Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
Algebraic treatment of connection reduction over a special disc.
problem Reduction theory for connections over a specific geometric structure.
method Purely algebraic approach for arbitrary groups, with quantitative results.
result New quantitative results in reduction theory.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Defines and classifies algebraic Schouten solitons in 3D Lorentzian Lie groups.
problem Classifying solitons in 3D Lorentzian Lie groups.
method Defined algebraic Schouten solitons and classified them for specific connections.
result Classified algebraic Schouten solitons for various connections on 3D Lorentzian Lie groups.
In this paper, we compute canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. We classify algebraic Ricci solitons assoc…
A new algebraic structure emerges from reductive homogeneous spaces.
problem Understanding the algebraic properties of tangent bundles.
method Defined a new algebraic structure based on connections and torsion.
result Post-Lie-Yamaguti algebra is a new algebraic structure related to Lie-Yamaguti algebras.
The paper studies curvatures in metric Jordan algebras, proving unique connections and curvature formulas.
problem Investigating curvatures in metric Jordan algebras.
method Defined the Jordan-Levi-Civita connection, introduced curvature tensors, and proved curvature formulas.
result Every formally real Jordan algebra admits a metric of non-positive Jordan curvature and a Jordan-Einstein metric of negative Jordan scalar curvature.
In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions us…
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
We prove that any connected proper Dupin hypersurface in Rn is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in Rn that satisfies a certain finiteness condition. Hence any taut submanifo…
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf G-algebras and coloring Kirby diagrams. result Invariants defined for 4-manifolds and connections.
In this paper, we compute the Wanas tensor associated to canonical connections on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Wanas solitons associated to canonical connections. We classify algebraic Wanas solitons associated to canonical connections on three-dimensional Lor…
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the …
Holonomy of Weyl connections in Lorentzian space classified.
problem Classifying holonomy algebras of Weyl connections in Lorentzian signature.
method Classification through construction of examples of Weyl connections with all possible holonomy algebras.
result Examples of Weyl connections with all possible holonomy algebras constructed.
On a manifold with a projective connection we canonically assign a second order differential operator acting on the algebra of all densities to any tensor density Sij of fixed weight λ. In particular, this implies that on any projectively connected manifold, a `bracket' (symmetric biderivation) on the algebra of…
The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.
problem Understanding universal enveloping algebras of Lie-Rinehart algebras.
method Extending a theorem to left Hopf algebroids and applying it to universal enveloping algebras of Lie-Rinehart algebras.
result Provides a crossed product decomposition of universal enveloping algebras for curved and flat connections.
Study of multidifferential operators and Dorfman connections on Courant algebroids.
problem Exploring multidifferential operators and Dorfman connections on Courant algebroids.
method Construction of an algebra and complex of multidifferential operators, study of Dorfman connections.
result Cartan calculus, curvatures of induced connections and basic differential geometric identities make sense in the constructed algebra.
Paper constructs connections on curves with specific Galois groups.
problem Constructing connections with prescribed differential Galois groups.
method Restricting to trivial vector bundles and using Lie algebra from regular forms.
result Differential Galois group is a closure of the Lie algebra.
Proves a stack of G-bundles with logarithmic connections is finite type.
problem Moduli of G-bundles with logarithmic connections over curves.
method Algebraic stack analysis and finite type proof.
result Proves the moduli stack is of finite type.
New algebraic structures for topological pairs.
problem No specific problem stated; generalization of knot quandles.
method Introducing multi-quandles for topological pairs.
result New algebraic structures for topological pairs.
The paper studies connections and Finsler geometry on JB-algebra structure groups.
problem Investigating geometric structures on JB-algebra structure groups.
method Endowing the structure group with a connection and Finsler metric, computing quantities, and proving minimality of paths.
result Established the Finsler metric and distance on the cone of a JB-algebra.
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
This work explores algebraic structures from curvature and torsion in affine connections.
problem Understanding algebraic structures from curvature and torsion in affine connections.
method Post-Lie algebra, D-algebra, and special polynomials.
result A particular class of geometrically special polynomials is generated by torsion and curvature.
