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. Local classification of 4D Ricci solitons with specific algebra properties.
problem Classifying 4D Ricci solitons with a 2D Abelian Killing algebra.
method Local classification under specific curvature and symmetry conditions.
result Classification of Ricci solitons with orthogonally intransitive 2D Abelian Killing algebra.
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.
The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold (M,g) and the structure of its full isometry group. The Lie algebra of the full isometry group of (M,g) is identified with the Lie algebra of Killing fields g on (M,g). We…
We consider the Lie algebra consisting of all derivations on the free associative algebra, generated by the first homology group of a closed oriented surface, which kill the symplectic class. We find the first non-trivial abelianization of this Lie algebra and discuss its relation to unstable cohomology classes of the …
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.
Characterizes symmetric Killing tensors on specific Lie groups.
problem Understanding Killing tensors on specific Lie groups.
method Completely characterized left-invariant symmetric Killing tensors on almost abelian Lie groups.
result All such tensors are decomposable into polynomial expressions of Killing vector fields and metric.
The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.
problem Constructing Killing spinors on pseudo-Riemannian solvmanifolds.
method Using nilsolitons and pseudo-Iwasawa condition, the paper constructs families of pseudo-Iwasawa solvmanifolds with Killing spinors.
result All pseudo-Iwasawa solvmanifolds admitting a Killing spinor belong to a specific family.
We provide a generalization of the Lie algebra of conformal Killing vector fields to conformal Killing-Yano forms. A new Lie bracket for conformal Killing-Yano forms that corresponds to slightly modified Schouten-Nijenhuis bracket of differential forms is proposed. We show that conformal Killing-Yano forms satisfy a gr…
Researchers found indecomposable Killing tensor fields on quaternionic projective spaces.
problem Characterizing Killing tensor fields on projective spaces.
method Analyzing Killing tensor fields on quaternionic and complex projective spaces, proving algebraic properties.
result Generated algebras of Killing tensor fields on quaternionic and complex projective spaces.
We present an algebraic procedure that finds the Lie algebra of the local Killing fields of a smooth metric. In particular, we determine the number of independent local Killing fields about a given point on the manifold. Spaces of constant curvature and locally symmetric spaces are also discussed. Furthermore, we obtai…
We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the 2-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means…
We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integra…
Defines Killing (super)algebras for spin manifolds, including gauge transformations.
problem Understanding deformations of spin structures on manifolds.
method Introduces a new algebraic structure, studies its deformations using Spencer cohomology.
result Identifies subclasses of deformations and reconstructs supersymmetric backgrounds.
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.
Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras
problem Killing vectors and Bochner-type theorems in pseudo-Riemannian Hom-Lie algebras
method Hom-versions of Bochner theorems
result Space of Killing vectors forms a totally geodesic Hom-Lie subalgebra
Study on the existence of Dirac complements for Dirac structures.
problem Existence of Dirac complements for Dirac structures.
method Cohomological techniques and Lie-theoretical methods.
result Non-existence of Dirac complements for certain Lie algebras.
The paper studies deformations of Kundt metrics using nil-Killing vector fields.
problem Deformations of Kundt metrics in the direction of type III tensors.
method Characterizations within the Kundt class using nil-Killing vector fields.
result Theorem classifying algebraic stability of tensors and sufficient criteria for preserving spi's.
We prove a formula that relates the Euler-Poincaré characteristic of a closed semi-algebraic set to its Lipschitz-Killing curvatures
Extends Killing superalgebras to higher dimensions and signatures.
problem Generalizing Killing superalgebras to higher dimensions and signatures.
method Definition of Killing superalgebras for connections on spinor bundles, sufficient conditions for existence, abstract study using Spencer cohomology.
result Existence of Killing superalgebras as filtered deformations of graded subalgebras of the Poincaré superalgebra.
Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additiona…
Study transcendence of abelian differential periods from bi-algebraic perspective.
problem Arithmetic and functional transcendence of periods of abelian differentials.
method Bi-algebraic structure on strata of abelian differentials.
result Characterization of arithmetic points and proof of linear bi-algebraic curves.
In this paper, we consider a connected Riemannian manifold M where a connected Lie group G acts effectively and isometrically. Assume X∈g=Lie(G) defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition X=Xr+Xs according to a Levi decompositio…
Study of abelian structures on odd-dimensional Lie algebras and their geometric properties.
problem Characterizing abelian structures on odd-dimensional Lie algebras.
method Introducing and analyzing abelian almost contact and almost 3-contact structures, and their compatibility conditions.
result Classification of 5-dimensional Sasakian Lie algebras and 7-dimensional abelian almost 3-contact Lie algebras.
