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

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.