A Levi-Malcev type decomposition for 2-step solvable Lie algebras with a complex structure
problem Decomposition of 2-step solvable Lie algebras with a complex structure method Proving a Levi-Malcev type decomposition
result Fino-Vezzoni conjecture holds for 2-step solvable unimodular Lie algebras Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.
Paper generalizes results from nilpotent Lie algebras to broader types.
problem Generalizing results from nilpotent Lie algebras to broader types.
method Examined Einstein Lorentzian unimodular and solvable Lie algebras.
result Key results from nilpotent Lie algebras still hold in broader settings.
Classifies solvable symplectic Lie algebras via extensions and proves structural theorems.
problem Characterizing solvable symplectic Lie algebras.
method Symplectic double extension process.
result Classifies Lie algebras of dimensions up to 6 and proves structural theorems.
Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.
problem Investigate the existence of complex curves in hypercomplex nilmanifolds.
method Analyze quaternionic-solvable hypercomplex structures on nilpotent Lie algebras and prove the absence of complex curves in complex manifolds.
result Prove the non-existence of complex curves in complex manifolds associated with quaternionic-solvable hypercomplex structures.
Study SKT and Kähler structures on specific Lie algebras.
problem Characterize SKT and Kähler structures on solvable Lie algebras with codimension two nilradical.
method Classify and construct new examples of SKT solvable Lie algebras.
result Provide a classification of SKT Lie algebras in dimension six and extend SKT nilpotent Lie algebras to higher dimensions.
Study on symplectic Lie algebras with specific dimensions.
problem Understanding symplectic structures on solvable Lie algebras.
method Analysis of Lie algebras over R or C with specific properties. result Description of complete Lie algebras with dim nilradical ≤ 6 and symplectic structure.
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…
Characterizes non-degenerate cyclic metric Lie algebras.
problem Understanding the structure of non-solvable cyclic metric Lie algebras.
method Using sufficient conditions, cyclic quadruples, and double extension method.
result Complete characterization of non-degenerate cyclic metric Lie algebras.
Classifies 2-solvable Frobenius Lie algebras based on endomorphisms.
problem Classifying 2-solvable Frobenius Lie algebras.
method Semidirect sum decomposition, Jordan form, classification of MASAs.
result Complete classification of 2-solvable Frobenius Lie algebras in low dimensions.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-…
The paper constructs Einstein Sasaki metrics on solvable Lie groups.
problem Constructing left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups.
method Characterizing pseudo-Kähler structures and derivations giving rise to Sasaki-Einstein metrics.
result Classification of z-standard Sasaki solvable Lie algebras of dimension ≤7. 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…
This paper studies actions of solvable Lie groups on nilpotent Lie groups.
problem Characterizing which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Using Lie algebra properties and semisimple splitting, the paper provides methods to check for such actions.
result A full description of possibilities for actions up to dimension 4.
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.
Let ωg be a Lie algebra valued differential 1-form on a manifold M satisfying the structure equations dωg+21ωg∧ωg=0 where g is solvable. We show that the problem of finding a smooth map ρ:M→G, where G is an n-dimensional so…
Classifies two-step solvable Lie groups with SKT structures.
problem Classifying Lie groups with SKT structures.
method Shear construction and analysis of SKT shear data on Abelian Lie algebras.
result Large part of the classification for two-step solvable SKT algebras of dimension six.
New Einstein metrics found on specific Lie algebras.
problem Finding special Einstein metrics on solvable Lie algebras.
method Concrete procedure to construct Einstein pseudo-Kähler and para-Kähler metrics.
result Existence of Einstein pseudo-Riemannian metrics on specific Lie algebras.
Flat hypercomplex nilmanifolds have a specific solvability property.
problem Characterizing solvability in hypercomplex nilmanifolds.
method Proving solvability through a sequence of subalgebras.
result Flat hypercomplex nilmanifolds are H-solvable.
Study locally conformally balanced metrics on specific Lie algebras.
problem Characterize and classify locally conformally balanced metrics on almost abelian Lie algebras.
method Characterizations and classifications based on specific properties of Lie algebras.
result Classification of six-dimensional almost abelian Lie algebras with locally conformally balanced metrics.
Study on Lie groups with negative Ricci curvature, including open questions and a new cone.
problem Understanding Lie groups with negative Ricci curvature metrics.
method Overview and introduction of a new cone C(n) for solvable Lie algebras.
result Introduction of a new open cone C(n) that parametrizes solvable Lie algebras with negative Ricci curvature metrics.
Study classifies LC Kahler structures on 4D solv Lie alg, with applications.
problem Classifying LC Kahler structures on 4D solvable Lie algebras.
method Investigation through linear equivalence and geometric interpretation.
result Produces many examples, including lcK structures on Oeljeklaus-Toma manifolds.
New Lie algebras from quivers lead to rigid Ricci solitons.
problem Constructing Lie algebras from quivers to study geometric structures.
method Using finite quivers without cycles to construct solvable Lie algebras and proving their geometric properties.
result Simply-connected Lie groups corresponding to these Lie algebras admit left-invariant Ricci solitons, and when quivers are oriented multi-trees, these groups are rigid.
