Classifies complex structures on specific nilpotent Lie algebras.
problem Classifying complex structures on nilpotent Lie algebras with minimal center.
method Classification through algebraic and geometric methods.
result Space of complex structures on specific Lie algebras up to isomorphism.
New findings on complex structures in nilpotent Lie algebras and pseudo-Kähler geometry.
problem Classifying complex structures on nilpotent Lie algebras.
method Analysis of Lie algebras in dimensions eight and higher, identifying structures with pseudo-Kähler metrics.
result Identification of pseudo-Kähler nilmanifolds with invariant complex structures, providing counterexamples and topological restrictions.
We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex pro…
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined H-nilpotent structures for Lie subgroups of SO(2n,2n) related to neutral hyperKähler structures. result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an H-nilpotent structure. Completes classification of G2-structures on specific nilpotent Lie groups.
problem Classifying seven-dimensional nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing nilpotent Lie groups of various steps and dimensions.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
This paper completes the classification of certain nilpotent Lie groups with specific geometric structures.
problem Classifying nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing seven-dimensional nilpotent Lie groups of various steps.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups with these structures.
The paper characterizes and examines nilpotent complex structures on stratified Lie algebras.
problem Characterizing and understanding nilpotent complex structures on stratified Lie algebras.
method Introduced a new descending series pj to prove a new characterization of nilpotent complex structures and examined whether these structures preserve the strata. result Found that there exists a J-invariant stratification on a step 2 nilpotent Lie algebra with a complex structure. The study characterizes G₂-structures on 2-step nilpotent Lie groups.
problem Characterizing G₂-structures on 2-step nilpotent Lie groups.
method Analyzing left-invariant purely coclosed G₂-structures on 7-dimensional 2-step nilpotent Lie groups.
result Criteria for Riemannian metrics induced by these structures and determination of isomorphism classes.
Classifies nilpotent Lie groups with specific G2-structures.
problem Identifying 7D nilpotent Lie groups with purely coclosed G2-structures. method Examined all 7D nilpotent Lie algebras, classified them, and verified the existence or non-existence of G2-structures. result Provided a complete classification of 7D nilpotent Lie groups with purely coclosed G2-structures. Study para-complex structures on specific Lie groups, finding explicit forms and properties.
problem Characterizing para-Kähler structures on six-dimensional nilpotent Lie groups.
method Examined left-invariant para-complex structures on six-dimensional nilpotent Lie groups, obtained explicit expressions and investigated curvature properties.
result Para-complex structures are nilpotent and para-Kähler metrics are Ricci-flat.
We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional n…
The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
Proves a complex structure conjecture for a specific type of Lie groups.
problem Proving a conjecture about left-invariant complex structures on nilpotent Lie groups.
method Analyzes simply connected, nilpotent Lie groups of dimension 2n.
result Proves biholomorphism to C^n for the specified Lie groups.
The study examines nilpotent similarity structures on manifolds and their properties.
problem Characterizing closed manifolds with nilpotent similarity structures.
method Generalizes convexity arguments to geodesic segments in nilpotent Lie groups.
result Closed manifolds with nilpotent similarity structures are either complete or radiant.
We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on R8 which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilp…
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.
Study on existence of p-Kähler structures on nilmanifolds with nilpotent complex structures.
problem Existence of p-Kähler structures on nilmanifolds with nilpotent complex structures. method Determine optimal p for existence of p-Kähler structures and analyze the relationship between balanced metrics and degeneracy steps of the Frölicher spectral sequence. result No p-Kähler structures exist for an optimal p on nilmanifolds with nilpotent complex structures. The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. We prove that if two metrics on a n-manifold of bounded sectional curvature are L0-bi-Lipchitz equivalent and sufficient collapsed (depending on L0 and n), then up to a diffeo…
Characterizes complex structures on specific Lie groups.
problem Identifying Lie groups with left-invariant complex structures.
method Analyzing Lie algebras and their corresponding Lie groups, considering different nilpotency levels.
result Conditions for the existence of left-invariant complex structures and pluriclosed metrics on 2-step nilpotent Lie groups.
We present the classification of real nilpotent quasi-filiform Lie algebras endowed with a complex structure. A nilpotent Lie algebra g is called quasi-filiform is the nilindex is equal to dim(n)-2. We recall that the filiform case (nilindex =dim(g)-1) has already been studied.
We classify real 6-dimensional nilpotent Lie algebras for which the corresponding Lie group has a left-invariant complex structure, and estimate the dimensions of moduli spaces of such structures.
