A note on tame/compatible almost complex structures on four-dimensional Lie algebras

TL;DR

This paper classifies four-dimensional Lie algebras based on their ability to admit tame and compatible almost complex structures, providing new examples of structures that are tamed but not compatible.

Contribution

It offers a complete description of four-dimensional Lie algebras satisfying the tame-compatible question for all almost complex structures, including new examples of non-unimodular cases.

Findings

Classified four-dimensional Lie algebras satisfying the tame-compatible question

Constructed examples of non-unimodular Lie algebras with tamed but not compatible structures

Identified conditions under which almost complex structures are tame or compatible

Abstract

Four-dimensional, oriented Lie algebras $\mathfrak{g}$ which satisfy the tame-compatible question of Donaldson for all almost complex structures $J$ on $\mathfrak{g}$ are completely described. As a consequence, examples are given of (non-unimodular) four-dimensional Lie algebras with almost complex structures which are tamed but not compatible with symplectic forms.