Poisson Homology Research Articles

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.

Poisson cohomology, Koszul duality, and Batalin–Vilkovisky algebras

We study the noncommutative Poincaré duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich’s deformation quantization as well as Koszul duality preserve the corresponding Poincaré duality. As a corollary, the Batalin– Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.

On a Batalin–Vilkovisky operator generating higher Koszul brackets on differential forms

We introduce a formal $\hbar$-differential operator $\Delta$ that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a $P_{\infty}$-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in \texttt{arXiv:1808.10049}. (This operator is an analogue of the Koszul--Brylinski boundary operator $\partial_P$ which defines Poisson homology for an ordinary Poisson structure.) Here we introduce $\Delta=\Delta_P$ by a different method and establish its properties. We show that this BV type operator generating higher Koszul brackets can be included in a one-parameter family of BV type formal $\hbar$-differential operators, which can be understood as a quantization of the cotangent $L_{\infty}$-bialgebroid. We obtain symmetric description on both $\Pi TM$ and $\Pi T^*M$. For the purpose of the above, we develop in detail a theory of formal $\hbar$-differential operators and also of operators acting on densities on dual vector bundles. In particular, we have a statement about operators that can be seen as a quantization of the Mackenzie--Xu canonical diffeomorphism. Another interesting feature is that we are able to introduce a grading, not a filtration, on our algebras of operators. When operators act on objects on vector bundles, we obtain a bi-grading.

A note on the duality between Poisson homology and cohomology

For a Poisson algebra A, by studying its universal enveloping algebra Ape, we prove a duality theorem between Poisson homology and cohomology of A.

Poisson (Co)homology of a Class of Frobenius Poisson Algbras

In this paper, we study the truncated polynomial algebra L in n variables, and discuss the following four problems in detail: 1) Homology complex and homology group of Poisson algebra L; 2) Given a new Poisson bracket by calculation modular derivation of Frobenius Poisson algebra; 3) Calculate the twisted homology group of Poisson algebra L; 4) Verify the theorem of twisted Poincaré duality between twisted Poisson homology and Poisson Cohomology.

Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds

We prove that for regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group and we give some applications. In particular, we show that for regular unimodular Poisson manifolds top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what is a perfect Poisson manifold. We use these Poisson homology computations to provide families of perfect Poisson manifolds.

Twisted Poisson Homology of Truncated Polynomial Algebras in Four Variables

In this paper, we study the twisted Poisson homology of truncated polynomials algebra A in four variables, and we calculate exactly the dimension of i-th (i = 1, 2, 3, 4) twisted Poisson homology group over A by the induction on the length. The calculation methods provided in this paper can also solve truncated polynomials algebra in a few variables.

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.

Filtrations on Springer fiber cohomology and Kostka polynomials

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

Homological unimodularity and Calabi–Yau condition for Poisson algebras

In this paper, we show that the twisted Poincar\'e duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincar\'e duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi-Yau algebra if the Poisson structure is unimodular.

Twisted Poincaré duality for Poisson homology and cohomology of affine Poisson algebras

This paper investigates the Poisson (co)homology of affine Poisson algebras. It is shown that there is a twisted Poincaré duality between their Poisson homology and cohomology. The relation between the Poisson (co)homology of an affine Poisson algebra and the Hochschild (co)homology of its deformation quantization is also discussed, which is similar to Kassel’s result (1988) for homology and is a special case of Kontsevich’s theorem (2003) for cohomology.

Invariants of Hamiltonian flow on locally complete intersections

We consider the Hamiltonian flow on complex complete intersection surfaces with isolated singularities, equipped with the Jacobian Poisson structure. More generally we consider complete intersections of arbitrary dimension equipped with Hamiltonian flow with respect to the natural top polyvector field, which one should view as a degenerate Calabi–Yau structure. Our main result computes the coinvariants of functions under the Hamiltonian flow. In the surface case this is the zeroth Poisson homology, and our result generalizes those of Greuel, Alev and Lambre, and the authors in the quasihomogeneous and formal cases. Its dimension is the sum of the dimension of the top cohomology and the sum of the Milnor numbers of the singularities. In other words, this equals the dimension of the top cohomology of a smoothing of the variety. More generally, we compute the derived coinvariants, which replaces the top cohomology by all of the cohomology. Still more generally we compute the \({\mathcal{D}}\)-module which represents all invariants under Hamiltonian flow, which is a nontrivial extension (on both sides) of the intersection cohomology \({\mathcal{D}}\)-module, which is maximal on the bottom but not on the top. For cones over smooth curves of genus g, the extension on the top is the holomorphic half of the maximal extension.

