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Journal of Lie Theory 31 (2021), No. 1, 063--092
Copyright Heldermann Verlag 2021

Flat Affine Symplectic Lie Groups

Fabricio Valencia
Instituto de Matemáticas, Universidad de Antioquia, Medellín 050010, Colombia

We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a flat affine symplectic Lie group with bi-invariant symplectic connection contains a nontrivial one parameter subgroup formed by central translations. We give two methods for constructing flat affine symplectic Lie groups, thus obtaining all those having bi-invariant symplectic connections. We get nontrivial examples of simply connected flat affine symplectic Lie groups in every even dimension.

Keywords: Flat affine symplectic structure, flat affine symplectic Lie group, bi-invariant symplectic connection, geodesic completeness.

MSC: 53D05, 53A15; 22E60, 22F30.

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