Poisson Varieties Research Articles

On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

Abstract The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$, where $G$ is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic $p> 3$, $U$ is the uniradical of a Borel subgroup of $G$, and $\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$. Then we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.

Bott–Samelson Varieties and Poisson Ore Extensions

Abstract We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

Complete families of commuting functions for coisotropic Hamiltonian actions

Let G be an algebraic group defined over a field F of characteristic zero with g=LieG. The dual space g⁎ equipped with the Lie–Poisson bracket is a Poisson variety and each irreducible G-invariant subvariety X⊂g⁎ carries the induced Poisson structure. We prove that there is a set {f1,...,fl}⊂F[X] of algebraically independent polynomial functions, which pairwise commute with respect to the Poisson bracket, such that l=(dim⁡X+tr.degF(X)G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varieties of corank zero and 2.

Hessenberg varieties, Slodowy slices, and integrable systems

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group $G$ and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $X(H_0)$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko--Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $X(H_0)$. We also show $X(H_0)$ to have an open dense symplectic leaf isomorphic to $G/Z \times S_{\text{reg}}$, where $Z$ is the centre of $G$ and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. This allows us to invoke results about integrable systems on $G\times S_{\text{reg}}$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.

On the local structure of quantizations in characteristic p

Let $A$ be a central quantization of an affine Poisson variety $X$ over a field of characteristic $p>0.$ We show that the completion of $A$ with respect to a closed point $y\in X$ is isomorphic to the tensor product of the Weyl algebra with a local Poisson algebra. This result can be thought of as a positive characteristic analogue of results of Losev and Kaledin about slice algebras of quantizations in characteristic 0.

Poisson traces, D-modules, and symplectic resolutions

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.

Constructions and classifications of projective Poisson varieties

This paper is intended both as an introduction to the algebraic geometry of holomorphic Poisson brackets, and as a survey of results on the classification of projective Poisson manifolds that have been obtained in the past 20 years. It is based on the lecture series delivered by the author at the Poisson 2016 Summer School in Geneva. The paper begins with a detailed treatment of Poisson surfaces, including adjunction, ruled surfaces and blowups, and leading to a statement of the full birational classification. We then describe several constructions of Poisson threefolds, outlining the classification in the regular case, and the case of rank-one Fano threefolds (such as projective space). Following a brief introduction to the notion of Poisson subspaces, we discuss Bondal’s conjecture on the dimensions of degeneracy loci on Poisson Fano manifolds. We close with a discussion of log symplectic manifolds with simple normal crossings degeneracy divisor, including a new proof of the classification in the case of rank-one Fano manifolds.

Donaldson–Thomas transformations of moduli spaces of G-local systems

Kontsevich and Soibelman defined Donaldson–Thomas invariants of a 3d Calabi–Yau category equipped with a stability condition [41]. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Given a stability condition, the DT-transformation allows to recover the DT-invariants.Let S be an oriented surface with punctures and a finite number of special points on the boundary, considered modulo isotopy. It gives rise to a moduli space XPGLm,S, closely related to the moduli space of PGLm-local systems on S, which carries a canonical cluster Poisson variety structure [13]. For each puncture of S, there is a birational Weyl group action on the space XPGLm,S. We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution ⁎ of XPGLm,S induced by dualizing a local system on S.Let μ be the total number of punctures and special points, and g(S) the genus of S. We assume that μ>0. We say that S is admissible if either g(S)+μ≥3 and μ>1 when S has only punctures, or S is an annulus with a special point on each boundary circle.Using a combinatorial characterization of a class of DT transformations due to B. Keller [38], we describe the DT-transformation of the space XPGLm,S for any admissible S.We show that the Weyl group and the involution ⁎ act by cluster transformations of the dual moduli space ASLm,S, and describe the DT-transformation of the space ASLm,S.If S admissible, combining our work with the work of Gross, Hacking, Keel and Kontsevich [35] we get a canonical basis in the space of regular functions on the cluster variety XPGLm,S, and in the Fomin–Zelevinsky upper cluster algebra with principal coefficients [23] related to the pair (SLm,S), as predicted by Duality Conjectures [15].

