Related papers: Totally nonnegative cells and matrix Poisson varie…
We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of…
We study the equivalence of Poisson structures around a given symplectic leaf of nonzero dimension. Some criteria of Poisson equivalence are derived from a homotopy argument for coupling Poisson structures. In the case when the transverse…
We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave…
We introduce a natural nondegeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds are privileged by the fact that their deformation…
This paper investigates the Poisson geometry associated to a cluster algebra over the complex numbers, and its relationship to compatible torus actions. We show, under some assumptions, that each Noetherian cluster algebra has only finitely…
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…
Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all maximal by inclusion weakly separated collections of minors have the…
This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of…
Zamolodchikov periodicity is a property of certain discrete dynamical systems and was one of the primary motivations for the creation of cluster algebras. It was first observed by Zamolodchikov in his study of thermodynamic Bethe ansatz,…
We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact…
We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in…
Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the…
We study the totally nonnegative part of the Peterson variety in arbitrary Lie type and establish its connection to the strongly dominant weight polytope. In particular, we prove that the totally nonnegative part of the Peterson variety is…
Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory.…
We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables…
Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B \supset T$. For an arbitrary automorphism $\theta$ of $G$, we introduce a holomorphic Poisson structure $\pi_\theta$ on $G$ which is…
We study totally nonnegative parts of critical varieties in the Grassmannian. We show that each totally nonnegative critical variety Crit$^{\ge0}_f$ is the image of an affine poset cyclohedron under a continuous map and use this map to…
Fundamental representations of real simple Poisson Lie groups are Poisson actions with a suitable choice of the Poisson structure on the underlying (real) vector space. We study these (mostly quadratic) Poisson structures and corresponding…
We present and expand some existing results on the Zariski closure of cyclic groups and semigroups of matrices. We show that, with the exclusion of isolated points, their irreducible components are toric varieties. Additionally, we…
Postnikov constructed a decomposition of a totally nonnegative Grassmannian into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point belongs to and to determine affine coordinates of the…