Narrow Results

Let k be a field of characteristic 0, A a noncommutative Poisson k-algebra, U(A) the ordinary enveloping algebra of A, 𝒞 a quasi-Poisson A-coring that is projective as a left A-module, *𝒞 the left dual ring of 𝒞 (it is a right U(A)-module algebra) and Λ a right quasi-Poisson 𝒞-comodule that is finitely generated as a right U(A)#*𝒞-module. The vector space End^𝒫,𝒞(Λ) of right quasi-Poisson 𝒞-colinear maps from Λ to Λ is a ring. We give necessary and sufficient conditions for projectivity and flatness of a module over End^𝒫,𝒞(Λ). If 𝒞 contains a fixed quasi-Poisson grouplike element, we can replace Λ with A.

We introduce quasi-Poisson cohomology groups for a Poisson algebra, which can be computed by its quasi-Poisson complex. Moreover, there exists a Grothendieck spectral sequence relating quasi-Poisson cohomology to Hochschild cohomology and Lie algebra cohomology.