Almost K\"ahler structures on four dimensional unimodular Lie algebras

TL;DR

This paper investigates the conditions under which almost complex structures on 4-dimensional unimodular Lie algebras admit taming and compatible symplectic forms, introducing cohomological decompositions and applications to homogeneous structures.

Contribution

It establishes an equivalence between taming and compatible symplectic forms, introduces cohomological groups for J-invariant forms, and characterizes tameness via cohomology dimensions.

Findings

Taming and compatibility are equivalent for these structures.

Cohomological J-decomposition theorem for H^2(g).

Characterization of tameness through dimensions of H_J^±(g).

Abstract

Let $J$ be an almost complex structure on a 4-dimensional and unimodular Lie algebra $\mathfrak{g}$. 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 $H^+_J(\mathfrak{g})$ and $H^-_J(\mathfrak{g})$ as the subgroups of the Chevalley-Eilenberg cohomology classes which can be represented by $J$-invariant, respectively $J$-anti-invariant, 2-forms on $\mathfrak{g}$. and we prove a cohomological $J-$decomposition theorem following \cite{DLZ}: $H^2(\mathfrak{g})=H^+_J(\mathfrak{g})\oplus H^-_J(\mathfrak{g})$. We discover that tameness of $J$ can be characterized in terms of the dimension of $H^{\pm}_J(\mathfrak{g})$, just as in the complex surface case. We also describe the tamed and compatible symplectic cones respectively. Finally, two applications to homogeneous $J$ on 4-manifolds are obtained.