It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compat…
We construct families of functions in involution for transverse Poisson structures at nilpotent elements of Lie-Poisson structures on simple Lie algebras by using the argument shift method. Examples show that these families contain completely integrable systems that consist of polynomial functions. We provide a uniform…
In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.
The paper explores maximal nilpotent complex structures on Lie algebras.
problem Understanding maximal nilpotent complex structures on Lie algebras.
method Analyzing properties of nilpotent Lie algebras and their complex structures.
result For a nilpotent Lie algebra with step 2, there are only 2-3 maximal complex structures.
Study semi-Kähler structures on specific Lie groups without symplectic structures.
problem Exploring structures on Lie groups without symplectic structures.
method Defined semi-Kähler and almost para-semi-Kähler structures on specific Lie groups.
result Geometric properties of these structures are studied.
Finite 2-step nilpotent groups studied by Kim and Manturov.
problem Understanding the structure of groups defined by Kim and Manturov.
method Analyzing the groups Γn4 for all n≥6. result The groups are finite and 2-step nilpotent.
Left-invariant metrics force 2-step nilpotent groups, preserving Kähler-like conditions.
problem Existence of left-invariant pluriclosed Hermitian metrics on Lie groups.
method Analyzing left-invariant metrics on unimodular Lie groups with abelian complex structures.
result Pluriclosed flow preserves Strominger Kähler-like conditions on 2-step nilpotent Lie groups.
This paper shows how post-Lie algebra structures can be induced by simply transitive NIL-affine actions.
problem Understanding which solvable Lie groups can act simply transitively on nilpotent Lie groups.
method Introducing post-Lie algebra structures and showing their correspondence with simply transitive actions.
result Simply transitive NIL-affine actions induce complete post-Lie algebra structures in the 2-step nilpotent case.
Study of complex structures on specific solvmanifolds, proving existence and non-existence results.
problem Classification and properties of complex structures on six-dimensional solvmanifolds.
method Classification of Lie algebras, analysis of complex structures, study of Hermitian metrics.
result Determination of new balanced solvmanifolds and confirmation of conjectures.
CR embeddings in complex spaces for specific Lie groups.
problem Embedding specific Lie groups in complex spaces.
method Using integrable complex structures on subbundles of tangent bundles.
result CR embeddings possible as the edge of wedges in complex domains.
Higher nilpotent analogues of the A−∞-structure are explicitly defined on arbitrary simplicial complexes, generalizing explicit construction of /hep-th/0704.2609. These structures are associated with the higher nilpotent differential dn, satisfying dnn=0, which is naturally defined on triangulated manifo…
We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization,…
We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theo…
Study G2-instantons on specific Lie groups, finding conditions and structures.
problem Characterize G2-instantons on 2-step nilpotent Lie groups.
method Analyze connections arising from characteristic connections, use Lie group structure and torsion.
result Establish necessary and sufficient conditions for G2-instantons, define naturally reductive structures.
Abstract classifies Lie algebras with complex or symplectic structures.
problem Classifying Lie algebras with specific structures.
method Analyzing Jordan normal form and restrictions on matrix A. result Classification reduces to nilpotent case, with specific structure implications.
Unique complex structures on specific Lie algebras.
problem Existence and uniqueness of complex structures on nilpotent Lie algebras.
method Analysis of complex structures on nilpotent almost abelian Lie algebras.
result Full control over cohomology and deformations of almost abelian complex nilmanifolds.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. In this paper we study contact structure on 2-step nilpotent, Heisenberg type Lie groups. We decompose this Lie groups to center and orthogonal complement, then investigate properties of both orthogonal Lie subgroups. Finally, we provide a connection between matchings in groups and field extensions and 2-step nilpotent…
We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra g under the presence of a complex structure J. In particular, we find a bound for the dimension of the center of g when it does not contain any non-trivial J-invariant ideal. Thanks to thes…
The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.
problem Investigating invariant SKT, balanced, and generalized Kähler structures on compact quotients of almost nilpotent Lie groups.
method Characterization and classification of Hermitian almost nilpotent Lie algebras, study of structures under flows, and non-existence results.
result Construction of new compact SKT manifolds and examples of non-split generalized Kähler structures.
Constructs bihamiltonian structures from Lie algebras for specific types of nilpotent elements.
problem Constructing bihamiltonian structures from Lie algebras.
method Using classical W-algebras and non-regular nilpotent elements of semisimple type. result Leading terms define Dubrovin-Frobenius manifolds and calculate central invariants.
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.
We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent…
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…
Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.
problem Rigidity of invariant Ricci flow blowdown limits on nilpotent bundles with zero curvature.
method Construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature.
result Proves blowdown limit is locally an expanding Ricci soliton for three-dimensional Heisenberg group structure group.