Hessel Posthuma

Homology of formal deformations of proper étale Lie groupoids

Abstract In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a non-commutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.

On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

Abstract In our recent paper, we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of partial differential equations (PDEs) associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin and Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental–Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we give an alternative derivation using a deformation formula for the weak quasi-Miura transformation that relates our hierarchy of PDEs with its dispersionless limit.

The cyclic theory of Hopf algebroids

We give a systematic description of the cyclic cohomology theory of Hopf alge\-broids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and etale groupoids.

Orbifold cup products and ring structures on Hochschild cohomologies

In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an

Deformations of semisimple poisson pencils of hydrodynamic type are unobstructed

S^1

Bihamiltonian Cohomology of KdV Brackets

-equivariant version of the Chen--Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology.

The bi-Hamiltonian cohomology of a scalar Poisson pencil

We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

On the Algebraic Index for Riemannian Étale Groupoids

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular, this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.

A polynomial bracket for the Dubrovin-Zhang hierarchies

We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that

Geometry of orbit spaces of proper Lie groupoids

BH^p_d(\hat{F}, d_1,d_2)