Related papers: Lattices in some Symplectic or Affine Nilpotent Li…

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This…

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…

We classify solvable Lie groups admitting left invariant symplectic half-flat structure. When the Lie group has a compact quotient by a lattice, we show that these structures provide solutions of supersymmetric equations of type IIA.

$k$-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant $k$-symplectic…

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…

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.…

We study left invariant contact forms and left invariant symplectic forms on Lie groups. We give the classification of all symplectic structures on nilpotent Lie algebras up the dimension 6.

A special symplectic Lie group is a triple $(G,\omega,\nabla)$ such that $G$ is a finite-dimensional real Lie group and $\omega$ is a left invariant symplectic form on $G$ which is parallel with respect to a left invariant affine structure…

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal K\"ahler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric…

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice…

We study the geometry of a family of Lie groups, which contained the classical affine Lie groups, endowed with an exact left invariant symplectic form. We show that this family is closed by symplectic reduction and symplectic double…

We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre…

Let $\Gamma$ be a lattice in a simply-connected nilpotent Lie group $N$ whose Lie algebra $\mathfrak{n}$ is $p$-filiform. We show that $\Gamma$ is either abelian or 2-step nilpotent if $\Gamma$ is isomorphic to the fundamental group of a…

We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal K\"ahler metrics and all the structures…

Let $(G,\Omega)$ be a symplectic Lie group, i.e, a Lie group endowed with a left invariant symplectic form. If $\G$ is the Lie algebra of $G$ then we call $(\G,\omega=\Om(e))$ a symplectic Lie algebra. The product $\bullet$ on $\G$ defined…

Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent…

We derive explicit formulae for the subalgebra zeta functions of all higher Heisenberg Lie algebras over an arbitrary compact discrete valuation ring $\mathfrak{o}$. To this end, we develop Hecke-theoretic techniques for the enumeration, by…

We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie…

We study the hard Lefschetz property on compact symplectic solvmanifolds, i.e., compact quotients $M=\Gamma\backslash G$ of a simply-connected solvable Lie group $G$ by a lattice $\Gamma$, admitting a symplectic structure.

We are interested in the classification of left-invariant symplectic structures on Lie groups. Some classifications are known, especially in low dimensions. In this paper we establish a new approach to classify (up to automorphism and…