The main aim of this paper is the description of a large class of lattices in some nilpotent Lie ... more The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiforms, carrying a flat left invariant linear connection and often a left invariant symplectic form. As a consequence we obtain an infinity of, non homemorphic, compact affine or symplectic nilmanifolds. We review some new facts about the geometry of compact symplectic nilmanifolds and we describe symplectic reduction for these manifolds. For the Heisenberg-Lie group, defined over a local associative and commutative finite dimensional real algebra, a necessary and sufficient condition for the existence of a left invariant symplectic form, is given. Finally in the symplectic case we show that a lattice in the group determines naturally lattices in the double Lie group corresponding to any solution of the classical Yang-Baxter equation.
This paper deals essentially with affine or projective transformations of Lie groups endowed with... more This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. We give necessary and sufficient conditions for the existence of flat left invariant projective structures on Lie groups. We also determine Lie groups admitting flat bi-invariant affine or projective structures. These groups could play an essential role in the study of homogeneous spaces M = G/H having a flat affine or flat projective structures invariant under the natural action of G on M . A. Medina asked several years ago if the group of affine transformations of a flat affine Lie group is a flat projective Lie group. In this work we provide a partial positive answer to this question.
This paper deals with affine connections on real manifolds. We give a new characterization of fla... more This paper deals with affine connections on real manifolds. We give a new characterization of flat affine connections on real manifolds by means of certain affine representations of the Lie group of automorphisms preserving the connection. Then we specialize the characterization to the case of a left invariant connection on a Lie group. In the last case, we show the existence of a Lie group endowed with a flat affine bi-invariant connection whose Lie algebra contains the Lie algebra of complete infinitesimal affine transformations of the given Lie group. We also prove some results about flat affine manifolds whose group of diffeomorphisms admit a flat affine bi-invariant structure. The paper is illustrated with several examples.
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannia... more To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannian metric. We give a useful necessary and sufficient condition that guaranties the flatness of a left invariant semi Riemannian metric defined on a quadratic Lie group. All these semi Riemannian metrics are complete. We show that there are no Riemannian or Lorentzian flat left invariant metrics on non Abelian quadratic Lie groups, and that every quadratic 3 step nilpotent Lie group admits a flat left invariant semi Riemannian metric. The case of quadratic 2 step nilpotent Lie groups is also addressed.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same ti... more We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T *-extension of a nilpotent algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra. Finally, we prove that every symplectic quadratic Lie algebra is a special symplectic Manin algebra and we give an inductive classification in terms of symplectic quadratic double extensions.
We give a characterization of flat affine connections on manifolds by means of a natural affine r... more We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental form of the bundle of linear frames of the manifold. We show that the group of affine transformations of a real flat affine n-dimensional manifold, acts on R n leaving an open orbit when its dimension is greater than n. Moreover, when the dimension of the group of affine transformations is n, this orbit has discrete isotropy. For any given Lie subgroup H of affine transformations of the manifold, we show the existence of an associative envelope of the Lie algebra of H, relative to the connection. The case when M is a Lie group and H acts on G by left translations is particularly interesting. We also exhibit some results about flat affine manifolds whose group of affine transformations admits a flat affine bi-invariant structure. The paper is illustrated with several examples.
Symmetry, Integrability and Geometry: Methods and Applications
In this work it is shown that a necessary condition for the completeness of the geodesics of left... more In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.
In this paper we study the geometry of oscillator groups: they are the only non commutative simpl... more In this paper we study the geometry of oscillator groups: they are the only non commutative simply connected solvable Lie groups which have a biinvariant Lorentzian metric. We first study curvature and geodesics, and then give a full analysis of lattices - i.e. discrete co-compact subgroups - getting examples of compact Lorentzian homogeneous varieties.
We describe the structure of the Lie groups endowed with a leftinvariant symplectic form, called ... more We describe the structure of the Lie groups endowed with a leftinvariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber.This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We specialize our principal results to the cases of simply connected Hamiltonian symplectic nilpotent Lie groups or Frobenius symplectic Lie groups. Finally we pursue the study of the classical affine Lie group as a symplectic Lie group.
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Papers by Alberto MEDINA