Left-invariant symplectic structures on diagonal almost abelian Lie groups*

Abstract

We are interested in the classification or finding conditions for the existence of left-invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. We approach this problem by studying the “moduli space of left-invariant nondegenerate $2$-forms”, which is a certain orbit space in the set of all nondegenerate $2$-forms on a Lie algebra. In this paper, using this approach, we give a classification of left-invariant symplectic structures on all almost abelian Lie algebras determined by diagonal matrices.