Maximal commutative subalgebras, Poisson geometry and Hochschild homology

TL;DR

This paper explores the relationship between maximal commutative subalgebras, Poisson geometry, and Hochschild homology, introducing a spectral sequence that links these concepts and computes Hochschild homology in this context.

Contribution

It introduces a spectral sequence for Hochschild homology associated with maximal commutative subalgebras and Poisson structures, providing explicit computations of its E^{2} groups.

Findings

Spectral sequence converges to Hochschild homology with specific coefficients.

E^{2} groups are computed using canonical homology and Poisson modules.

The approach depends on the choice of a maximal commutative subalgebra.

Abstract

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E^{2}_{p,q}-groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and a maximal commutative subalgebra.