Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants

TL;DR

This paper proves Shoikhet's conjecture relating Poisson and Hochschild homology via tangent isomorphism, leading to a Duflo isomorphism on coinvariants, advancing understanding of deformation quantization.

Contribution

It establishes Shoikhet's conjecture that the tangent isomorphism on homology is an algebra module isomorphism, and derives a Duflo isomorphism on coinvariants.

Findings

Proves Shoikhet's conjecture on homology isomorphism.

Shows the tangent isomorphism respects module structures.

Derives a Duflo isomorphism on coinvariants.

Abstract

In this paper we prove a conjecture of B. Shoikhet. This conjecture states that the tangent isomorphism on homology, between the Poisson homology associated to a Poisson structure on $\mathbb{R}^d$ and the Hochschild homology of its quantized star-product algebra, is an isomorphism of modules over the (isomorphic) respective cohomology algebras. As a consequence, we obtain a version of the Duflo isomorphism on coinvariants.