Poisson enveloping algebras and the Poincaré–Birkhoff–Witt theorem

Poisson algebras are, just like Lie algebras, particular cases of Lie–Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper, where he also introduces the notion of an enveloping algebra and proves — under some mild conditions — that the enveloping algebra of a Lie–Rinehart algebra satisfies a Poincaré–Birkhoff–Witt theorem (PBW theorem). In the case of a Poisson algebra (A,⋅,{⋅,⋅}) over a commutative ring R (with unit), Rinehart's result boils down to the statement that if A is smooth (as an algebra), then gr(U(A)) and Sym(Ω(A)) are isomorphic as graded algebras; in this formula, U(A) stands for the Poisson enveloping algebra of A and Ω(A) is the A-module of Kähler differentials of A (viewing A as an R-algebra). In this paper, we give several new constructions of the Poisson enveloping algebra in some general and in some particular contexts. Moreover, we show that for an important class of singular Poisson algebras, the PBW theorem still holds. In geometrical terms, these Poisson algebras correspond to (singular) Poisson hypersurfaces of arbitrary smooth affine Poisson varieties. Throughout the paper we give several examples and present some first applications of the main theorem; applications to deformation theory and to Poisson and Hochschild (co-) homology will be worked out in a future publication.

Derivations of quantizations in characteristic p

Let k be an algebraically closed field of odd characteristic. We describe derivations of a large class of quantizations of affine normal Poisson varieties over k.

Poisson deleting derivations algorithm and Poisson spectrum

ABSTRACTCauchon [5] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.

Cluster Poisson varieties at infinity

A positive space is a space with a positive atlas, i.e., a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes Thurston’s compactification of a Teichmuller space. A cluster Poisson variety, originally called cluster $${\mathcal X}$$ -variety [2], is covered by a collection of coordinate tori $$({{\mathbb C}}^*)^n$$ , which form a positive atlas of a specific kind. We define a special completion $$\widehat{\mathcal X}$$ of $${\mathcal X}$$ . It has a stratification whose strata are cluster Poisson varieties. The coordinate tori of $${\mathcal X}$$ extend to coordinate affine spaces $${\mathbb A}^n$$ in $$\widehat{\mathcal X}$$ . We define completions of Teichmuller spaces for decorated surfaces $${\mathbb S}$$ with marked points at the boundary. The set of positive points of the special completion of the cluster Poisson variety $${\mathcal X}_{ PGL _2, {\mathbb S}}$$ related to the Teichmuller theory on $${\mathbb S}$$ [1] is a part of the completion of the Teichmuller space (see Fig. 1 on the next page).

Equivariant Slices for Symplectic Cones

The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqueness, study examples (quotient singularities and hypertoric varieties), and applications to noncommutative algebra (their quantization). We also pose some natural questions on existence and quantization of C*-actions on slices to conical symplectic leaves.

Co-invariants of Lie algebras of vector fields on algebraic varieties

We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural D-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us.

Hochschild Cohomology of Deformation Quantizations over ℤ/p n ℤ

Let A 1 be an Azumaya algebra over a smooth affine symplectic variety X over Spec F p , where p is an odd prime. Let A be a deformation quantization of A 1 over the p-adic integers. In this note we show that for all n ≥ 1, the Hochschild cohomology of A/p n A is isomorphic to the de Rham-Witt complex \(W_{n}{\Omega }^{\ast }_{X}\) of X over \(\mathbb {Z}/p^{n}\mathbb {Z}\). We also compute the center of deformations of certain affine Poisson varieties over F p .

Vanishing cycles on Poisson varieties

We slightly extend results of Evens and Mirkovic and “compute” the characteristic cycles of intersection cohomology sheaves on transversal slices in a double affine Grassmannian. We propose a conjecture relating the hyperbolic stalks and microlocalization at a torus-fixed point in a Poisson variety.

Motion in a Symmetric Potential on the Hyperbolic Plane

AbstractWe study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.

Poisson varieties from Riemann surfaces

Short survey based on talk at the Poisson 2012 conference. The main aim is to describe and give some examples of wild character varieties (naturally generalising the character varieties of Riemann surfaces by allowing more complicated behaviour at the boundary), their Poisson/symplectic structures (generalising both the Atiyah–Bott approach and the quasi-Hamiltonian approach), and the wild mapping class groups.

Geometry and braiding of Stokes data; Fission and wild character varieties

A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.

Isomorphisms of quantizations via quantization of resolutions

In this paper, we prove the existence of isomorphisms between certain non-commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain Poisson algebras. We show that quantizations of Kleinian singularities obtained by three different constructions are isomorphic to each other. The constructions are via symplectic reflection algebras, quantum Hamiltonian reduction, and W-algebras. Next, we prove that parabolic W-algebras in type A are isomorphic to quantum Hamiltonian reductions associated to quivers of type A. Finally, we show that the symplectic reflection algebras for wreath-products of the symmetric group and a Kleinian group are isomorphic to certain quantum Hamiltonian reductions. Our results involving W-algebras are new, while for those dealing with symplectic reflection algebras we just find new proofs. A key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate Poisson varieties.