| We introduce the concept of strongly r-matrix induced (SRMI) Poisson structure, report on the ralation of this proprerty 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 cohomology procedure for classifying SRMI Poisson structures in arbitrary dimension. This approch allows the decomposition of Poisson cohomology into , basically, a Koszul cohomology and 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 aproblem 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. |
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