Open Access
2017 Flat Affine and Symplectic Geometries on Lie Groups
Andrés Villabón
J. Geom. Symmetry Phys. 46: 95-121 (2017). DOI: 10.7546/jgsp-46-2017-95-121


In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina. On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of $\mathbb{R}^n$. In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe by Medina-Saldarriaga-Giraldo, we determine the affine symplectomorphisms. Finally we exhibit the Hess connection, associated to a Lagrangian bi-foliation, which is flat left invariant affine.


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Andrés Villabón. "Flat Affine and Symplectic Geometries on Lie Groups." J. Geom. Symmetry Phys. 46 95 - 121, 2017.


Published: 2017
First available in Project Euclid: 14 February 2018

MathSciNet: MR3791933
Digital Object Identifier: 10.7546/jgsp-46-2017-95-121

Rights: Copyright © 2017 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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