Classification of almost abelian Lie groups admitting left-invariant complex or symplectic structures

TL;DR

This paper classifies almost abelian Lie algebras based on their ability to admit complex or symplectic structures, revealing that nilpotent cases with complex structures also admit symplectic structures, with broader implications.

Contribution

It provides a classification of almost abelian Lie algebras admitting complex or symplectic structures, especially focusing on the nilpotent case and the restrictions on the matrix A.

Findings

Nilpotent A implies complex structures also admit symplectic structures.

Classification reduces to analyzing Jordan normal form of A.

Several consequences derived from the classification theorems.

Abstract

We classify the almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0 \ltimes_A \mathbb R^{2n-1}$ admitting complex or symplectic structures. The matrix $A\in M(2n-1,\mathbb R )$ encodes the adjoint action of $e_0$ on the abelian ideal $\mathbb R^{2n-1}$, and the existence of complex or symplectic structures on $\mathfrak g_A$ imposes restrictions on the Jordan normal form of $A$. The classification essentially reduces to the case when $A$ is nilpotent, so we start by considering this case. It turns out that if $A$ is nilpotent and $\mathfrak g_A$ admits a complex structure, then $\mathfrak g_A$ necessarily admits a symplectic structure. This is not true in general when $A$ is non-nilpotent. Finally, several consequences of the classification theorems are obtained.