Lie bialgebras of complex type and associated Poisson Lie groups

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 GG whose corresponding duals G∗G∗ are complex Lie groups. We also prove that a Hermitian structure on gg with ad-invariant metric induces a structure of the same type on the double Lie algebra Dg=g⊕g∗Dg=g⊕g∗, with respect to the canonical ad-invariant metric of neutral signature on DgDg. We show how to construct a 2n2n-dimensional Lie bialgebra of complex type starting with one of dimension 2(n−2),n≥22(n−2),n≥2. This allows us to determine all solvable Lie algebras of dimension ≤6 admitting a Hermitian structure with ad-invariant metric. We present some examples in dimensions 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.