arXiv subjects

Gabriela P. Ovando

Publications and source records attributed to Gabriela P. Ovando.

24 records · Page 2Linked to original sources

arXiv2010

Naturally reductive pseudo Riemannian 2-step nilpotent Lie groups

This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved here that the simply connected Lie group $N$ can be constructed starting from a real representation $(π,\vv)$ of a certain Lie algebra $\ggo$. We study the geometry of $(N, \la \,,\,\ra_N)$ and we find the corresponding naturally reductive homogeneous structure. On the other hand, related to the case of degenerate center we provide another family of naturally reductive spaces, both non compact and also compact examples.

math.DG↗

arXiv2010

Weak Mirror Symmetry of Complex Symplectic Algebras

A complex symplectic structure on a Lie algebra $\lie h$ is an integrable complex structure $J$ with a closed non-degenerate $(2,0)$-form. It is determined by $J$ and the real part $Ω$ of the $(2,0)$-form. Suppose that $\lie h$ is a semi-direct product $\lie g\ltimes V$, and both $\lie g$ and $V$ are Lagrangian with respect to $Ω$ and totally real with respect to $J$. This note shows that $\lie g\ltimes V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of $Ω$ and $J$ are isomorphic. The geometry of $(Ω, J)$ on the semi-direct product $\lie g\ltimes V$ is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra $\lie g$. By further exploring a relation between $(J, Ω)$ with hypersymplectic algebras, we find an inductive process to build families of complex symplectic algebras of dimension $8n$ from the data of the $4n$-dimensional ones.

math.DG↗

arXiv2009

Complex structures on tangent and cotangent Lie algebras of dimension six

This paper deals with complex structures on Lie algebras $\ct_π \hh=\hh \ltimes_π V$, where $π$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\ct_π \hh$ for $\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constrain $J\hh=V$. Whenever $π$ is the adjoint representation this kind of complex structures are associated to non singular derivations of $\hh$. This fact derives different kind of applications. Finally an approach to the pseudo Kähler geometry was done.

math.DG↗

arXiv2008

The Adler Kostant Symes scheme in physics

The purpose of this material is to review the Adler Kostant Symes scheme as a theory which can be developped succesfully in different contexts. It was useful to describe some mechanical systems, the so called generalized Toda, and now it was proved to be a tool for the study of the linear approach to the motion of n uncoupled harmonic oscillators. The complete integrability of these systems has an algebraic description. In the original theory this is related to ad-invariant functions, but new examples show that new conditions should be investigated.

math-ph↗

arXiv2004

Invariant pseudo Kaehler metrics in dimension four

Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,ω)$ are parametrized up to complex isomorphism (where $J$ is a complex structure and $ω$ is a symplectic structure). Such structure gives rise to a pseudo Riemannian metric $g$ for which $J$ is parallel. It is proved that most of these complex homogeneous spaces admit a pseudo Kähler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular Kähler Lie algebras are flat in dimension four. Other algebraic and geometric features are treated. A general construction of Ricci flat pseudo Kähler structures in higher dimensions on some affine Lie algebras is given. Walker and hypersymplectic metrics on Lie algebras are compared.

math.DG↗

arXiv2004

Four dimensional symplectic Lie algebras

In this paper we deal with symplectic Lie algebras. All symplectic structures are determined for dimension four and the corresponding Lie algebras are classified up to equivalence. Symplectic four dimensional Lie algebras are described either as solutions of the cotangent extension problem or as symplectic double extension of $\Bbb RR^2$ by $\Bbb RR$. The difference in the choice of a certain model lies on the existence or not of a lagrangian ideal. Moreover all extensions of a two dimensional Lie algebra are determined, and so all solutions (up to equivalence) of the cotangent extension problem are given in dimension two. By studying the adjoint representation we generalize to higher dimensions finding obstructions to the existence of symplectic forms. Finally, as an appendix we compute the authomorphisms of four dimensional symplectic Lie algebras and the cohomology over $\Bbb RR$ of the solvable real four dimensional Lie algebras.

math.DG↗