Related papers: Stokes manifolds and cluster algebras

We geometrize the constructions of twisted Poisson modules introduced by Luo-Wang-Wu, and Poisson chain complexes with coefficients in Poisson modules defined in the algebraic setting to the geometric setting of Poisson manifolds. We then…

Crawley-Boevey introduced the definition of a noncommutative Poisson structure on an associative algebra A that extends the notion of the usual Poisson bracket. Let V be a symplectic manifold and G be a finite group of symplectimorphisms of…

Scattering symplectic manifolds are (closed) manifolds with a mildly degenerate Poisson structure. In particular they can be viewed as symplectic structures on a Lie algebroid which is almost everywhere isomorphic to the tangent bundle. In…

The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two…

Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, $b^k$-, scattering and…

Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates…

To show that certain wild character varieties are multiplicative analogues of quiver varieties, Boalch introduced colored multiplicative quiver varieties. They form a class of (nondegenerate) Poisson varieties attached to colored quivers…

In a recent work, we constructed a rational map from a simple Lie group $\mathcal G$ to itself that intertwines the standard Poisson--Lie structure on $\mathcal G$ with a Poisson homogeneous one defined by a pair of quasi-triangular…

Let $C$ be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to $C$. We prove that the decorated double…

A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this…

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations…

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that…

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld…

Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely…

We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their…

There is a canonical derived Poisson structure on the universal enveloping algebra $\mathcal{U}\mathfrak{a}$ of a (DG) Lie algebra $\mathfrak{a}$ that is Koszul dual to a cyclic cocommutative (DG) coalgebra. Interesting special cases of…

A.A. Kirillov introduced the family algebras in 2000. In this paper we study the noncommutative Poisson bracket P on the classical family algebra. We show that P is the first-order deformation from the classical family algebra to the…

We study Poisson-flat connections with logarithmic poles along a simple normal crossings divisor on a holomorphic Poisson manifold, where flatness is required only along the symplectic foliation. After identifying the relevant logarithmic…

Let $M$ be an exact symplectic manifold with $c_1(M)=0$. Denote by $\mathrm{Fuk}(M)$ the Fukaya category of $M$. We show that the dual space of the bar construction of $\mathrm{Fuk}(M)$ has a differential graded noncommutative Poisson…

We show that, on a smoothly paracompact convenient manifold $M$ modeled on a convenient space with the bornological approximation property, the dual map of a Poisson bracket factors as a smooth section of the vector bundle…