Poisson Varieties Research Articles

Stable envelopes for slices of the affine Grassmannian

The affine Grassmannian associated to a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}$$\\end{document} is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of Maulik and Okounkov (Astérisque 408:ix+209, 2019) in this family. We construct an explicit recursive relation for the stable envelopes in the G=PSL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}= \ extbf{PSL}_{2}$$\\end{document} case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.

The Cluster Complex for Cluster Poisson Varieties and Representations of Acyclic Quivers

Abstract Let $\mathcal{X}$ be a skew-symmetrizable cluster Poisson variety. The cluster complex $\Delta ^{+}(\mathcal{X})$ was introduced in [ 12]. It codifies the theta functions on $\mathcal{X}$ that restrict to a character of a seed torus. Every seed $\textbf{s}$ for $\mathcal{X}$ determines a fan realization $\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$ of $\Delta ^{+}(\mathcal{X})$. For every $\textbf{s}$ we provide a simple and explicit description of the cones of $\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$ and their facets using c-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$ in terms of $F$-polynomials. In case $\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to $\textbf{s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $\underline{\Delta ^{+}_{\textbf{s}}(\mathcal{X})}$ using g-vectors of (non-necessarily rigid) objects in $\textsf{K}^{\textrm{b}}(\textrm{proj} \; kQ)$.

Q-Painlevé equations on cluster Poisson varieties via toric geometry

We provide a relation between the geometric framework for q-Painlevé equations and cluster Poisson varieties by using toric models of rational surfaces associated with q-Painlevé equations. We introduce the notion of seeds of q-Painlevé type by the negative semi-definiteness of symmetric bilinear forms associated with seeds, and classify the mutation equivalence classes of these seeds. This classification coincides with the classification of q-Painlevé equations given by Sakai. We realize q-Painlevé systems as automorphisms on cluster Poisson varieties associated with seeds of q-Painlevé type.

Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

Abstract We develop techniques to construct isomorphisms between simple affine W-algebras and affine vertex algebras at admissible levels. We then apply these techniques to obtain many new, and conjecturally all, admissible collapsing levels for affine W-algebras. In short, if a simple affine W-algebra at a given level is equal to its affine vertex subalgebra generated by the centraliser of an ${\mathfrak {sl}}_2$ -triple associated with the underlying nilpotent orbit, then that level is said to be collapsing. Collapsing levels are important both in representation theory and in theoretical physics. Our approach relies on two fundamental invariants of vertex algebras. The first one is the associated variety, which, in the context of admissible level simple affine W-algebras, leads to the Poisson varieties known as nilpotent Slodowy slices. We exploit the singularities of these varieties to detect possible collapsing levels. The second invariant is the asymptotic datum. We prove a general result asserting that, under appropriate hypotheses, equality of asymptotic data implies isomorphism at the level of vertex algebras. Then we use this to give a sufficient criterion, of combinatorial nature, for an admissible level to be collapsing. Our methods also allow us to study isomorphisms between quotients of W-algebras and extensions of simple affine vertex algebras at admissible levels. Based on such examples, we are led to formulate a general conjecture: for any finite extension of vertex algebras, the induced morphism between associated Poisson varieties is dominant.

Quantization of Deformed Cluster Poisson Varieties

Quantization of Deformed Cluster Poisson Varieties

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

Bases for upper cluster algebras and tropical points

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland’s representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

Configuration Poisson Groupoids of Flags

Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

ON THE SINGULARITIES OF MISHCHENKO–FOMENKO SYSTEMS

To each complex semisimple Lie algebra \( \mathfrak{g} \) and regular element a ∈ \( \mathfrak{g} \)reg, one associates a Mishchenko–Fomenko subalgebra \( \mathcal{F} \)a ⊆ ℂ[\( \mathfrak{g} \)]. This subalgebra amounts to a completely integrable system on the Poisson variety \( \mathfrak{g} \), and as such has a bifurcation diagram Σa ⊆ Spec(\( \mathcal{F} \)a). We prove that Σa has codimension one in Spec(\( \mathcal{F} \)a) if a ∈ \( \mathfrak{g} \)reg is not nilpotent, and that it has codimension one or two if a ∈ \( \mathfrak{g} \)reg is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of Σa.

TAMARKIN–TSYGAN CALCULUS AND CHIRAL POISSON COHOMOLOGY

AbstractWe construct and study some vertex theoretic invariants associated with Poisson varieties, specializing in the conformal weight $0$ case to the familiar package of Poisson homology and cohomology. In order to do this conceptually, we sketch a version of the calculus, in the sense of [12], adapted to the context of vertex algebras. We obtain the standard theorems of Poisson (co)homology in this chiral context. This is part of a larger project related to promoting noncommutative geometric structures to chiral versions of such.

Brane Quantization of Toric Poisson Varieties

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 tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric K\"ahler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.

