On Cohomological Decomposability of Almost-K\"ahler Structures

We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-K"ahler structure (J, \omega, g). In particular, almost-K"ahler manifolds satisfying a Lefschetz type property, and solvmanifolds endowed with left-invariant almost-complex structures are investigated.