  We study the structure of Lie groups admitting left invariant abelian complex
structures in terms of commutative associative algebras. If, in addition, the
Lie group is equipped with a left invariant Hermitian structure, it turns out
that such a Hermitian structure is K\"ahler if and only if the Lie group is the
direct product of several copies of the real hyperbolic plane by a euclidean
factor. Moreover, we show that if a left invariant Hermitian metric on a Lie
group with an abelian complex structure has flat first canonical connection,
then the Lie group is abelian.
