Strongly r-matrix induced tensors, Koszul cohomology, and arbitrary-dimensional quadratic Poisson cohomology

TL;DR

This paper introduces strongly r-matrix induced Poisson structures, explores their cohomology via Koszul methods, and classifies structures within the Dufour-Haraki framework, providing a linear algebra approach to compute Poisson cohomology.

Contribution

It defines SRMI Poisson structures, develops a cohomological decomposition, and applies the method to classify and compute cohomology of specific structures.

Findings

Decomposition of Poisson cohomology into Koszul and relative cohomology.

Connection between Koszul cohomology and Spectral Theory.

Linear algebra reduction for cohomology computation.

Abstract

We introduce the concept of strongly $r$-matrix induced ({\small SRMI}) Poisson structure, report on the relation of this property with the stabilizer dimension of the considered quadratic Poisson tensor, and classify the Poisson structures of the Dufour-Haraki classification (DHC) according to their membership of the family of {\small SRMI} tensors. One of the main results of our work is a generic cohomological procedure for {\small SRMI} Poisson structures in arbitrary dimension. This approach allows decomposing Poisson cohomology into, basically, a Koszul cohomology and a relative cohomology. Moreover, we investigate this associated Koszul cohomology, highlight its tight connections with Spectral Theory, and reduce the computation of this main building block of Poisson cohomology to a problem of linear algebra. We apply these upshots to two structures of the DHC and provide an exhaustive description of their cohomology. We thus complete our list of data obtained in previous works, see \cite{MP} and \cite{AMPN}, and gain fairly good insight into the structure of Poisson cohomology.