  In this work we study a particular class of Lie bialgebras arising from
Hermitian structures on Lie algebras such that the metric is ad-invariant. We
will refer to them as Lie bialgebras of complex type. These give rise to
Poisson Lie groups G whose corresponding duals G* are complex Lie groups. We
also prove that a Hermitian structure on the Lie algebra $\mathfrak{g}$ with
ad-invariant metric induces a structure of the same type on the double Lie
algebra ${\mathcal D}\mathfrak{g}= \mathfrak{g}\oplus\mathfrak{g}^*$, with
respect to the canonical ad-invariant metric of neutral signature on ${\mathcal
D}\mathfrak{g}$. We show how to construct a 2n-dimensional Lie bialgebra of
complex type starting with one of dimension 2(n-2). This allows us to determine
all solvable Lie algebras of dimension $\leq 6$ admitting a Hermitian structure
with ad-invariant metric. We exhibit some examples in dimension 4 and 6,
including two one-parameter families, where we identify the Lie-Poisson
structures on the associated simply connected Lie groups, obtaining also their
symplectic foliations.
