arXiv research

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals…

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for th…

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial pr…

This paper contains all computations supporting the classification of 7-dimensional Einstein nilradicals given in the article "Classification of 7-dimensional Einstein nilradicals" (arXiv). Each algebra is analyzed in detail here.

A Riemannian Einstein solvmanifold is called standard, if the orthogonal complement to the nilradical of its Lie algebra is abelian. No examples of nonstandard solvmanifolds are known. We show that the standardness of an Einstein metric solvable Lie algebra is completely detected by its nilradical and prove that many c…

Study SKT and Kähler structures on specific Lie algebras.

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, …

The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues…

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the ni…

In this note we are concerned with the distribution of Einstein and non-Einstein nilradicals among all nilpotent Lie groups. A nilpotent Lie group is called an Einstein, resp. non-Einstein, nilradical if it is a nilpotent Lie group which does, resp. does not, admit a left-invariant Ricci soliton metric. Using technique…

The paper analyzes Lie symmetries in a specific geometric context.

We show that every non-compact simple real Lie algebra not isomorphic to so(n,1) has a unique conjugacy class of parabolic subalgebras whose nilradical is of Heisenberg type, or non-singular, and give some applications.

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures. The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained fo…

Every real simple non-compact Lie algebra not isomorphic to $\mathfrak{so}(1,n)$ contains a unique standard parabolic subalgebra whose nilradical is a generalized Heisenberg algebra. Here we discuss the associated parabolic geometries and the riemannian geometry of the harmonic spaces having the former as conformal inf…

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. …

Study on symplectic Lie algebras with specific dimensions.

This thesis was concerned with classifying the real indecomposable solvable Lie algebras with codimension one nilradicals of dimensions two through seven. This thesis was organized into three chapters. In the first, we described the necessary concepts and definitions about Lie algebras as well as a few helpful theorems…

We construct many examples of Lie groups with compact Levi factor admitting a left-invariant metric with negative Ricci curvature. We start with a Lie algebra with Levi factor su(n) or so(n) acting on an abelian nilradical via the representation on the space of homogeneous polynomials. In the case of su(2) we obtain a …

Study explores solvable Lie groups' actions on closed Lorentzian manifolds.

This article can be viewed as a continuation of the articles arXiv:0912.3486 and arXiv:1012.3714 where the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradical which ad…

New Einstein metric found on non-standard solvmanifold.

It is natural to ask whether solvsolitons are global maxima for the Ricci pinching functional F:=scal^2/|Ric|^2 on the set of all left-invariant metrics on a given solvable Lie group S, as it is to ask whether they are the only global maxima. A positive answer to both questions was given in a recent paper by the same a…

We analyze symplectic forms on six dimensional real solvable and non-nilpotent Lie algebras. More precisely, we obtain all those algebras endowed with a symplectic form that decompose as the direct sum of two ideals or are indecomposable solvable algebras with a four dimensional nilradical.

We consider the question of whether a given solvable Lie group admits a left-invariant metric of strictly negative Ricci curvature. We give necessary and sufficient conditions of the existence of such a metric for the Lie groups the nilradical of whose Lie algebra is either abelian or Heisenberg or standard filiform, a…

The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.

For finite dimensional real Lie algebras, we investigate the existence of an inner product having a basis comprised of geodesic elements. We give several existence and non-existence results in certain cases: unimodular solvable Lie algebras having an abelian nilradical, algebras having an abelian derived algebra, algeb…

We give an overview of what is known on Lie groups admitting a left-invariant metric of negative Ricci curvature, including many natural questions and conjectures in the solvable case. We also introduce an open and convex cone C(n) of derivations attached to each nilpotent Lie algebra n, which is defined as the image o…

Let $G$ be a simply connected solvable Lie group with a lattice $Γ$ and $N$ the nilradical of $G$. For a complex valued representation $ρ: G\to GL(V_ρ)$ such that the restriction $ρ_{|_{N}}$ is unipotent, as an advanced variation of cohomology computation of solvmanifolds by using Lie algebra cohomology, we construct a…

In this work we introduce an obstruction for the existence of symplectic structures on nilpotent Lie algebras. Indeed, a necessary condition is presented in terms of the cohomology of the Lie algebra. Using this obstruction we obtain both positive and negative results on the existence of symplectic structures on a larg…

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

Study rigidifies Einstein manifolds with symmetry, proving conjecture.

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the mom…

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct inf…

There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit…

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation; for instance the action of the nilradical of the isometry group does not need to…

A Riemannian manifold (M,g) is said to be Einstein if its Ricci tensor satisfies ric(g) = cg, for some real number c. In the homogeneous case, a problem that is still open is the so called Alekseevskii Conjecture. This conjecture says that any homogeneous Einstein space with negative scalar curvature (i.e. c < 0) is a …

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the sp…

We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply …

We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie algebra $\mathfrak{n}$. It turns out that such a metric always exists, except for in the two cases, wh…

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture …

We study the existence of strong Kähler with torsion (SKT) metrics and of symplectic forms taming invariant complex structures $J$ on solvmanifolds $G/Γ$ providing some negative results for some classes of solvmanifolds. In particular, we show that if either $J$ is invariant under the action of a nilpotent complement o…

Based on the Lie theoretical methods of algebraic Fourier transformation, we classify in the case of generic values of inducing parameters the scalar singular vectors corresponding to the diagonal branching rules for scalar generalized Verma modules in the case of orthogonal Lie algebra and its conformal parabolic suba…

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of "stepwise square integrable" representations on …

We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compac…

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…

A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einst…

The paper classifies foliations formed by generic coadjoint orbits of specific Lie groups.

The abstract describes a foliation of orbits for a specific class of Lie groups.