type groups.     

On classical artinian localizations

ABSTRACT

We present some results about Lie algebras, which can be written as the sum of two subalgebras in two cases: where both subalgebras are simple or both are nilpotent. In the first case we suggest new examples of simple Lie algebras admitting decomposition into the sum of simple subalgebras and give explicit realizations where the existence of such decompositions was established earlier. We single out cases where such decomposition is not possible. We also construct examples of solvable Lie algebras, which are the sums of two nilpotent subalgebras, and the derived length of the sum is greater than the sum of the nilpotent indexes of the summands.     

Left invariant contact structures on Lie groups

Amongst other results, we perform a ‘contactization’ method to construct, in every odd dimension, many contact Lie groups with a discrete center, unlike the usual (classical) contactization which only produces Lie groups with a non-discrete center. We discuss some applications and consequences of such a construction, construct several examples and derive some properties. We give classification results in low dimensions. A complete list is supplied in dimension 5. In any odd dimension greater than 5, there are infinitely many locally non-isomorphic solvable contact Lie groups. We also characterize solvable contact Lie algebras whose derived ideal has codimension one. For simplicity, most of the results are given in the Lie algebra version.     

Lie algebras with almost dimensionally nilpotent inner derivations

The structure of the Lie algebras with almost dimensionally nilpotent inner derivations is studied. It is proved that, if the base field is of characteristic 0, then, when d>6 is odd, there exist just two d-dimensional Lie algebras; when d>6 is even, there exists just one d-dimensional Lie algebra such that these Lie algebras are nonsolvable and have some almost dimensionally nilpotent inner derivations.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

Lie Algebras with Nilpotent Length Greater than that of each of their Subalgebras

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call minimal non- \({\mathcal N}\). To facilitate this we investigate solvable Lie algebras of nilpotent length k, and of nilpotent length ≤k, and extreme Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-\({\mathcal N}\) Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index ≤3.     

Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.     

On the Ricci curvature of solvable metric lie algebras with two-step nilpotent derived algebras

We prove that the Ricci operator of any nonunimoular solvable metric Lie algebra having a two-step nilpotent derived Lie algebra of dimension 6 has at least two negative eigenvalues.     

On the Chern‐Ricci flow and its solitons for Lie groups

This paper is concerned with Chern‐Ricci flow evolution of left‐invariant hermitian structures on Lie groups. We study the behavior of a solution, as t is approaching the first time singularity, by rescaling in order to prevent collapsing and obtain convergence in the pointed (or Cheeger‐Gromov) sense to a Chern‐Ricci soliton. We give some results on the Chern‐Ricci form and the Lie group structure of the pointed limit in terms of the starting hermitian metric and, as an application, we obtain a complete picture for the class of solvable Lie groups having a codimension one normal abelian subgroup. We have also found a Chern‐Ricci soliton hermitian metric on most of the complex surfaces which are solvmanifolds, including an unexpected shrinking soliton example.     

Finding Einstein solvmanifolds by a variational method

We use a variational approach to prove that any nilpotent Lie algebra having a codimension-one abelian ideal, and anyone of dimension , admits a rank-one solvable extension which can be endowed with an Einstein left-invariant riemannian metric. A curve of -dimensional Einstein solvmanifolds is also given. Received: 29 May 2001; in final form: 4 October 2001 / Published online: 4 April 2002