Study on flat connections with controlled irregularity.
problem Boundedness of algebraic flat connections with limited irregularity.
method Analysis of families of algebraic flat connections and holonomic D-modules.
result Established boundedness of families of algebraic flat connections with controlled irregularity.
In this article, we present an integration of any real finite-dimensional Leibniz algebra as a Lie rack which reduces in the particular case of a Lie algebra to the ordinary connected simply connected Lie group. The construction is not functorial.
Introduces symplectic flatness for connections over symplectic manifolds.
problem Flatness conditions for connections over symplectic manifolds.
method Introduces symplectic flatness condition and twisting of differential complexes.
result Symplectic flat connections represent a subclass of Yang-Mills connections.
We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.
New algebraic structure for vector bundles with special properties.
problem Developing new algebraic structures for vector bundles.
method Introducing para-associative algebroids and showing local triviality conditions.
result Existence of a differential connection is necessary and sufficient for local triviality.
New algebraic presentations for 3D cobordisms and 4D 2-handlebodies.
problem Finite algebraic presentations for cobordism and handlebody categories.
method Direct proof using a new functor construction.
result Algebraic presentations of 3Cob and 4HB categories.
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.
Research explores Lie algebras in Riemannian manifolds.
problem Understanding Lie algebras in Riemannian manifolds.
method Analyzes the Lie algebra of infinitesimal isometries.
result Identifies two commutative ideals in the Lie algebra.
The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
Locally conformally product Lie algebras are characterized and constructed.
problem Characterizing and constructing compact locally conformally product Lie algebras.
method Characterization via closed 1-forms and non-unimodular Lie algebras acting on abelian ones.
result Explicit examples of compact LCP manifolds that are not solvmanifolds.
Study special affine connections on symmetric spaces and their products.
problem Characterize special affine connections on symmetric spaces.
method Analyze canonical affine connections, introduce special products, and study holonomy Lie algebras.
result Established a correspondence between special affine connections and special products on the Lie algebra.
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.
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.
Study finds abnormal paths on specific Lie groups using algebraic structures.
problem Identifying abnormal extremals on Lie groups with quasimetrics.
method Analyzing Lie algebras and seminorms to determine abnormal extremals.
result Established criterion for strong abnormality of extremals.
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
Study invariant Poisson structures on homogeneous manifolds, algebraically and geometrically.
problem Characterize and understand invariant Poisson structures on homogeneous manifolds.
method Algebraic characterization and bijective correspondence with Lie subalgebras, symplectic foliation, and invariant contravariant connections.
result Established a connection between invariant Poisson tensors and Lie subalgebras with a 2-cocycle.
We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.
Let G be a connected Lie group and g its Lie algebra. We denote by ∇0 the torsion free bi-invariant linear connection on G given by ∇X0Y=21[X,Y], for any left invariant vector fields X,Y. A Poisson structure on g is a commutative and associative product on $\mathfra…
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreduci…
New algebra defined for Legendrian submanifolds, preserving key invariants.
problem Defining a new algebra to preserve invariants of Legendrian submanifolds.
method Combining string topology techniques with combinatorial methods to count holomorphic disks.
result The new algebra PDA is a filtered, differential graded algebra that captures invariants of Legendrian submanifolds. Study of Tannakian categories for integrable connections on Kaehler manifolds.
problem Understanding Tannakian categories for integrable connections on Kaehler manifolds.
method Analyzing pairs (E, D) where E is a trivial holomorphic vector bundle and D is an integrable holomorphic connection.
result The pro-algebraic affine group scheme uniquely determines the isomorphism class of compact Riemann surfaces.
Abstract: Review and definitions of generalised spin structures, their connections, and symmetry algebra.
problem Understanding and characterizing generalised spin structures and their properties.
method Definitions, basic notions, connections, covariant Lie derivative, covariant Cartan calculus, symmetry algebra.
result Characterization of homogeneous generalised spin structures.
Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.
problem Classifying Lorentzian symmetric spaces with Einstein-Yang-Mills properties.
method Classification based on invariant metric connections and diagonal metrics.
result Four-dimensional symmetric spaces with nontrivial isotropy groups are classified.