arXiv research

New invariant real rank identifies constant real Lie algebroids.

Study Hom-Lie algebroid connections on complex manifolds.

Study families of Lie algebroids on complex spaces, introducing unfoldings.

The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are…

In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. A…

Study Lie algebroid connections on principal bundles over complex projective varieties.

Study of Hermitian metrics on Lie algebroids over complex spaces.

A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a compl…

Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.

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…

The paper explores linear generalised complex structures over vector bundles.

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 …

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

We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a natural generalization of Poisson quasi-Nijenhuis manifolds and show that any such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an associated Courant algebroid is obtained. We introduce the notion of a morphism of quasi-Lie…

The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.

This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (gro…

Horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on prolongation of a Lie algebroid are considered. Then using exact sequences, semisprays are constructed. Moreover, important geometrical objects such as classical distinguished connections, torsions and partial curvatures are…

Develops theory of para-holomorphic algebroids on Calabi-Yau manifolds.

A Q-algebroid is a Lie superalgebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the BRST model of a Lie algebroid, which generalizes the BRST model for equivariant co…

For any Lie algebroid A, its 1-jet bundle JA is a Lie algebroid naturally and there is a representation π: JA ->DA. Denote by dJ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex (Γ(…

Article proves tangent complex structure of Lie n-groupoid.

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…

In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids …

The $L_\infty$-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one $L_\infty$-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We a…

Study of real and quaternionic Lie algebroid connections on manifolds.

Complex Finsler vector bundles have been studied mainly by T. Aikou, who defined complex Finsler structures on holomorphic vector bundles. In this paper, we consider the more general case of a holomorphic Lie algebroid E and we introduce Finsler structures, partial and Chern-Finsler connections on it. First, we recall …

Regularisation method studies Lie algebroids via foliated structures.

Maps geometric deformations to algebraic classes in Lie groupoids and algebroids.

New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.

Study connections on complex Riemann surfaces for Lie algebroid structures.

Constructing $3$-Lie algebroids via connections

Researchers compute the cohomology of an elliptic tangent bundle.

Study infinitesimal deformations of Lie algebroid pairs.

The paper extends a theorem for complex structures on Lie groups to Courant algebroids.

In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distributio…

We apply the Atiyah-Singer index theorem and tensor products of elliptic complexes to the cohomology of transitive Lie algebroids. We prove that the Euler characteristic of a representation of a transitive Lie algebroid $A$ over a compact manifold $M$ vanishes unless $A=TM$, and prove a general Künneth formula. As appl…

We define the Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that to any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid is associated such a structure. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background…

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

Introduces new construction for Courant algebroids and curved structures.

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 studies deformations of Nijenhuis structures in Lie algebras and algebroids.

We shall prove that a moduli space of flat irreducible Lie algebroid connections over a compact manifold has locally a natural structure of a smooth differentiable space. This is a generalization of some well known results for the moduli space of holomorphic structures on a complex vector bundle over a compact complex …

Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive …

Lie algebroids are like infinitesimal Lie groupoids.

Extends T-duality to non-principal torus actions with elliptic tangent bundles.

This paper establishes an equivalence between transitive double Lie algebroids and core diagrams.

By studying the Frölicher-Nijenhuis decomposition of cohomology operators (that is, derivations $D$ of the exterior algebra $Ω(M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$), we describe new examples of Lie algebroid structures on the tangent bundle $TM$ (and its complexification $T^{\mathbb{C}}M$) constructed from pre-…

A general form for the boundary coupling of a Lie algebroid Poisson sigma model is proposed. The approach involves using the Batalin-Vilkovisky formalism in the AKSZ geometrical version, to write a BRST-invariant coupling for a representation up to homotopy of the target Lie algebroid or its subalgebroids. These consid…