Classification of Almost Abelian Lie Groups Admitting Left-Invariant Complex or Symplectic Structures

Abstract

We classify the almost abelian Lie algebras gA = Re0 ⋉A R 2n−1 admitting complex or symplectic structures. The matrix A ∈ M(2n − 1, R) encodes the adjoint action of e0 on the abelian ideal R 2n−1 , and the existence of complex or symplectic structures on gA 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 gA admits a complex structure, then gA 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.