Almost Kähler structures on four dimensional unimodular Lie algebras

Abstract

Let J be an almost complex structure on a 4-dimensional and unimodular Lie algebra g. We show that there exists a symplectic form taming J if and only if there is a symplectic form compatible with J. We also introduce groups HJ+(g) and HJ−(g) as the subgroups of the Chevalley–Eilenberg cohomology classes which can be represented by J-invariant, respectively J-anti-invariant, 2-forms on g. and we prove a cohomological J-decomposition theorem following Draˇghici et al. (2010) [12]: H2(g)=HJ+(g)⊕HJ−(g). We discover that tameness of J 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 J on 4-manifolds are obtained.