2006/10/12 by Adrián Andrada, A. Andrada, M. L. Barberis +5
Mathematics · Physics and Astronomy · #17B62 #53D17 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.DG #math.MP #math.QA #msc:17B62 #msc:53D17
paper · pdf · doi:10.48550/arxiv.math/0610415
arxiv created 2006/10/12 · openalex publication_date 2006/10/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathfrakg with ad-invariant metric induces a structure of the same type on the double Lie algebra \mathcal D\mathfrakg= \mathfrakg⊕\mathfrakg^*, with respect to the canonical ad-invariant metric of neutral signature on \mathcal D\mathfrakg. 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 ≤ 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.