Some restrictions on existence of abelian complex structures

We describe the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. More precisely, we consider a distinguished class of Lie algebras admitting abelian complex structures given by abelian double products. The structure of these Lie algebras can be described in terms of a pair of commutative associative algebras satisfying a compatibility condition. We will show that when $g$ is a Lie algebra with an abelian complex structure $J$, and $g$ decomposes as $g=u+Ju$, with $u$ an abelian subalgebra, then $g$ is an abelian double product.
Joint work with A. Andrada and M. L. Barberis.