The paper defines and explores Hom-Lie algebroids and related structures.
problem Defining and studying Hom-Lie algebroids and related algebraic structures.
method Modifying and extending the definitions of Lie algebroids and introducing new structures like Hom-Poisson manifolds, Hom-Lie bialgebroids, and Hom-Courant algebroids.
result Hom-Courant algebroids are shown to have an underlying algebraic structure of Hom-Leibniz algebras or Hom-Lie 2-algebras.
Paper constructs representations up to homotopy for hom-Lie algebroids.
problem Hom-Lie algebroids are a twisted version of Lie algebroids.
method Uses representations up to homotopy of Lie algebroids to define a similar structure for hom-Lie algebroids.
result Establishes a connection between representations up to homotopy of length 1 and extensions of hom-Lie algebroids.
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
The paper develops structures on Hom-Lie algebroids and Hom-Courant algebroids.
problem Exploring new algebraic structures on Hom-Lie algebroids and Hom-Courant algebroids.
method Introducing Hom-Poisson, Hom-Nijenhuis, and Hom-Poisson-Nijenhuis structures on Hom-Lie algebroids and Hom-Dirac structures on Hom-Courant algebroids.
result Established properties and relationships among these structures, including a hierarchy and correspondence.
We define hom-Lie algebroids, a definition that may seem cumbersome at first, but which is justified, first, by a one-to-one corespondence with hom-Gerstenhaber algebras, a notion that we also introduce, and several examples, including hom-Poisson structures.
Introduces new algebraic structures and their cohomology.
problem Developing algebraic models for hom-Lie algebroids.
method Definition of hom-Lie-Rinehart algebras and their cohomology.
result Characterization of low-dimensional cohomology spaces.
Introduces Hom-Lie groups and their integrability, defining Hexp map and adjoint representation.
problem Integrability of Hom-Lie algebras and associated Hom-Lie groups.
method Definition of Hom-Lie groups and algebras, integration of Hom-Lie algebras, Hexp map definition.
result Every regular Hom-Lie algebra is integrable, Hexp map is universal.
The study examines conditions for completeness and simplicity in hom-Lie superalgebras.
problem Conditions for completeness and simplicity in hom-Lie superalgebras.
method Equivalent conditions and derivations analysis.
result Conditions for completeness and simplicity in hom-Lie superalgebras.
The paper defines Z-graded hom-Lie superalgebras and explores their properties.
problem Understanding the structure and properties of Z-graded hom-Lie superalgebras.
method Definition and exploration of Z-graded hom-Lie superalgebras, invariant bilinear forms, and simplicity conditions.
result Maximal and minimal Z-graded hom-Lie superalgebras for local hom-Lie superalgebras are identified, and conditions for simplicity are checked.
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.
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
The paper defines and characterizes para-Kahler structures on hom-Lie algebras.
problem Defining and characterizing para-Kahler structures on hom-Lie algebras.
method Introducing pseudo-Riemannian, para-Hermitian, and para-Kahler structures on hom-Lie algebras, providing examples, and defining phase spaces.
result Para-Kahler hom-Lie algebras give phase spaces and conversely, can be constructed from phase spaces.
Complex and Kahler structures defined on hom-Lie algebras.
problem Defining structures on hom-Lie algebras.
method Introducing complex and Hermitian structures on hom-Lie algebras, providing examples, and constructing phase spaces.
result No proper complex (Hermitian) hom-Lie algebra of dimension two exists.
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. Simplified definition of LA-Courant algebroids and Poisson Lie 2-algebroids.
problem Defining and characterizing LA-Courant algebroids and Poisson Lie 2-algebroids.
method Using split Lie 2-algebroids and self-dual 2-representations to define LA-Courant algebroids, and studying geometric examples and induced structures.
result New examples of Poisson Lie 2-algebroids and a new construction of Courant algebroids.
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.
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
problem Generalizing Dirac pairs to Jacobi algebroids.
method Introducing Dirac pairs on Jacobi algebroids and showing their relationship to Lie algebroids.
result Dirac pairs on Jacobi algebroids characterize compatible structures.
Paper constructs various algebroids using n-systems and metric n-systems.
problem Creating algebroids from n-systems and metric n-systems.
method Using n-systems and metric n-systems to construct algebroids.
result Explicit computations for all resulting structure maps.
Paper generalizes representations of Lie algebroids to weighted Lie algebroids.
problem Representations of Lie algebroids and their generalizations.
method Introducing and studying weighted Lie algebroids, showing relations to VB-algebroids and generalizing the van Est theorem.
result New natural examples of higher term representations up to homotopy of Lie algebroids uncovered.
Constructing 3-Lie algebroids via connections
problem Constructing Lie algebroids and 3-Lie algebroids method Using connections generated by finite families of differential operators and dual sections
result Providing sufficient conditions for generating families to determine Lie algebroid and 3-Lie algebroid structures Paper examines pre-Courant algebroids and their properties.
problem Defining and working with pre-Courant algebroids.
method Examination of supermanifold description and definition of structures.
result Definition and simplification of weighted pre-Courant algebroids.
Involution algebroids extend Lie algebroids to tangent categories.
problem Extending Lie algebroid theory to tangent categories.
method Defining involution algebroids that replace the Jacobi identity with a Yang-Baxter-like equation.
result Every Lie algebroid is an involution algebroid and every involution algebroid admits a Lie bracket.
The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.
problem Categorification of Lie and Courant algebroids to better understand geometric structures.
method Introducing and studying new algebraic structures like VB-Lie 2-algebroids and VB-LWX 2-algebroids.
result Established correspondences and new constructions between Lie and Courant algebroids.
