Lie bialgebras of complex type and associated Poisson Lie groups

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dc.creatorAndrada, A.
dc.creatorBarberis, M. L.
dc.creatorOvando, G.
dc.date2006-10-12
dc.date.accessioned2026-07-07T07:29:02Z
dc.date.available2026-07-07T07:29:02Z
dc.descriptionIn 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.
dc.identifierhttps://arxiv.org/abs/math/0610415
dc.identifierhttp://arxiv.org/abs/math/0610415
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/117948
dc.subjectDifferential Geometry
dc.subjectMathematical Physics
dc.subjectQuantum Algebra
dc.subject17B62; 53D17
dc.titleLie bialgebras of complex type and associated Poisson Lie groups
dc.typetext

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