Classifies metric symplectic Lie algebras with quadratic extensions.
problem Classifying metric symplectic Lie algebras.
method Standard model, quadratic cohomology sets, isomorphism classes.
result Complete list of metric symplectic Lie algebras in special cases.
Classifies complex symplectic structures on Lie algebras with large abelian ideals.
problem Classifying complex symplectic structures on Lie algebras with large abelian ideals.
method Two constructions of complex symplectic structures on Lie algebras with large abelian ideals, considering compact quotients of Lie groups.
result Complete classification of complex symplectic structures on almost abelian Lie algebras.
A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω is ∂∂ˉ-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a 2n-dimensional SKT Lie algebra $\mathfr…
The paper classifies para-Kähler structures on Lie groups.
problem Classifying para-Kähler structures on Lie groups.
method Classification based on symplectic Lie algebras, finding compatible para-complex structures and pseudo-Riemannian metrics.
result Explicit forms of para-complex structures and pseudo-Riemannian metrics are found.
The Streets-Tian conjecture is confirmed for Lie algebras with specific abelian ideals.
problem The Streets-Tian conjecture on compact complex manifolds admitting Hermitian-symplectic metrics.
method Detailed case analysis of Lie algebras with abelian ideals of codimension 2, explicit construction of Hermitian-symplectic metrics and pathways to Kähler metrics.
result The Streets-Tian conjecture is confirmed for Lie algebras containing abelian ideals of codimension 2.
Study pseudo-Kähler structures on almost abelian solvmanifolds.
problem Classify pseudo-Kähler structures on almost abelian Lie algebras.
method Analyzing invariant structures on solvmanifolds with specific Lie algebra properties.
result Classification of pseudo-Kähler structures on almost abelian Lie algebras.
Classifies symplectic Lie algebras with degenerate center.
problem Understanding symplectic Lie algebras with degenerate center.
method Standard model, quadratic cohomology sets, classification scheme.
result Complete list of 6-dimensional nilpotent symplectic Lie algebras.
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a com…
Research explores Kähler and semi-para-Kähler structures on specific Lie groups.
problem Existence of Kähler and semi-para-Kähler structures on six-dimensional unsolvable Lie groups.
method Examines four specific Lie algebras and their structures.
result One Lie algebra admits Kähler metrics, others admit semi-para-Kähler and semi-Kähler structures.
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.
The study of flat symplectic Lie algebras and groups.
problem Characterizing and understanding flat symplectic Lie algebras and groups.
method Analyzing the derived ideal, curvature, and double extension process.
result Every flat symplectic Lie algebra is obtained by a sequence of double extensions starting from the trivial algebra.
Method constructs complex symplectic Lie algebras from simpler ones.
problem Classifying complex symplectic Lie algebras of various dimensions.
method Complex symplectic oxidation method
result Classification of eight-dimensional nilpotent complex symplectic Lie algebras.
The study classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
problem Classifying and constructing symplectic structures on Lie algebras and solvmanifolds.
method Analyzes locally conformally symplectic Lie algebras and solvmanifolds.
result Classifies and constructs symplectic structures on Lie algebras and solvmanifolds.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a nilpotent algebra admitting an invertiblederivation and also as the double extension of…
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.
We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie …
Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs (J,ω) are parametrized up to complex isomorphism (where J is a complex structure and ω is a symplectic structure). Such structure gives rise to a pseu…
The paper classifies orbit closures of symplectic Lie algebras.
problem Classifying orbit closures of symplectic Lie algebras under the action of Sp(4,R). method Analyzing the natural action of Sp(4,R) on the set of 4-dimensional Lie algebras with symplectic structures. result A complete classification of orbit closures of 4-dimensional symplectic Lie algebras.
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
New abelian quotient found in symplectic derivation Lie algebra.
problem Understanding the structure of symplectic derivation Lie algebra.
method Computational approach to abelianization of the weight 12 part.
result 1-dimensional weight 12 part for g≥8. 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.
In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double exten…
Identifies groupoid analogue of symplectic Lie groups.
problem Understanding symplectic structures on Lie groupoids.
method Introduces t-symplectic Lie groupoids and explores their properties. result Establishes a correspondence between Lie algebroid structures and t-symplectic Lie groupoid structures. Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
The paper finds symplectic compactifications of coadjoint orbits.
problem Understanding symplectic structures on coadjoint orbits.
method Defined real analytic symplectomorphisms on subsets of coadjoint orbits.
result Coadjoint orbits of compact Lie algebras are symplectic compactifications of domains of cotangent bundles.
We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.
