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.
The paper defines left-symmetric bialgebroids and their Manin triples.
problem No specific problem stated; focuses on definitions and constructions.
method Introduced left-symmetric bialgebroids and Manin triples, constructed from pseudo-Hessian manifolds.
result Established a relation between Maurer-Cartan type equations and Dirac structures.
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…
Introduces comomentum sections and proves they are Poisson maps.
problem Generalizing Poisson maps to Hamiltonian Lie algebroids.
method Introduces comomentum sections and proves they are Lie algebroid morphisms and Poisson maps.
result Comomentum sections are Poisson maps between proper Poisson manifolds.
Generalizes Hamiltonian structures to Dirac structures for new mechanics models.
problem Extending Hamiltonian structures to non-symplectic manifolds.
method Introduces Hamiltonian Lie algebroids and momentum sections over Dirac structures.
result Constructs new sigma models based on these structures.
The paper studies Lie algebroid and groupoid quotients with forms, applying to Poisson and Dirac structures.
problem Understanding quotients of Lie algebroids and groupoids with compatible differential forms.
method Identifying Lie theoretic conditions for forms to be basic, characterizing induced forms on quotients, and applying results to Poisson and Dirac structures.
result Recovery and generalization of known results on Poisson reduction.
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.
Introduces homotopy momentum sections on multisymplectic manifolds.
problem No specific problem stated; focuses on introducing a new concept.
method Introduces a new concept of homotopy momentum sections on multisymplectic manifolds.
result Shows that a gauged nonlinear sigma model with Wess-Zumino term has homotopy momentum section structure.
Explains how pre-symplectic structures can be changed.
problem Understanding how pre-symplectic structures can be deformed.
method Uses Dirac geometry to explain the geometric origin of L∞-algebra controlling deformations. result Discovers the geometric origin of the L∞-algebra controlling deformations of pre-symplectic structures. Study on deformations of pre-symplectic structures using an L-infinity algebra.
problem Deformation theory of pre-symplectic structures.
method Parametrization of deformations using Koszul L-infinity algebra.
result A quotient of the Koszul L-infinity algebra is isomorphic to the L-infinity algebra controlling foliations.
Study shows equivalence in foliations and pre-symplectic forms aligns with gauge equivalence.
problem Deformation theory of foliations and pre-symplectic forms.
method Proved geometric equivalence agrees with algebraic gauge equivalence using L∞-algebras. result Gauge equivalences for foliations and pre-symplectic structures are consistent.
Alternative proof of coisotropic embedding theorem for pre-symplectic manifolds.
problem Proving the coisotropic embedding theorem for pre-symplectic manifolds.
method Recast geometric choice of connection as algebraic embedding into cotangent bundle, identify symplectic thickening as submanifold of Hamiltonian momenta conjugate to kernel directions.
result Alternative proof of the coisotropic embedding theorem.
The paper explores symmetries and conserved charges on pre-symplectic manifolds.
problem Analyzing conserved charges on solutions of Hamiltonian field theories.
method Using pre-symplectic structures and Gotay's coisotropic embedding theorem, the paper deals with gauge theories and examples like Electrodynamics and Klein-Gordon theory.
result Emergence of the energy-momentum tensor algebra of conserved currents.
We give a local normal form for Dirac structures. As a consequence, we show that the dimensions of the pre-symplectic leaves of a Dirac manifold have the same parity. We also show that, given a point m of a Dirac manifold M, there is a well-defined transverse Poisson structure to the pre-symplectic leaf P through…
Study shows bi-Hamiltonian structure for polygon evolutions in centro-affine space.
problem Understanding bi-Hamiltonian structure for polygon evolutions.
method Geometric realizations of discrete flows, lifting to pre-symplectic forms.
result Compatibility of two Hamiltonian structures proved straightforward.
Study well-posedness of Faraday tensor problem on specific spacetime manifolds.
problem Well-posedness of the Cauchy problem for the Faraday tensor on globally hyperbolic manifolds with timelike boundary.
method Existence of Green operators for the operator d+δ and a suitable pre-symplectic structure on the space of solutions. result Existence of Green operators and pre-symplectic structure for the operator d+δ. Solves inverse problem for Maxwell equations using vector fields.
problem Inverse problem for Maxwell equations in vacuum.
method Abstract theory of implicit differential equations over pre-symplectic manifolds.
result Provides solution for Maxwell equations using vector fields.
The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions n=4r−2. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…
Given a Poisson (or more generally Dirac) manifold P, there are two approaches to its geometric quantization: one involves a circle bundle Q over P endowed with a Jacobi (or Jacobi-Dirac) structure; the other one involves a circle bundle with a (pre-) contact groupoid structure over the (pre-) symplectic groupoid…
The paper constructs a moduli space for opers and proves its symplectic properties.
problem Understanding the moduli space of opers and its symplectic structure.
method Constructing the moduli space of marked oper structures and proving properties of the holonomy map.
result The symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space of marked opers.
We construct the odd symplectic structure and the equivariant even (pre)symplectic one from it on the space of differential forms on the Riemann manifold. The Poincare -- Cartan like invariants of the second structure define the equivariant generalizations of the Euler classes on the surfaces.
New method integrates Poisson homogeneous spaces to symplectic groupoids.
problem Integrating Poisson homogeneous spaces to symplectic structures.
method Using Dirac geometry and explicit constructions, integrates Poisson homogeneous spaces to symplectic groupoids.
result Every Poisson homogeneous space of a Poisson Lie group integrates to a symplectic groupoid.
We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…
We construct integrable hierarchies of flows for curves in centroaffine R3 through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. We show that the induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and …
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.
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…
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.
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.