arXiv research

The paper defines left-symmetric bialgebroids and their Manin triples.

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

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 …

The paper introduces new structures for left-symmetric algebroids.

Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.

Study Nijenhuis operators and their linearization problem using left-symmetric algebras.

Left-symmetric algebras help define affine spheres.

Classifies 3D non-degenerate left-symmetric algebras.

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

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G,G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x,y].z=0. We derive some basic characterizations of such left-symmetric …

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialg…

Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the mult…

Introduces Lie-Nijenhuis bialgebroids for Poisson-Nijenhuis groupoids.

Extends Drinfel'd doubles to include twists and generalizes Lie bialgebroids.

Extends Manin triples to Lie bialgebroids over Lie groupoids.

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissoniza…

Extends Lie bialgebroids to homotopy theory.

This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-…

LA-Courant algebroids link double Lie bialgebroids via Manin triples.

We describe infinitesimally Dirac groupoids via geometric objects that we call Dirac bialgebroids. In the two well-understood special cases of Poisson and presymplectic groupoids, the Dirac bialgebroids are equivalent to the Lie bialgebroids and IM-$2$-forms, respectively. In the case of multiplicative involutive distr…

Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.

The paper defines pre-symplectic algebroids and their applications.

We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order…

We study the local structure of Lie bialgebroids at regular points. In particular, we classify all transitive Lie bialgebroids. In special cases, they are connected to classical dynamical $r$-matrices and matched pairs induced by Poisson group actions

Defines new structures on Lie groupoids and Lie algebroids.

The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.

The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.

An alternative proof of the duality of generalized Lie bialgebroid is given and proved a canonical Jacobi structure can be defined on the base of it. We also introduce the notion of morphism between generalized Lie bialgebroids and proved that the induced Jacobi structure is unique upto a morphism.

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, Poisson-Nijenhuis structures, and triangular …

Deforms Poisson quasi-Nijenhuis manifolds using closed 2-forms.

Locally convex bialgebroids reconstruct Lie groupoids of orbits.

We study twisted Jacobi manifolds, a concept that we had introduced in a previous Note. Twisted Jacobi manifolds can be characterized using twisted Dirac-Jacobi, which are sub-bundles of Courant-Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce t…

We use the supergeometric formalism, more precisely, the so-called "big bracket" (for which brackets and anchors are encoded by functions on some graded symplectic manifold) to address the theory of Jacobi algebroids and bialgebroids (following mainly Iglesias-Marrero and Grabowski-Marmo as a guideline). This formalism…

The paper defines and explores Hom-Lie algebroids and related structures.

Jacobi groupoids are introduced as a generalization of Poisson and contact groupoids and it is proved that generalized Lie bialgebroids are the infinitesimal invariants of Jacobi groupoids. Several examples are discussed.

Extends Lie bialgebroids for string and M theories with new calculus framework.

We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.

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…

Solves a challenging case of Nijenhuis operator linearization in 2D.

The study investigates linearizability of Poisson structures on groupoids.

Characterizes Vaisman solvmanifolds and connects them to other geometric structures.

In these lecture notes, we give a quick account of the theory of Poisson groupoids and Lie bialgebroids. In particular, we discuss the universal lifting theorem and its applications including integration of quasi-Lie bialgebroids, integration of Poisson Nijenhuis structures and Alekseev and Kosmann-Schwarzbach's theory…

We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional $\mathbb Z$-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like $TTM$, e…

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…

New type of manifolds derived from Poisson structures.

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

Study properties of coisotropic submanifolds and generalize Nambu structures.

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,Λ,E)$ for which 1 is an admiss…