A special symplectic Lie group is a triple (G,ω,∇) such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ∇. In this paper starting from a special symplectic Lie group we show how to ``defo…
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…
Study on a specific type of Lie algebras with Kähler and contact properties.
problem Characterizing and classifying transversely Kähler almost contact metric Lie algebras.
method Analyzing properties of Lie algebras with contact forms and Kähler structures, considering center dimensions and quotient properties.
result Classification of 5-dimensional η-Einstein transversely Kähler almost contact metric Lie algebras.
Three new types of graded Lie groups are constructed and analyzed.
problem Generalizing Lie theory to Z-graded geometry. method Direct geometric construction and functor-of-points perspective.
result Isomorphic Lie algebras of the new graded Lie groups.
Study left-invariant metrics on Lie groups, finding Einstein metrics on nilpotent Lie groups of low dimensions.
problem Characterize left-invariant Einstein metrics on Lie groups.
method Introduce a natural GL(n,R) action, use moment map, gauge transformations, and Lie algebra derivations.
result First example of a left-invariant Einstein metric with non-zero scalar curvature on a nilpotent Lie group.
This paper extends Riemannian geometry concepts to Hom-ρ-commutative algebras.
problem Extending Riemannian geometry concepts to Hom-ρ-commutative algebras. method Recalling Hom-ρ-commutative algebras, developing metric, connection, torsion, curvature, and differential operators. result Established differential calculus and symplectic/Poisson structures on Hom-ρ-commutative algebras. Study LCS structures of the second kind on Lie algebras.
problem Characterize and construct LCS Lie algebras.
method Construct new examples using representations and characterize existing ones.
result Characterize all LCS Lie algebras obtained with the construction.
We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the correspon…
In this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G* are complex Lie groups. We also prove that a He…
The paper explores structures on 3-Lie algebras, including product, complex, and symplectic.
problem Exploring structures on 3-Lie algebras.
method Introducing phase spaces, product structures, complex structures, and compatibility conditions.
result Four types of special integrability conditions for product, complex, and symplectic structures on 3-Lie algebras.
The paper defines pre-symplectic algebroids and their applications.
problem Understanding the geometric structure of symplectic Lie algebroids.
method Introducing pre-symplectic algebroids and establishing their correspondence with symplectic Lie algebroids.
result Pre-symplectic algebroids are geometric structures underlying symplectic Lie algebroids.
Deform symplectic structures using moment maps and Lie algebra elements.
problem Dealing with symplectic structure deformations on manifolds.
method Using quasi-Poisson theory and Lie algebra elements to deform symplectic structures.
result Concrete examples of symplectic structure deformations on complex projective and Grassmannian spaces.
On a Poisson manifold endowed with a Riemannian metric we will construct a vector field that generalizes the double bracket vector field defined on semi-simple Lie algebras. On a regular symplectic leaf we will construct a generalization of the normal metric such that the above vector field restricted to the symplectic…
Generalizes kinematical Lie algebras for isotropic spacetimes.
problem Classifying relativity algebras in isotropic spacetimes.
method Elementary proof and Lie group analysis.
result Symplectic involutive Lie algebras always exist.
Study of Type IIA flow on symplectic Lie algebras for geometric structures.
problem Detecting geometric structures like Lagrangian torus fibrations and harmonic almost complex structures.
method Investigate F-harmonic forms and the long-time behavior of the Type IIA flow. result The Type IIA flow helps in detecting desired geometric structures.
Study LCS structures on Lie algebras of type I, proving trivial Morse-Novikov cohomology and constructing solvmanifolds.
problem Locally conformal symplectic structures on Lie algebras of type I.
method Analyzing Lie algebras of type I, proving trivial Morse-Novikov cohomology, and constructing solvmanifolds.
result LCS structures on Lie algebras of type I are of the first kind and can be used to construct compact solvmanifolds.
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.
Study second-Chern-Einstein metrics on 4D manifolds, finding Killing vector fields and examples.
problem Investigate second-Chern-Einstein metrics on 4D almost-Hermitian manifolds.
method Analyze compact and unimodular almost-abelian Lie algebras, use Killing vector fields and parallel non-zero Lee forms.
result Describe 4D compact second-Chern-Einstein locally conformally symplectic manifolds and classify unimodular almost-abelian Lie algebras with second-Chern-Einstein metrics.
We study Lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure. We prove that each of those Lie algebras is completely determined by a pair (U,H) where U is a complex commutative associative algebra and H is a sesquilinear hermitian form on U which ve…
In this work we study the problem of existence of symplectic structures on free nilpotent Lie algebras. Necessary and sufficient conditions are given for even dimensional ones. The one dimensional central extension for odd dimensional free nilpotent Lie algebras is also considered.
We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure tha…
The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.
problem Classifying and constructing complex symplectic structures on 4D Lie algebras.
method Interpreting complex symplectic and pseudo-Kähler structures, developing a method for constructing hypersymplectic structures.
result Obtained an example of a hypersymplectic structure on a 4-step nilmanifold.