Lie bialgebras of complex type and associated Poisson Lie groups

JOURNAL OF GEOMETRY AND PHYSICS

卷:58

Article

Andrada, A.¹  Barberis, M. L.¹  Ovando, G.² 

[1] Univ Nacl Cordoba, FAMAF, CIEM, RA-5000 Cordoba, Argentina

[2] Univ Nacl Rosario, CONICET, FCEIA, RA-2000 Rosario, Santa Fe, Argentina

关键词: Lie bialgebra;    Poisson Lie group;    Complex structure;    Ad-invariant metric;   

DOI  :  10.1016/j.geomphys.2008.05.006

来源: Elsevier

【 摘 要 】

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 Dg = g circle plus g*, with respect to the canonical ad-invariant metric of neutral signature on Do. We show how to construct a 2n-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. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free