Abelian complex structures on solvable Lie algebras

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$,where$A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\mathfrak{a}\mathfrak{f}\mathfrak{f}(\mathbb{C})$ and the corresponding Lie groups are complex affine manifolds. It turns out that all 4-dimensional Lie algebras carrying abelian complex structures are central extensions of such affine Lie algebras.