Almost Kähler structures on four dimensional unimodular Lie algebras (Q423701)

The main subject of this article is the study of tameness of almost complex structures on unimodular four-dimensional Lie algebras. First, the authors address the ``tame to compatible'' question of \textit{S. K. Donaldson} [Nankai Tracts Math. 11, 153--172 (2006; Zbl 1140.58018)]. Let \(J\) be an almost complex structure on a \(4\)-dimensional unimodular Lie algebra \(\mathfrak{g}\). They show that in this situation there is a symplectic form taming \(J\) if and only if there is a symplectic form compatible with \(J\). Then, they prove a decomposition of the Chevalley-Eilenberg cohomology group \(H^2(\mathfrak{g})=H^+_J(\mathfrak{g})\oplus H^-_J(\mathfrak{g})\) in terms of \(J\)-invariant and anti-invariant forms, as in [\textit{T. Draghici} et al., Int. Math. Res. Not. 2010, No. 1, 1--17 (2010; Zbl 1190.32021)]. Tameness of \(J\) is characterized by the dimension of \(H^-_J(\mathfrak{g})\). Finally, a description of tamed and compatible symplectic cones is given. The authors provide examples to illustrate their results and give applications to compact quotients of \(4\)-dimensional Lie groups.