The paper constructs structures for Lie pairs and their Atiyah classes.
problem Understanding structures of Lie pairs and their Atiyah classes.
method Constructs dg-manifolds and dg-Lie algebroids for Lie pairs, showing quasi-isomorphisms and Atiyah classes.
result Induces a quasi-isomorphism between dg-Lie algebroids and Atiyah classes of Lie pairs.
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. New maps help understand deformations of modules over Lie algebroids.
problem Understanding deformations of modules over Lie algebroids.
method Introduce semiregularity maps and use DG-Lie algebra control.
result Semiregularity maps annihilate obstructions under certain conditions.
New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.
problem Constructing Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.
method Using homological perturbation lemma and contraction, constructing isomorphisms between cochain complexes and Chevalley-Eilenberg cohomologies.
result Identifies Atiyah and Todd classes of Lie pair pullback dg Lie algebroids with those of the Lie pair.
New algebra structure derived from Lie pairs.
problem Constructing A∞-algebras from Lie pairs. method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
New algebraic structure derived from Kähler manifolds.
problem Understanding algebraic structures on differential forms.
method Introducing L∞[1] R-algebras and proving linearization theorems. result Induced L∞[1] R-algebra structures on Γ(L) are linearizable under certain conditions. The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.
problem Formal geometry of dg manifolds.
method Construction of Fedosov dg foliation and homotopy contractions.
result Isomorphism of Cartan and noncommutative calculi.
This paper studies Hopf algebras from dg manifolds.
problem Understanding Hopf algebras from the perspective of dg manifolds.
method Analyzes the universal enveloping algebra of Lie algebra objects in homotopy categories of dg modules.
result The universal enveloping algebra of the Lie algebra object is a Hopf algebra.
We study the dg-Lie algebra f_n generated by the coefficients of the universal translation invariant flat dg-connection on the n-dimensional affine space. We describe its "semiabelianization" (in particular, the universal quotient which is a crossed module of Lie algebras) in terms of closed differential forms of arbit…
New approach to principal groupoid bundles with connections using dg-Lie groupoids.
problem Developing a new perspective on principal bundles with connections.
method Using dg-Lie groupoids and additional adjustment data for Lie groupoids.
result Adjusted connections provide a global formulation of curved Yang-Mills-Higgs theories.
Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.
problem Finite-dimensional group of conformal transformations in higher dimensions.
method Derived deformation theory of ambitwistor space of complex null-geodesics.
result Infinite-dimensional dg-Lie algebra incorporating symmetries and conformal structure deformations.
Generalizes Hodge correlators using quantum master equation concepts.
problem Developing a mathematical framework for non-acyclic Chern-Simons theory.
method Introduces a DG Lie algebra of uni-trivalent graphs with loops satisfying a Maurer-Cartan equation.
result Arithmetic analogue of effective action and quantum master equation.
We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) suc…
Models for self-equivalences and diffeomorphisms of manifolds.
problem Classifying spaces of self-equivalences and diffeomorphisms of manifolds.
method Construct rational models using equivariant algebraic methods.
result Formula for rational cohomology of classifying spaces.
Constructs local superconformal algebras on supermanifolds.
problem Deformations of odd distributions on supermanifolds.
method Local super dg Lie algebra construction.
result Returns conformal supergravity multiplet in various dimensions.
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered L∞-model. result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
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.
We find a minimal differential graded (dg) operad whose generic representations in Rn are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to Rn which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar constru…
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.
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.
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…
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 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.
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 (…