On the Lichnerowicz-Poisson cohomology complex and quasi-Poisson algebras

This article explores F-commutative algebra (A,·) with Poisson structure defined by a non-Lie algebra bivector Q. We analyse the Lichnerowicz-Poisson coboundary δ*Q and its implications as a quasi-Poisson structure, where the Schouten bracket does not vanish, complicating cohomology computations. The examined open question is whether the deviating term is a coboundary of Poisson cohomology. Our main objectives are to construct a Poisson subalgebra on which the twisted term δ2Q(Q) vanishes, and to study the existence of Poisson structures P on a Poisson ideal satisfying the condition δ2P(·) = δ2Q(Q).