Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
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.
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
Lie groups with bi-invariant distance are products of abelian and compact groups.
problem Characterizing Lie groups with bi-invariant distances.
method Analyzing the structure of Lie groups and introducing a Finsler norm.
result The sectional curvature of bi-invariant distances is non-negative and vanishes only for abelian subalgebras.
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
The paper studies a flow on complex Lie groups, showing convergence to solitons.
problem The study of curvature flows on complex Lie groups.
method Positive Hermitian curvature flow on left-invariant metrics.
result The flow converges to solitons in both nilpotent and almost-abelian cases.
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
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.
Characterizes hypercomplex Lie groups and their solvmanifolds.
problem Understanding hypercomplex structures on Lie groups and their solvmanifolds.
method Characterization of almost abelian Lie groups with hypercomplex structures, analysis of Obata and Bismut connections, classification of hypercomplex Lie groups, and construction of solvmanifolds.
result Classification of hypercomplex almost abelian Lie groups in dimension 8 and properties of their solvmanifolds.
Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.
problem Classify balanced Hermitian structures on almost abelian Lie algebras.
method Classify six-dimensional almost abelian Lie algebras with balanced structures, investigate flow of balanced metrics and anomaly flow.
result Prove conjecture for compact almost abelian solvmanifolds with left-invariant complex structures.
Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
problem Characterizing simply connected indefinite homogeneous spaces of finite volume.
method Analyzing Lie algebras with abelian solvable radical and symmetric bilinear form.
result Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. 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…
Study Weyl-Einstein structures on conformal solvmanifolds, proving Einstein property and classifying metrics.
problem Characterize Weyl-Einstein structures on conformal solvmanifolds.
method Analyzing left-invariant metrics and using conformal Lie group structures.
result Every conformal solvmanifold with Weyl-Einstein structure is Einstein.
Let K be a compact Lie group. We compute the abelianization of the Lie algebra of equivariant vector fields on a smooth K-manifold X. We also compute the abelianization of the Lie algebra of strata preserving smooth vector fields on the quotient X/K.
The note confirms a conjecture for specific Lie groups.
problem The conjecture about constant holomorphic sectional curvature in non-Kähler geometry.
method Compact quotients of Lie groups with specific properties.
result The conjecture is confirmed for almost abelian Lie algebras and those with certain abelian ideals.
Study lattices in specific Lie groups for geometric structures.
problem Existence of lattices in Lie groups with certain geometric structures.
method Analyzing left invariant locally conformal Kähler or symplectic structures.
result Existence of lattices only in dimension 4 for Kähler structures, and in any even dimension for symplectic structures.
Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.
problem Existence and classification of LCSKT structures on Lie groups and their quotients.
method Introducing LCSKT structures and studying their properties on Lie groups and their quotients.
result Existence of non-trivial LCSKT structures on 6-dimensional nilpotent Lie algebras and almost abelian Lie algebras.
Uniform doubling property proven for specific Lie groups.
problem Proving uniform doubling property for Lie groups.
method Analyzing quotient groups of SU(2) x R^n.
result Uniform doubling property holds for SU(2) x R^n and its quotients.
Classifies geodesic orbit spaces with abelian isotropy subgroups.
problem Characterizing and classifying geodesic orbit spaces with specific isotropy subgroups.
method Simplified study of geodesic orbit metrics on G/S by reducing to submanifolds and generalized flag manifolds, using properties of root systems.
result Geodesic orbit spaces of the form (G/S,g) are naturally reductive.
Global group laws connect equivariant bordism rings to formal group laws.
problem Establishing connections between equivariant bordism rings and formal group laws.
method Global homotopy theory framework; proving isomorphisms and universal properties.
result Equivariant bordism rings are isomorphic to Lazard rings for abelian Lie groups.
Constructs Lie groups with negative Ricci curvature.
problem Finding Lie groups with negative Ricci curvature.
method Constructs Lie groups with specific algebraic structures and representations.
result Proves existence of Lie groups with negative Ricci curvature for various Levi factors.
Algebraic dimension is zero for generic complex structures on hypercomplex nilmanifolds.
problem Understanding the algebraic dimension of complex subvarieties in hypercomplex nilmanifolds.
method Analyzing the algebraic dimension of complex subvarieties of hypercomplex nilmanifolds using properties of hypercomplex structures and Lie algebras.
result For generic complex structures, the algebraic dimension of complex subvarieties in hypercomplex nilmanifolds is zero.
