Related papers: Thick morphisms, higher Koszul brackets, and $L_{\…

We generalize to the homotopy case a result of K. Mackenzie and P. Xu on relation between Lie bialgebroids and Poisson geometry. For a homotopy Poisson structure on a supermanifold $M$, we show that $(TM, T^*M)$ has a canonical structure of…

It is well known that the chain map between the de Rham and Poisson complexes on a Poisson manifold also maps the Koszul bracket of differential forms into the Schouten bracket of multivector fields. In the generalized case of a…

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbol k[x_1,x_2,\dots, x_n]/I$ where $\boldsymbol k$ is a field of characteristic $0$ and $I\subseteq \boldsymbol k[x_1,x_2,\dots, x_n]$ is an ideal. Assume that there is a…

We show that $L_{\infty}$-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…

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the…

We study Maurer-Cartan elements on homotopy Poisson manifolds of degree $n$. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and…

The notion of a \emph{higher-order algebroid}, as introduced by J\'o\'zwikowski and Rotkiewicz in their work \emph{Higher-order analogs of Lie algebroids via vector bundle comorphisms} (SIGMA, 2018), generalizes the concepts of a…

We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an $L_\infty$-algebra, which we…

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in…

We study the shifted analogue of the "Lie--Poisson" construction for $L_\infty$ algebroids and we prove that any $L_\infty$ algebroid naturally gives rise to shifted derived Poisson manifolds. We also investigate derived Poisson structures…

The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie…

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…

In this paper we propose a noncommutative generalization of the relationship between compact K\"ahler manifolds and complex projective algebraic varieties. Beginning with a prequantized K\"ahler structure, we use a holomorphic Poisson…

Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras…

Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie…

A Poisson structure on a manifold is characterized by the Schouten bracket. The graded algebra of the tangent bundle with the Schouten bracket is a prototype of Lie superalgebra. The Poisson condition means that a cycle in the 2-chain…

A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of…

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $\mathbb{C}[X_{1},..., X_{n}]$ using exterior calculus. After presenting some non homogeneous Poisson brackets on this algebra, we compute…

In this note we show how to construct a homotopy BV-algebra on the algebra of differential forms over a higher Poisson manifold. The Lie derivative along the higher Poisson structure provides the generating operator.

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures.…