Almost Kähler structures on four dimensional unimodular Lie algebras

Let JJ be an almost complex structure on a 44-dimensional and unimodular Lie algebra gg. We show that there exists a symplectic form taming JJ if and only if there is a symplectic form compatible with JJ. We also introduce groups HJ+(g) and HJ−(g) as the subgroups of the Chevalley–Eilenberg cohomology classes which can be represented by JJ-invariant, respectively JJ-anti-invariant, 22-forms on gg. and we prove a cohomological JJ-decomposition theorem following Draˇghici et al. (2010) [12]: H2(g)=HJ+(g)⊕HJ−(g). We discover that tameness of JJ can be characterized in terms of the dimension of HJ±(g), just as in the complex surface case. We also describe the tamed and compatible symplectic cones. Finally, two applications to homogeneous JJ on 44-manifolds are obtained.