We give the classification of solvable and splitting Lie triple system and it turn that, up to isomorphism there exist 7 non isomorphic canonical Lie triple systems and 6 non isomorphic splitting canonical Lie triple systems and find the solvable Lie algebras associated.
Study on uniqueness of ad-invariant metrics in Lie algebras.
problem Uniqueness of ad-invariant metrics in Lie algebras up to automorphisms.
method Analysis of Lie algebras, cotangent Lie algebras, and specific conditions for uniqueness.
result Uniqueness of ad-invariant metric on T∗g implies solvability of g, but not conversely. Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
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…
Defines and analyzes the holonomy Lie algebra of geometric lattices.
problem Holonomy Lie algebra of geometric lattices.
method Combinatorial definition and analysis of solvable pairs of lattices.
result Holonomy Lie algebra structure of hypersolvable lattices.
We define a solvable extension of the graph 2-step nilpotent Lie algebras of [5] by adding elements corresponding to the 3-cliques of the graph. We study some of their basic properties and we prove that two such Lie algebras are isomorphic if and only if their graphs are isomorphic. We also briefly discuss some metric …
New concept of metric Lie algebras helps classify Lie groups.
problem Classifying Lie groups based on conformal Killing symmetric tensors.
method Introducing metric Lie algebras of Killing type and proving conditions for these algebras.
result Conditions for Lie algebras to be of Killing type with respect to any positive definite metric.
We show that the Cheeger isoperimetric constant of a solvable simply connected Lie group G with Lie algebra $\G$ is $h(G)=\max_{H\in\G,||H||=1} \tr(\ad (H))$.
Study derivations for nilpotent Lie algebras with negative Ricci curvature.
problem Characterize derivations leading to solvable extensions with negative Ricci curvature.
method Investigate the space of diagonalizable derivations for specific Lie algebras.
result Prove conjecture about derivations in dimension 5 and for Heisenberg and standard filiform Lie algebras.
Flat Hermitian Lie algebras are always Kähler.
problem Classifying Lie groups with Hermitian structures that are flat.
method Analysis on the Hermitian geometry of 2-step solvable Lie groups.
result Flat Hermitian Lie algebras are Kähler.
The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.
problem Understanding negatively curved homogeneous Finsler manifolds.
method Generalizing Heintze-Kobayashi-Wolf theory to homogeneous Finsler geometry, proving two main theorems.
result Negatively curved homogeneous Finsler manifolds are isometric to Lie groups with specific properties.
In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…
Study on metrics on specific nilmanifolds, finding new examples and properties.
problem Characteristically solvable nilmanifolds and their metrics.
method Explicit determination of left-invariant metrics and their properties.
result First known examples of Lie groups without positive index of symmetry.
Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.
problem Determining the index of symmetry for solvable 3D Lie groups with left-invariant metrics.
method Examined all solvable three-dimensional Lie groups, combined with previous work on unimodular groups.
result Index of symmetry is positive for every solvable 3D Lie algebra with a left-invariant metric, and is never 2.
Study LCP structures on solvmanifolds, complete list up to 5 dimensions.
problem Characterize LCP structures on solvable Lie groups.
method Classify LCP structures on Lie groups, focus on solvable unimodular case.
result Complete list of solvable unimodular Lie algebras up to dimension 5 with LCP structures.
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 investigate a certain class of solvable metric Lie algebras. For this purpose a theory of twofold extensions associated to an orthogonal representation of an abelian Lie algebra is developed. Among other things, we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre an…
We investigate the existence of left-invariant closed G2-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to $\mathfrak{sl}(2,\mathb…
We study a type of left-invariant structure on Lie groups, or equivalently on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain associate…
We obtain structure results for locally conformally symplectic Lie algebras. We classify locally conformally symplectic structures on four-dimensional Lie algebras and construct locally conformally symplectic structures on compact quotients of all four-dimensional connected and simply connected solvable Lie groups.
It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the thr…
Study left-invariant pseudo-Riemannian metrics on Lie groups using moving bracket approach.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups with vanishing scalar curvature invariants.
method Using the moving bracket approach, analyze Lie algebras of dimensions ≤ 6 and semi-simple Lie algebras.
result All Lie algebras of dimension ≤ 6 except 3-dimensional solvable Lie algebra are in the null cone, leading to VSI metrics.
The twist construction is a method to build new interesting examples of geometric structures with torus symmetry from well-known ones. In fact it can be used to construct arbitrary nilmanifolds from tori. In our previous paper, we presented a generalization of the twist, a shear construction of rank one, which allowed …
New non-solvable Lie groups found with negative Ricci curvature.
problem Finding new Lie groups with negative Ricci curvature.
method Using a general construction from a previous article, the authors produce metric Lie algebras with negative Ricci curvature for compact semisimple Lie algebras.
result The constructed Lie algebras have negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of the Lie algebra.
In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.