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 $\mathfrak{g}$. 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 $H_J^{\pm }(\mathfrak{g})$, 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.

References in corpus (3)

• On Cohomological Decomposition of Almost-Complex Manifolds and Deformations
• On the algebraic independence of Hamiltonian characteristic classes
• Special Hermitian metrics and Lie groups

Cited by in corpus (5)

• Special Hermitian metrics on compact solvmanifolds
• Symplectic, product and complex structures on 3-Lie algebras
• Cohomological aspects of non-Kähler manifolds
• Almost Kahler forms on rational 4-manifolds
• A note on tame/compatible almost complex structures on four-dimensional Lie algebras