Lie bialgebras of complex type and associated Poisson Lie groups

• Published in: Journal of geometry and physics 2008-10, Vol.58 (10), p.1310-1328
• Main Authors: Andrada, A., Barberis, M.L., Ovando, G.
• Format: Article
• Language: English
• Subjects: Ad-invariant metric; Complex structure; Lie bialgebra; Poisson Lie group
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/j.geomphys.2008.05.006

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 g with ad-invariant metric induces a structure of the same type on the double Lie algebra D g = g ⊕ g ∗ , with respect to the canonical ad-invariant metric of neutral signature on D g . We show how to construct a 2 n -dimensional Lie bialgebra of complex type starting with one of dimension 2 ( 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.