Related papers: Plethora of cluster structures on $GL_n$

In this paper we study left invariant CR structures on Lie groups which are compatible with geometric properties as Poisson and kahler properties.

Let $G$ be a finite-dimensional Poisson algebraic, Lie or formal group. We show that the center of the quantization of $G$ provided by an Etingof-Kazhdan functor is isomorphic as an algebra to the Poisson center of the algebra of functions…

For a connected abelian Lie group T acting on a Poisson manifold (Y,{\pi}) by Poisson isomorphisms, the T-leaves of {\pi} in Y are, by definition, the orbits of the symplectic leaves of {\pi} under T, and the leaf stabilizer of a T-leaf is…

We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all $n_1+m$ sources are separated from all $n_2+m$ sinks, we can construct a cluster-algebra realization of elements of an affine…

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra…

Multivariate count data are commonly encountered through high-throughput sequencing technologies in bioinformatics, text mining, or in sports analytics. Although the Poisson distribution seems a natural fit to these count data, its…

Leclerc constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing…

We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which $G$ acts, then the…

We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to…

Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structures for the corresponding Lie bialgebroids. Applications include Drinfel'd's classification in the case of Poisson groups and a description of leaf…

The moduli space of Anosov representations of a surface group in a semisimple group, which is an open set in the character variety, admits many more natural functions than the regular functions. We will study in particular length functions…

Classical limits of quantum groups give rise to multiplicative Poisson structures such as Poisson-Lie and quasi-Poisson structures. We relate them to the notion of a shifted Poisson structure which gives a conceptual framework for…

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable…

The supersymmetric generalization of Pisson-Lie T-duality in N=2 superconformal WZNW models on the compact groups is considered. It is shown that the role of Drinfeld's doubles play the complexifications of the corresponding compact groups.…

We consider the symplectic groupoid of pairs $(B, A)$ with $A$ real unipotent upper-triangular matrix and $B\in GL_n$ being such that $\tilde A=BAB^T$ is also a unipotent upper-triangular matrix. Fock and Chekhov defined a Poisson map of…

For a complex algebraic torus $\mathbb{T}$, we study $\mathbb{T}$-invariant Poisson deformations of a $\mathbb{T}$-log-symplectic log-canonical Poisson structure $\pi_0$ on $\mathbb{C}^n$. We show that every $\mathbb{T}$-invariant…

An algebra with bracket ({\sf AWB} for short) is an associative algebra endowed with a bilinear bracket satisfying a Leibniz-type compatibility condition, as introduced in \cite{casas}. It can be viewed as a noncommutative generalization of…

The main aim of this article is to characterize inner Poisson structure on a quantum cluster algebra without coefficients. Mainly, we prove that inner Poisson structure on a quantum cluster algebra without coefficients is always a standard…

We introduce a new type of noncommutative Poisson structure on associative algebras. It induces Poisson structures on the moduli spaces classifying semisimple modules. Path algebras of doubled quivers and preprojective algebras have…

Stokes' manifolds, also known as wild character varieties, carry a natural symplectic structure. Our goal is to provide explicit log-canonical coordinates for these natural Poisson structures on the Stokes' manifolds of polynomial…