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.
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.
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 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.
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diff…
New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.
problem Extending Courant algebroid structures to higher multi-Courant algebroids.
method Constructing higher geometric versions of algebraic structures defined by Keller and Waldmann.
result Higher multi-Courant algebroids form a Poisson algebra.
The paper classifies symplectic Lie and L∞ algebroids and their applications.
problem Classifying symplectic Lie and L∞ algebroids and their higher gauge symmetries. method Classifying zero-, one-, and two-shifted symplectic algebroids using classical geometric higher structures.
result New examples of twisted Courant algebroids from codimension-two cycles and symplectic interpretations of higher structures.
We solve higher-order morphisms for twisted Courant algebras.
problem Construct canonical L∞-morphisms for higher Courant algebroids. method Develop a general framework for arbitrary r. result Affirmative answer to Zambon's question for higher degrees.
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…
The paper linearizes higher Courant algebroids using jet and differential operators.
problem Understanding the structure of higher Courant algebroids.
method Isomorphic spaces of sections and linearization of jet and differential operator bundles.
result Higher Courant algebroids can be linearized as pseudo-linearization and Weinstein-linearization.
We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.
We show how the relation between Q-manifolds and Lie algebroids extends to ``higher'' or ``non-linear'' analogs of Lie algebroids. We study the identities satisfied by a new algebraic structure that arises as a replacement of operations on sections of a Lie algebroid. When the base is a point, we obtain a generalizat…
We show that L∞-algebroids, understood in terms of Q-manifolds can be described in terms of certain higher Schouten and Poisson structures on graded (super)manifolds. This generalises known constructions for Lie (super)algebras and Lie algebroids.
Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle.…
The paper connects higher-dimensional mechanics to Lie n-algebroids.
problem Understanding interactions in higher-dimensional gauge systems.
method Comparing BV/BRST formalism with Lie n-algebroids and defining polytorsion.
result Relates topological n-branes to differential geometry on Lie n-algebroids.
We present the notion of higher Kirillov brackets on the sections of an even line bundle over a supermanifold. When the line bundle is trivial we shall speak of higher Jacobi brackets. These brackets are understood furnishing the module of sections with an L∞-algebra, which we refer to as a homotopy Kirillov …
This paper explores string principal bundles and their relation to Courant algebroids.
problem Understanding the infinitesimal symmetries of string structures and their geometric counterparts.
method Constructing and analyzing stacks of principal 2-bundles of string groups and their connections.
result The process of associating higher Atiyah algebroids loses some information and not all transitive Courant algebroids come from string bundles.
Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
problem Addressing problems in higher differential geometry using derived Lie ∞-groupoids and algebroids.
method Construct CFO structures, study L∞-algebroids, homotopical algebras, and homotopy-coherent representations.
result Construct Atiyah classes for L∞-algebroids pairs and study singular foliations and their holonomies.
The paper studies Lie n-algebroids and their representations up to homotopy.
problem Understanding Lie n-algebroids and their representations.
method Analyzes differential graded modules and representations up to homotopy, describes adjoint and coadjoint modules, and computes the Weil algebra.
result Alternative characterisation of non-degeneracy of higher Poisson structures.
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM⊕∧nT∗M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n+1)-vector fi…
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.
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…
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 contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of…
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
Uniform criteria for stability of fixed points in various geometric structures.
problem Stability of fixed points in Poisson geometry and higher Lie theory.
method Uniform approach to criteria for stability, using differential graded Lie algebras and cohomology.
result Vanishing of a finite-dimensional cohomology group implies stability of fixed points.
The paper studies deformations of Filippov algebroids using cohomology and DGLA.
problem Deformations of Filippov algebroids.
method Defined a DGLA for Filippov algebroids and used low-dimensional cohomology to discuss deformations. Characterized trivial deformations using Nijenhuis operators and defined finite order deformations.
result Characterized trivial deformations of Filippov algebroids using Nijenhuis operators.