The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and releva…
Paper confirms conjecture for specific Lie algebras.
problem Fino-Vezzoni conjecture on Lie algebras with abelian ideals of codimension two.
method Analyzes unimodular Lie algebras with abelian ideals of codimension two.
result Confirms the Fino-Vezzoni conjecture for this specific class of Lie algebras.
Study abelian factors in Lie algebras from graph edge labels.
problem Understanding abelian factors in Lie algebras from graph edge labels.
method Analyzing 2-step nilpotent Lie algebras constructed from graphs, computing abelian factors, and studying singularity properties.
result Explicit computation of abelian factors for various graph families.
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 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.
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.
This research bridges Killing vectors and Lie algebras through induced vector fields.
problem Understanding the relationship between Killing vector fields and Lie algebras.
method Defining and exploring induced vector fields to connect Killing vector fields with isometry Lie groups.
result Established a new connection between Killing vector fields and Lie algebras.
Study on dimensions of Killing vector fields on gradient Ricci solitons.
problem Estimating dimensions of Killing vector fields on gradient Ricci solitons.
method Analyzes the structure of gradient Ricci solitons to estimate dimensions of Killing vector fields.
result Maximal dimension of Killing vector fields on irreducible non-trivial gradient Ricci solitons.
Paper proves injectivity of non-abelian X-ray transform on certain spaces.
problem Injectivity of non-abelian X-ray transform on asymptotically hyperbolic spaces.
method Gauge equivalence for unitary connections and skew-Hermitian Higgs fields.
result Injectivity result for non-abelian X-ray transform over skew-Hermitian Higgs fields.
Characterizes conformal Killing tensors and their Killing scales.
problem Characterizing conformal Killing tensors and their Killing scales.
method Differential prolongation using conformally invariant tractor calculus.
result Provides an invariant characterisation of Einstein Killing scales.
New framework for generalized Killing spinors on manifolds.
problem Formulating generalized Killing spinor equations as polyform systems.
method Developing a new framework using polyforms and algebraic relations in the Kähler-Atiyah bundle.
result Characterization of real spinor squaring map as a real algebraic variety.
Study on Hermitian metrics on Lie algebras with specific ideals.
problem Classifying Hermitian metrics on Lie algebras with abelian ideals.
method Examined unimodular Lie algebras with abelian ideals of codimension two, classified metrics.
result Classification of Bismut Kähler-like and Bismut torsion-parallel metrics.
We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras aff(A), where A is a commutative algebra. These affine Lie algebras are natural generalizations of aff(C) and the corresponding Lie grou…
Study of presymplectic forms on almost abelian Lie algebras, determining moduli spaces and their finiteness.
problem Determining conditions for the existence of presymplectic forms on almost abelian Lie algebras.
method Analyzing the moduli space of presymplectic forms and using matrix congruence to find canonical representatives.
result The moduli space of presymplectic forms on almost abelian Lie algebras is finite and all forms are permutations of a canonical 2-form.
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.
This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.
problem Characterizing LCSKT structures on almost abelian Lie algebras.
method Analyzing the LCSKT condition and its compatibility with other Hermitian structures.
result Classification of LCSKT almost abelian Lie algebras in dimension 6.
Characterizes almost abelian Lie algebras with integrable complex structure
problem Classifying almost abelian Lie algebras
method Using presentations consisting of a real number, an element in a vector space, and an endomorphism
result Classifies p-Kähler, p-pluriclosed, Kähler, balanced, pluriclosed, and Gauduchon metrics Symmetry algebras of Killing vector fields and conformal Killing vectors fields can be extended to Killing-Yano and conformal Killing-Yano superalgebras in constant curvature manifolds. By defining Z-gradations and filtrations of these superalgebras, we show that the second cohomology groups of them are triv…
We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Che…
We construct an abelian quotient of the symplectic derivation Lie algebra hg,1 of the free Lie algebra generated by the fundamental representation of Sp(2g,Q). More specifically, we show that the weight 12 part of the abelianization of hg,1 is 1-dimensional for $g…
The paper studies algebraic relations of first integrals on specific Lie groups.
problem Algebraic relations of first integrals on step-two and step-three nilpotent Lie groups.
method Analysis of isometry algebra and invariant first integrals.
result Complete families of first integrals can be constructed with Killing vector fields and symmetric Killing 2-tensor fields in low dimensions.
Valence two Killing tensors in the Euclidean and Minkowski planes are classified under the action of the group which preserves the type of the corresponding Killing web. The classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebr…
Study conformal Killing forms on specific nilpotent Lie groups.
problem Characterize conformal Killing forms on 2-step nilpotent Lie groups.
method Analyzing left-invariant forms on simply connected groups, proving properties of forms based on center dimension.
result Only specific forms exist under certain conditions.
We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.