It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …
The study calculates Poincaré series for character varieties of nilpotent groups.
problem Determining the cohomology of character varieties for nilpotent groups.
method Explicit formulas for Poincaré series and equivariant stable decompositions of subspaces.
result Explicit formulas for Poincaré series of character varieties for nilpotent groups.
Study on CKY forms on almost abelian Lie groups, proving parallelism and characterizing non-parallel cases.
problem Characterizing CKY forms on almost abelian Lie groups and proving parallelism.
method Analyzing almost abelian metric Lie algebras, proving parallelism for CKY forms, and classifying cases up to dimension 5.
result CKY forms are parallel on almost abelian Lie algebras, with exceptions for p=1 and p=n−1. In this paper we analyse the topological group cohomology of finite-dimensional Lie groups. We introduce a technique for computing it (as abelian groups) for torus coefficients by the naturally associated long exact sequence. The upshot in there is that certain morphisms in this long exact coefficient sequence can be a…
We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting gro…
We construct lattices on six dimensional not completely solvable almost abelian Lie groups, for which the Mostow condition does not hold. For the corresponding compact quotients, we compute the de Rham cohomology (which does not agree in general with the Lie algebra one) and a minimal model. We show that some of these …
The paper explores p-Kähler structures on Lie group quotients.
problem Existence of p-Kähler structures on compact quotients of Lie groups. method Analyzes p-Kähler structures on nilmanifolds and almost abelian solvmanifolds. result Proves that (n−2)-Kähler almost abelian solvmanifolds of complex dimension n≥3 are Kähler. 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.
Study group extensions and bundles on manifolds.
problem Understanding group extensions and their relation to bundles on manifolds.
method Establish and study a correspondence between extensions and equivariant bundles using non-abelian cohomology.
result A correspondence between G^-bundles and twisted Γ-equivariant bundles on Galois Γ-coverings. Study convexity properties of gradient map on probability measures.
problem Properties of gradient map on probability measures on submanifolds.
method Analysis of Kähler manifolds and Lie group actions.
result Convexity results for gradient map in Abelian case, extension to non-Abelian case.
We study the cohomology properties of the singular foliation $\F$ determined by an action Φ:G×M→M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $\lau{\IH}{*}{\per{p}}{\mf}$ is finite dimensiona…
The paper constructs a new Ricci-flat metric on almost abelian Lie groups.
problem Finding Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups.
method Constructing left-invariant metrics on almost abelian Lie groups, focusing on dimensions four or higher.
result A new Ricci-flat metric that generalizes the Petrov solution to higher dimensions.
Complete left-invariant metrics on Lie groups with specific properties.
problem Completeness of left-invariant semi-Riemannian metrics on Lie groups.
method Introducing bi-Lipschitz Riemannian Clairaut metrics and proving completeness conditions.
result All left-invariant metrics are complete for certain Lie groups.
We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct p…
The paper provides conditions for factoring equivariant spectral triples in unbounded KK-theory.
problem Factoring equivariant spectral triples in unbounded KK-theory.
method Sufficient conditions for factorization of equivariant spectral triples as a Kasparov product.
result Equivariant Dirac-type spectral triples on torus principal bundles always factorize.
It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.
A recent preprint of Csikós, Pyber and Szabó (arXiv:1411.7524) proves that the diffeomorphism group of T2×S2 is not Jordan. The purpose of this paper is to generalize the arguments of Csikós, Pyber and Szabó in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jor…
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
Study on spectral sequence for abelian Lie group actions, with bounds and applications.
problem Understanding the spectral sequence for abelian Lie group actions.
method Provided upper bounds and examples to show these bounds are sharp.
result Sharp bounds on the degeneration page of spectral sequence.
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.
We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that an…
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
New framework for equivariant neural networks using Lie group decompositions.
problem Limitations of existing equivariant neural network methods for Lie groups.
method Lie group structure and geometry, decomposition into subgroups and submanifolds.
result Equivariant neural networks for affine transformations outperform previous methods.
This paper resolves equivariant K-theory for abelian actions.
problem Understanding equivariant K-theory for abelian group actions.
method Using iterated spaces and twisted deRham forms, the paper describes equivariant K-theory in terms of bundles over the base.
result A direct proof of the equivariant Atiyah-Hirzebruch isomorphism is provided.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.