Flat Affine and Symplectic Geometries on Lie Groups

TL;DR

This paper explores flat affine and symplectic structures on Lie groups, particularly focusing on the double Lie group of the oscillator Lie group, and investigates affine symplectomorphisms and Hess connections.

Contribution

It constructs new flat affine structures on specific Lie groups and analyzes affine symplectomorphisms and Hess connections within this geometric framework.

Findings

Existence of flat left invariant affine structures on the double Lie group of the oscillator Lie group.

Identification of affine symplectomorphisms for certain affine symplectic connections.

Construction of the Hess connection associated with Lagrangian bi-foliations.

Abstract

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 for 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.