Introduces new connections in higher geometry.
problem Defining connections in higher geometry.
method Develops formal differentiation and integration of maps to derived stacks.
result Establishes new L∞-algebras of higher symmetries. Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale L…
This thesis generalizes structures on Q-manifolds and Lie n-algebroids.
problem Representation theory and linear structures of Q-manifolds and Lie n-algebroids. method Introduces differential graded modules and representations up to homotopy, defines Weil algebra, and studies VB-Lie n-algebroids. result Establishes an equivalence between VB-Lie n-algebroids and (n+1)-term representations up to homotopy of Lie n-algebroids. The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. A universal Lie ∞-algebroid is constructed for singular foliations.
problem Describing the geometry and structure of singular foliations.
method Construction of a Lie ∞-algebroid for every resolution of a singular foliation.
result The universal Lie ∞-algebroid uniquely encodes the geometry of singular foliations.
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical…
The paper studies Atiyah classes for Lie algebroid representations and homotopies.
problem Understanding Atiyah classes for Lie algebroid representations and homotopies.
method Constructing cocycles from higher connection forms of representations up to homotopy and studying their cohomology.
result There exists a cohomology class independent of extensions, vanishing for compatible extensions, and related to quasi-isomorphisms of Atiyah classes.
Generalizes sigma model with Lie algebroid structure and geometric conditions.
problem Consistency of constraints and gauge symmetry in topological sigma models.
method Analysis of geometric conditions and constraints in Hamiltonian and Lagrangian formalisms.
result Identifies universal compatibility condition between Lie algebroid and multi-symplectic structure.
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.
Generalizes momentum sections to higher-dimensional gauged sigma models.
problem Understanding momentum sections in Hamiltonian mechanics and sigma models.
method Introduces a generalization of momentum sections on pre-multisymplectic manifolds.
result Shows a connection between constrained Hamiltonian systems and gauged sigma models.
Constructs a homotopy Loday algebra from symplectic 2-manifolds.
problem Tackles the construction of algebraic structures from symplectic 2-manifolds.
method Uses higher derived brackets and Voronov's technique to construct a homotopy Loday algebra.
result Constructs a homotopy Loday algebra with a specific structure accommodating the Dorfman bracket.
Paper constructs L∞-algebroids from homotopy Poisson structures.
problem Homotopy Poisson structures and their algebraic properties.
method Introduces thick morphisms and L∞-morphisms. result Establishes an L∞-algebra structure on forms. Regularisation method studies Lie algebroids via foliated structures.
problem Geometric structures on Lie algebroids.
method Regularisation procedure applied to various Lie algebroids.
result Proof of Weinstein conjecture for specific Lie algebroids.
Reviewing Q-manifolds, modular classes, and applications.
problem Obtaining invariant volumes on Q-manifolds.
method Exploring Q-manifolds, modular classes, and applying to specific examples.
result Applications to L∞-algebroids and higher Poisson manifolds. The paper introduces a new form on Lie algebroids over multisymplectic manifolds.
problem Higher generalizations of Poisson structures and momentum maps.
method Introducing a compatible E-n-form on Lie algebroids.
result The introduced form satisfies a compatibility condition with Lie algebroid and multisymplectic structures.
Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.
problem Understanding connections on Lie and Courant algebroids and their compatibility.
method Revisit and define basic curvature for Lie algebroids, introduce basic curvature for Courant algebroids, and use Atiyah cocycle for gauge theory.
result Basic curvature tensor for Courant algebroids and its relation to the Atiyah cocycle.
We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to …
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.
problem Provides a Morita invariant definition of Lie and Courant algebroids over Lie groupoids.
method Views vector fields as Maurer-Cartan elements in a differential graded Lie algebra and as functors and natural transformations.
result Obtains a unifying conceptual framework for studying various algebraic structures.
New algebroids allow studying various geometries simultaneously.
problem Constructing geometries on a broad class of algebroids.
method Introduced `anti-commutable` pre-Leibniz algebroids and admissible connections.
result Proved properties of admissible connections on various algebroids.
In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are parametrized by gene…