Gabriela P. Ovando

Naturally reductive pseudo-Riemannian spaces

Abstract A family of naturally reductive pseudo-Riemannian spaces is constructed out of the representations of Lie algebras with ad-invariant metrics. We exhibit peculiar examples, study their geometry and characterize the corresponding naturally reductive homogeneous structure.

Weak mirror symmetry of complex symplectic Lie algebras

Abstract A complex symplectic structure on a Lie algebra 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 h is a semi-direct product g ⋉ V , and both g and V are Lagrangian with respect to Ω and totally real with respect to J . This note shows that g ⋉ 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 g ⋉ V is also shown to be equivalent to that of a torsion-free flat symplectic connection on the Lie algebra g . By further exploring a relation between ( J , Ω ) with hypersymplectic Lie algebras, we find an inductive process to build families of complex symplectic algebras of dimension 8 n from the data of the 4 n -dimensional ones.

Small oscillations and the Heisenberg Lie algebra

The Adler Kostant Symes (A-K-S) scheme is used to describe mechanical systems for quadratic Hamiltonians of R 2n on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra g that admits an ad- invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on R 2n . This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in g is related to the commutativity of a family of derivations of the 2n+1-dimensional Heisenberg Lie algebra hn. Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in hn. For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.

Extending invariant complex structures

We study the problem of extending a complex structure to a given Lie algebra 𝔤, which is firstly defined on an ideal 𝔥 ⊂ 𝔤. We consider the next situations: 𝔥 is either complex or it is totally real. The next question is to equip 𝔤 with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either 𝔥 is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of 𝔤. