arXiv research

Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.

A Lie algebroid classifies G-structures with connections.

Defines the algebroid structure of double field theory.

The paper develops structures on Hom-Lie algebroids and Hom-Courant algebroids.

We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simpler way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic stru…

Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.

Constructs Poisson structure on Banach Lie algebroid predual.

In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifo…

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…

New algebraic structures for Lie 2-algebroids and their connections.

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 (…

The paper introduces new structures for left-symmetric algebroids.

New Poisson structures defined from Lie algebroids, with conditions for existence.

In this thesis we develop the notion of LA-Courant algebroids, the infinitesimal analogue of multiplicative Courant algebroids. Specific applications include the integration of q- Poisson (d, g)-structures, and the reduction of Courant algebroids. We also introduce the notion of pseudo-Dirac structures, (possibly non-L…

Introduces new structures in geometry and algebra.

New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.

Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…

This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs …

Introduces new construction for Courant algebroids and curved structures.

The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.

Constructing $3$-Lie algebroids via connections

Study integrability of specific geometric structures on odd Courant algebroids.

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…

A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length …

New Poisson structures on hypersurface algebroids discovered.

In this thesis we study geometric structures from Poisson and generalized complex geometry with mild singular behavior using Lie algebroids. The process of lifting such structures to their Lie algebroid version makes them less singular, as their singular behavior is incorporated in the anchor of the Lie algebroid. We d…

Characterizes Filippov n-algebroids using connections and formulas.

Study the pullbacks and blowups of Lie algebroids and related structures.

Generalizes Hamiltonian structures to Dirac structures for new mechanics models.

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $Λ$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $Λ.$ We show that many known properties of th…

Paper constructs various algebroids using n-systems and metric n-systems.

We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly sympl…

A notion of an algebroid - a generalization of a Lie algebroid structure is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T^M can be obtained in the framework of a general algebroid. Also a compatibility condition which leads,…

We construct a generalization of Courant algebroids which are classified by the third cohomology group $H^3(A,V)$, where $A$ is a Lie Algebroid, and $V$ is an $A$-module. We see that both Courant algebroids and $\mathcal{E}^1(M)$ structures are examples of them. Finally we introduce generalized CR structures on a manif…

This work explores higher-order algebroids via vector bundle comorphisms.

We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A …

Split Courant algebroids linked to special algebra structures.

The paper introduces a new form on Lie algebroids over multisymplectic manifolds.

New invariant real rank identifies constant real Lie algebroids.

New framework tackles geometric structure existence and classification.

Develops theory of para-holomorphic algebroids with para-complex connections.

Surveying integrability of Lie algebroids and structures.

Regularisation method studies Lie algebroids via foliated structures.

Hypercomplex structures on Courant algebroids unify holomorphic symplectic structures and usual hypercomplex structures. In this note, we prove the equivalence of two characterizations of hypercomplex structures on Courant algebroids, one in terms of Nijenhuis concomitants and the other in terms of (almost) torsionfree…

Extending Jacobi and Riemannian compatibility to Lie algebroids.

Characterizes integrability of generalized structures on Courant algebroids.

This thesis generalizes structures on $\mathcal{Q}$-manifolds and Lie $n$-algebroids.

This paper shows the equivalence of the categories of $N$-manifolds of degree $2$ with the category of double vector bundles endowed with a linear metric. Split Poisson $N$-manifolds of degree $2$ are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an …