Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.
problem Properties of para-Kähler manifolds with conformal Einstein soliton metrics.
method Investigated curvature properties of para-Kähler manifolds admitting conformal Einstein soliton.
result Certain curvature properties of para-Kähler manifolds were studied.
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 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.
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.
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.
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.
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 (…
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.
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.
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.
Lie algebroids are like infinitesimal Lie groupoids.
problem None explicitly stated in the abstract.
method Overview article.
result Lie algebroids are infinitesimal counterparts of Lie groupoids.
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.
This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.
problem Understanding and characterizing transitive double Lie algebroids.
method Using core diagrams and equivalence of transitive core diagrams with transitive double Lie groupoids.
result Transitive double Lie algebroids are completely determined by their core diagrams.
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.
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
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.
Study first-order locally convex Lie algebroids in Bastiani calculus.
problem Define and study first-order locally convex Lie algebroids.
method Define sheaves of Lie algebroid forms and morphisms, prove category structure, study representations and cohomology.
result First-order locally convex Lie algebroids form a category and have applications in Lie II theorems.
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.
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.
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.
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.
Homotopy invariance proven for twisted Lie algebroid cohomologies.
problem Homotopy invariance of twisted Lie algebroid cohomologies.
method Lie algebroid homotopy-invariance proof with examples.
result Comprehensive systematic way to compute twisted Lie algebroid cohomologies.
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.
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this…
We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obta…
Integrates transitive Lie algebroids to Lie groupoids, explaining obstructions.
problem Integrating transitive Lie algebroids to Lie groupoids.
method Geometric explanation and explicit construction of integration, with obstructions considered.
result Obstructions explained and integration constructed when they vanish.
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.
New algebraic structures for Lie 2-algebroids and their connections.
problem Characterizing and understanding Lie 2-algebroids and their structures.
method Construction of homotopy Poisson algebra and introduction of Dirac structures.
result One-to-one correspondence between Manin triples and Lie 2-bialgebroids.
Study cohomology of Lie algebroids over algebraic spaces using derived functors and Čech cohomology.
problem Cohomology of Lie algebroids over algebraic spaces.
method Express hypercohomology as a derived functor, simplify via Čech cohomology, define Hochschild hypercohomology, present Hochschild-Kostant-Rosenberg theorem.
result Presented a version of Hochschild-Kostant-Rosenberg theorem for locally free Lie algebroids.
The paper defines a new algebraic structure and shows its relation to existing ones.
problem Exploring new algebraic structures related to existing ones.
method Introducing CLWX 2-algebroids and showing their relation to QP-structures and Lie 3-algebras.
result QP-structures of degree 3 give rise to CLWX 2-algebroids, a higher analogue of Courant algebroids.
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…
Defines a higher version of omni-Lie algebroid for new geometries.
problem No specific problem stated; focuses on definition.
method Proposes a new definition of higher omni-Lie algebroid.
result Studies isotropic and involutive subbundles of higher omni-Lie algebroid.
This thesis bridges Lie theory and sketch theory using tangent categories.
problem Two diverging lines of research in Lie theory.
method Developing involution algebroids and using tangent categories to connect Lie algebroids and Weil algebras.
result The category of Lie algebroids is a functor category, and the Lie functor is a composition with a tangent categorical functor.
Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of mo…
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.
Integrates quadratic Lie algebroids to Riemannian Cartan-Lie groupoids.
problem Defining metrics on Lie algebroids and groupoids.
method Analyzes ad-invariant metrics and bi-invariant metrics on Lie algebroids and groupoids.
result Determines conditions for integration of positive quadratic Lie algebroids.
Holomorphic Lie algebroid connections on Riemann surfaces are characterized.
problem Characterizing holomorphic Lie algebroid connections on Riemann surfaces.
method Analyzes conditions for holomorphic vector bundles to admit Lie algebroid connections based on Lie algebroid properties.
result Conditions for holomorphic vector bundles to admit holomorphic Lie algebroid connections are determined.
Extends Lie algebroids by Lie algebroids with specific conditions.
problem Generalizing Lie algebroid extensions with curvature.
method Using strict covariant adjustments and Cartan connections.
result Provides an obstruction theory for certain Cartan connections.
LA-Courant algebroids link double Lie bialgebroids via Manin triples.
problem Establishing a mathematical framework for double Lie bialgebroids.
method Verification of Manin triple framework for double Lie bialgebroids using LA-Courant algebroids.
result LA-Courant algebroids provide a correspondence with double Lie bialgebroids.
The paper presents the geometry of Lie algebroids and its applications to optimal control. The first part deals with the theory of Lie algebroids, connections on Lie algebroids and dynamical systems defined on Lie algebroids (mainly Lagrangian and Hamiltonian systems). In the second part we use the framework of Lie alg…
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie al…
Lie algebroids linked to L∞ spaces in derived geometry.
problem Relating Lie algebroids to L∞ spaces in derived geometry. method Constructing a faithful functor from Lie algebroids to L∞ spaces and showing the relationship between representations and vector bundles. result Lie algebroids provide an essentially unique L∞ space, and a shifted-symplectic structure on a dg Lie algebroid produces a shifted-symplectic structure on the associated L∞ space. Criterion found for Lie algebroid connections on parabolic bundles.
problem Conditions for parabolic vector bundles to have Lie algebroid connections.
method Necessary and sufficient condition based on Lie algebroid structure.
result Found criterion for existence of parabolic Lie algebroid connections.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Defines and extends Lie algebroid prolongations in convenient settings.
problem Adapting Lie algebroid prolongations to convenient settings.
method Defined and adapted Lie algebroid prolongations over fibred manifolds.
result Stability of prolongations under projective and direct limits.
Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.
problem Extending complex geometry results to Lie algebroids.
method Using local coordinate calculations to generalize Kahler identities.
result Kernel of Lie algebroid Laplace operator vanishes for sufficiently large p+q.
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
Corrected a misunderstanding about generalized Lie algebroids.
problem Misconception about generalized Lie algebroids.
method Proof
result Generalized Lie algebroids are not a valid geometric object.