Introduces new construction for Courant algebroids and curved structures.
problem Understanding and classifying Courant algebroids and their lifts.
method Introduces Courant algebroid lift and curved Courant algebroids, establishing connections to various geometric structures.
result Established a classification of exact curved Courant algebroids and related connections to various geometric structures.
In this paper, we introduce the notion of E-Courant algebroids, where E is a vector bundle. It is a kind of generalized Courant algebroid and contains Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its special cases. We explore novel phenomena exhibited by E-Courant algebroids and provid…
We introduce the category of generalized Courant algebroids and show that it admits a free object on any anchored vector bundle. The free Courant algebroid is built from two components: the generalized Courant algebroid associated to a symmetric Leibniz algebroid and the free symmetric Leibniz algebroid on an anchored …
Defines the algebroid structure of double field theory.
problem Identify the algebroid structure of double field theory.
method Doubling the target space of a canonical Courant algebroid and projecting down to a specific subbundle.
result The DFT algebroid is a special example of a relaxed Courant algebroid structure.
We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providin…
Almost Lie algebroids extend Lie algebroids with a Jacobiator, leading to characteristic classes.
problem Extending Lie algebroids to include more general structures.
method Constructing cohomology and characteristic classes for almost Lie algebroids.
result Characteristic classes of almost Lie algebroids are pull-backs of base space classes.
In this paper, we give the notion of a CLWX 2-algebroid and show that a QP-structure of degree 3 gives rise to a CLWX 2-algebroid. This is the higher analogue of the result that a QP-structure of degree 2 gives rise to a Courant algebroid. A CLWX 2-algebroid can also be viewed as a categorified Courant algebroid. We sh…
Defines a transgression functor for higher-dimensional Courant algebroids.
problem None explicitly stated; focuses on definition and properties.
method Definition of transgression functor for Courant algebroids.
result Established a connection between Courant algebroids and Lie algebroids.
In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from $\mathcal …
VB-algebroids control deformations of Lie algebroids structures.
problem Deformation of Lie algebroid structures.
method Attach differential graded Lie algebra to VB-algebroids to control deformations.
result Controlled deformations of VB-algebroids through DG Lie algebra.
New invariant real rank identifies constant real Lie algebroids.
problem Characterizing complex Lie algebroids with constant real rank.
method Introducing real rank and minimal complex subalgebroid.
result Local splitting and characterization of complex Lie algebroids.
Notes on relative algebroids for geometric problems.
problem Geometric problems and their solutions.
method Explains how relative algebroids arise from geometric problems and introduces their structural theory.
result Relative algebroids unify Lie algebroids with partial differential equations.
New generalized Lie algebroids solve optimal control problems.
problem Optimal control problems not solvable by Lie algebroids.
method Proved generalized Lie algebroids as distinguished examples, disproved a theorem, and provided a new framework.
result Generalized Lie algebroids solve problems Lie algebroids cannot.
This paper studies Loday algebroids, introducing new concepts and formulas.
problem Exploring Loday algebroids and their cohomology, nonlinear connections, and characteristic classes.
method Introducing action Loday algebroids, clarifying Loday algebroid morphisms, studying nonlinear connections, and defining secondary characteristic classes.
result Established a generalized Chern-Simons formula for nonlinear connections on Loday algebroids.
This work explores higher-order algebroids via vector bundle comorphisms.
problem Generalizing concepts of higher-order tangent bundles and Lie algebroids.
method Introduces a vector bundle comorphism approach to describe higher-order algebroids.
result Establishes a one-to-one correspondence between higher-order Lie algebroids and specific algebraic structures.
Geometrically explains Lie 2-algebroids and their connections.
problem Exploring Lie 2-algebroids and their geometric properties.
method Explains Li-Bland's correspondence and uses geometric equivalence.
result Proves bicrossproduct of matched pairs of 2-representations is a split Lie 2-algebroid.
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in…
The paper defines Laplace operators for algebroid spaces.
problem Developing mathematical tools for algebroid spaces.
method Introducing Laplace-type operators for functions and forms on algebroid prolongations.
result Locally expressed Laplace operators for algebroid spaces.
Introduces higher algebroids via vector bundle comorphisms.
problem Generalizing Lie algebroids and higher tangent bundles.
method Defines higher algebroids as vector bundle comorphisms of graded-linear bundles with specific axioms.
result Provides natural examples and applications in geometric mechanics.
The paper defines and studies the first Pontryagin class for quadratic Lie 2-algebroids.
problem Defining and studying the first Pontryagin class for quadratic Lie 2-algebroids.
method Detailed study of transitive Lie 2-algebroids, introduction of quadratic Lie 2-algebroids, definition of first Pontryagin class, construction of quadratic Lie 2-algebroids.
result The first Pontryagin class is the obstruction class for the existence of a CLWX-extension and trivial for certain quadratic Lie 2-algebroids.
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (…
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
problem Exploring Lie algebroids with homological sections.
method Derived bracket formalism to define an odd Loday-Leibniz bracket on sections.
result Sections of inner Q-algebroids come equipped with an odd Loday-Leibniz bracket.
Study infinitesimal automorphisms of VB-groupoids and algebroids.
problem Understand transformations preserving the structure of VB-groupoids and algebroids.
method Examine vector fields generating flows that preserve both linear and groupoid/algebroid structures.
result Infinitesimal automorphisms of a special class are multiplicative sections of a derivation groupoid/algebroid.
Introduces holomorphic string algebroids and classifies them.
problem Classifying holomorphic string algebroids.
method Using Courant extensions and inner morphisms of holomorphic Courant algebroids.
result Classification of string algebroids via Cech cohomology.
Extends T-duality to exotic Courant algebroids.
problem Lack of T-duality for exotic Courant algebroids.
method Extends T-duality isomorphism to exotic exact Courant algebroids.
result Exchange of momentum and winding numbers.