We define a graded twisted-coassociative coproduct on the tensor algebra TW of any Z n -graded vector space W . If W is the desuspension space ↓ V of a graded vector space V , the coderivations (resp. quadratic "degree 1" codifferentials, arbitrary odd codifferentials) of this coalgebra are 1to-1 with sequences π s , s ≥ 1, of s-linear maps on V (resp. Z n -graded Loday structures on V , sequences that we call Loday infinity structures on V ). We prove a minimal model theorem for Loday infinity algebras, investigate Loday infinity morphisms, and observe that the Lod ∞ category contains the L ∞ category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie "stem" bracket on the cochain spaces of graded Loday, Loday infinity, and 2n-ary graded Loday algebras (the latter extend the corresponding Lie algebras in the sense of Michor and Vinogradov). These algebraic structures have square zero with respect to the stem bracket, so that we obtain natural cohomological theories that have good properties with respect to formal deformations. The stem bracket restricts to the graded Nijenhuis-Richardson andup to isomorphism-to the Grabowski-Marmo brackets (the last bracket extends the Schouten-Nijenhuis bracket to the space of graded antisymmetric first order polydifferential operators), and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of 2n-ary graded Lie algebras.

One defines the notion of universal deformation quantization: given any manifold M, any Poisson structure Λ on M and any torsionfree linear connection ∇ on M, a universal deformation quantization associates to this data a star product on (M, Λ) given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor Λ, the curvature tensor R and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poisson cohomology. Mathematics Subject Classifications (2000): 53D55, 81S10

Quadratic Poisson tensors of the Dufour-Haraki classification read as a sum of an r-matrix induced structure twisted by a (small) compatible exact quadratic tensor. An appropriate bigrading of the space of formal Poisson cochains then leads to a vertically positive double complex. The associated spectral sequence allows to compute the Poisson-Lichnerowicz cohomology of the considered tensors. We depict this modus operandi, apply our technique to concrete examples of twisted Poisson structures, and obtain a complete description of their cohomology. As richness of Poisson cohomology entails computation through the whole spectral sequence, we detail an entire model of this sequence. Finally, the paper provides practical insight into the operating mode of spectral sequences.

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.