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

Mourad Ammar, Guy Kass, Mohsen Masmoudi and Norbert Poncin

Vol. 245 (2010), No. 1, 1–23

DOI: 10.2140/pjm.2010.245.1

Abstract

We introduce the concept of strongly r-matrix induced (SRMI) Poisson structure, report on the relation of this property to the stabilizer dimension of the considered quadratic Poisson tensor, and classify the Poisson structures of the Dufour–Haraki classification (DHC) according to their membership in the family of SRMI tensors. A main result is a generic cohomological procedure for classifying SRMI Poisson structures in arbitrary dimension. This approach allows the decomposition of Poisson cohomology into, basically, a Koszul cohomology and a relative cohomology. Also 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 work, and gain fairly good insight into the structure of Poisson cohomology.

Keywords

r-matrix, quadratic Poisson structure, Poisson cohomology, Lichnerowicz–Poisson cohomology, Koszul cohomology, relative cohomology, spectral theory

Mathematical Subject Classification 2000

Primary: 17B56, 17B63, 55N99

Milestones

Received: 20 January 2009

Revised: 29 May 2009

Accepted: 23 June 2009

Published: 20 January 2010