Related papers: A generalized cluster structure on $GL_n$ via bira…
The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson…
We associate a family of cluster X-varieties to the dual Poisson-Lie group G* of a complex semi-simple Lie group G of adjoint type given with the standard Poisson structure. This family is described by the W-permutohedron associated to the…
We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N_{u,v} in the disk that correspond to the choice of a…
Let $G$ be a connected complex semi-simple Lie group, and let $Z_{{\bf u}}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure…
We consider the symplectic groupoid of pairs $(B,\mathbb{A})$ with $\mathbb A$ unipotent upper-triangular matrices and $B\in GL_n$ being such that $\widetilde {\mathbb A}=B{\mathbb A} B^{\text{T}}$ are also unipotent upper-triangular…
We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a…
As is well-known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plucker relations, Desnanot--Jacobi identities and their generalizations. We present a construction…
Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert…
We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several…
In the prequel of this paper, we have associated a family of cluster X-varieties to the dual Poisson-Lie group(G*,\pi_*) of (G,\pi_G) when (G,\pi_G) is a complex semi-simple Lie group of adjoint type, given with the standard Poisson…
We discuss the Lie Poisson groups structures associated to splittings of the loop group LGL(N), due to Sklyanin. Concentrating on the finite dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is…
We introduce the notion of tropicalization for Poisson structures on $\mathbb{R}^n$ with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a constant Poisson bracket on this cone. There is a…
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…
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized…
If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means: roughly, one has that the formal Poisson group…
Let $\Sigma $ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $\Sigma$ which are…
Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a…
In this paper, we study a family of compatible quadratic Poisson brackets on gl(n), generalizing the Sklyanin one. For any of the brackets in the family, the argument shift determines the compatible linear bracket. The main interest for us…
We introduce coupled double Poisson brackets on an associative algebra $A$ as pairs consisting of a generalized Van den Bergh's double Poisson bracket and a generalized Fairon--McCulloch's right double Poisson bracket subject to a…