Twisted Poincaré duality between Poisson homology and Poisson cohomology

A version of the twisted Poincaré duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincaré duality reduces to the Poincaré duality in usual sense. The main result generalizes the work of Launois and Richard [8] for the quadratic Poisson structures and Zhu [25] for the linear Poisson structures.

On (co)homology of Frobenius Poisson algebras

AbstractIn this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

Hypertoric Poisson homology in degree zero

Etingof and Schedler formulated a conjecture about the degree zero Poisson homology of an ane cone that admits a projective symplectic resolution. We strengthen this conjecture in general and prove the strengthened version for hypertoric varieties. We also formulate an analogous conjecture for the degree zero Hochschild homology of a quantization of such a variety.

Smooth structures on pseudomanifolds with isolated conical singularities

In this note we introduce the notion of a smooth structure on a conical pseudomanifold M in terms of C ∞-rings of smooth functions on M. For a finitely generated smooth structure C ∞(M) we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of M, and the notion of characteristic classes of M. We prove the vanishing of a Nash vector field at a singular point for a special class of Euclidean smooth structures on M. We introduce the notion of a conical symplectic form on M and show that it is smooth with respect to a Euclidean smooth structure on M. If a conical symplectic structure is also smooth with respect to a compatible Poisson smooth structure C ∞(M), we show that its Brylinski–Poisson homology groups coincide with the de Rham homology groups of M. We show nontrivial examples of these smooth conical symplectic-Poisson pseudomanifolds.

Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Let X be a surface with an isolated singularity at the origin, given by the equation Q(x,y,z)=0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = C^2/G for G < SL(2,C) finite. Let Y be the n-th symmetric power of X. We compute the zeroth Poisson homology of Y, as a graded vector space with respect to the weight grading. In the Kleinian case, this confirms a conjecture of Alev, that the zeroth Poisson homology of the n-th symmetric power of C^2/G is isomorphic to the zeroth Hochschild homology of the n-th symmetric power of the algebra of G-invariant differential operators on C. That is, the Brylinski spectral sequence degenerates in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, for the parameter equal to all but countably many points of the elliptic curve.

Homological properties of certain Generalized Jacobian Poisson Structures in dimension 3

The unimodularity condition for a Poisson structure (i.e., a Poisson structure with a trivial modular class) induces a Poincaré duality between its Poisson homology and its Poisson cohomology. Therefore, information about the Poisson homology of this kind Poisson structures induces by duality information about its Poisson cohomology and vise versa. However, it is not longer true in the case of a non-trivial modular class, which is the case of Generalized Jacobian Poisson Structures (GJPS). In this paper, certain GJPS are considered in dimension 3 and the properties of their Poisson homological groups and their Poisson cohomological groups are obtained. More precisely, under some assumptions, the Poincaré series of these Poisson homological groups are obtained, and these Poisson cohomological groups are computed explicitly, except the second group, which seems to be more complicated to obtain.

Poisson homology in degree 0 for some rings of symplectic invariants

Let g be a finite-dimensional semi-simple Lie algebra, h a Cartan subalgebra of g , and W its Weyl group. The group W acts diagonally on V : = h ⊕ h ∗ , as well as on C [ V ] . The purpose of this article is to study the Poisson homology of the algebra of invariants C [ V ] W endowed with the standard symplectic bracket. To begin with, we give general results about the Poisson homology space in degree 0, denoted by HP 0 ( C [ V ] W ) , in the case where g is of type B n − C n or D n , results which support Alev's conjecture. Then we are focusing the interest on the particular cases of ranks 2 and 3, by computing the Poisson homology space in degree 0 in the cases where g is of type B 2 ( so 5 ), D 2 ( so 4 ), then B 3 ( so 7 ), and D 3 = A 3 ( so 6 ≃ sl 4 ). In order to do this, we make use of a functional equation introduced by Y. Berest, P. Etingof and V. Ginzburg. We recover, by a different method, the result established by J. Alev and L. Foissy, according to which the dimension of HP 0 ( C [ V ] W ) equals 2 for B 2 . Then we calculate the dimension of this space and we show that it is equal to 1 for D 2 . We also calculate it for the rank 3 cases, we show that it is equal to 3 for B 3 − C 3 and 1 for D 3 = A 3 .

Poisson (co)homology of polynomial Poisson algebras in dimension four: Sklyanin's case

In this paper, we compute the Poisson (co)homology of a polynomial Poisson structure given by two Casimir polynomial functions which define a complete intersection with an isolated singularity.