Fundamental Theorem of Poisson (A,H)-Hopf modules

Let k be a field, H a Hopf algebra with a bijective antipode and A an H-comodule Poisson algebra. We assume that there is an H-colinear algebra map from H to the Poisson-center of A. We generalize the Fundamental Theorem of (A,H)-Hopf modules to Poisson (A,H)-Hopf modules, and we deduce relative projectivity in the category of Poisson (A,H)-Hopf modules. In many applications, A could be a rational G-module Poisson algebra, where G is an affine algebraic group or the coordinate ring of an affine Poisson variety over C on which acts morphically an algebraic group via automorphisms of Poisson varieties.

Arc Spaces and Chiral Symplectic Cores

We introduce the notion of chiral symplectic cores in a vertex Poisson variety, which can be viewed as analogs of symplectic leaves in Poisson varieties. As an application we show that any quasi-lisse vertex algebra is a quantization of the arc space of its associated variety, in the sense that its reduced singular support coincides with the reduced arc space of its associated variety. We also show that the coordinate ring of the arc space of Slodowy slices is free over its vertex Poisson center, and the latter coincides with the vertex Poisson center of the coordinate ring of the arc space of the dual of the corresponding simple Lie algebra.

Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principal G-bundle on a surface. The moduli space is a Poisson variety with Atiyah–Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars–Schneider models.

Donaldson-Thomas transformation of Grassmannian

Let m and n be two integers such that 1<m<n−1. The configuration space Confn×(Pm−1), which is the moduli space of n points in the projective space Pm−1 satisfying certain linearly independent condition, is closely related to the Grassmannian Grm,n, and is birationally equivalent to the cluster Poisson variety XAm−1□An−m−1.A Donaldson-Thomas transformation is a special formal automorphism on a cluster Poisson variety and encodes the Donaldson-Thomas invariants of the moduli space of stability conditions on the associated 3d Calabi-Yau category. The existence of a cluster Donaldson-Thomas transformation is part of Gross-Hacking-Keel-Kontsevich's sufficient condition for the Fock-Goncharov cluster duality on the cluster ensemble. In this paper we construct the cluster Donaldson-Thomas transformation on the cluster Poisson variety XAm−1□An−m−1 and realize it as an isomorphism on the configuration space Confn×(Pm−1), proving a conjecture of Goncharov and Shen. Via our construction we also prove the Fock-Goncharov cluster duality conjecture on the associated cluster ensemble and obtain a periodicity result on the Donaldson-Thomas transformation associated to the Am−1□An−m−1 quiver.

Twistor space of a generalized quaternionic manifold

We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was initiated by Pantilie (Ann. Mat. Pura. Appl. 193 (2014) 633–641), and allows one to extend the Penrose correspondence from the quaternion to the generalized quaternion case. He showed that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space which comes naturally equipped with a tautological almost generalized complex structure. But he has left open the problem of the integrability. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperkähler manifolds using the generalized Bismut connection, introduced by Gualtieri (Branes on Poisson varieties, The many facets of geometry: a tribute to Nigel Hitchin (2010) (Oxford: Oxford University Press) pp. 368–395).

Deformations of nonsingular Poisson varieties and Poisson invertible sheaves

We study deformations of nonsingular Poisson varieties and Poisson invertible sheaves, which extend the flat deformation theory of nonsingular varieties and invertible sheaves. In the Appendix, we study deformations of Poisson vector bundles. We identify first-order deformations and obstructions.

A differential graded Lie algebra controlling the Poisson deformations of an affine Poisson variety

We construct a differential graded Lie algebra controlling the Poisson deformations of an affine Poisson variety. We analyze in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties.

Donaldson-Thomas transformation of double bruhat cells in semisimple Lie groups

Double Bruhat cells $G^{u,v}$ were studied by Fomin and Zelevinsky. They provide important examples of cluster algebras and cluster Poisson varieties. Cluster varieties produce examples of 3d Calabi-Yau categories with stability conditions, and their Donaldson-Thomas invariants, defined by Kontsevich and Soibelman, are encoded by a formal automorphism on the cluster variety known as the Donaldson-Thomas transformation. Goncharov and Shen conjectured in that for any semisimple Lie group $G$, the Donaldson-Thomas transformation of the cluster Poisson variety $H\backslash G^{u,v}/H$ is a slight modification of Fomin and Zelevinsky's twist map. In this paper we prove this conjecture, using crucially Fock and Goncharov's cluster ensembles and the amalgamation construction. Our result, combined with the work of Gross, Hacking, Keel, and Kontsevich, proves the duality conjecture of Fock and Goncharov in the case of $H\backslash G^{u,v}/H$.

Superintegrability of Generalized Toda Models on Symmetric Spaces

Abstract In this paper we prove superintegrability of Hamiltonian systems generated by functions on $K\backslash G/K$, restricted to a symplectic leaf of the Poisson variety $G/K$, where $G$ is a simple Lie group with the standard Poisson Lie structure, and $K$ is its subgroup of fixed points with respect to the Cartan involution.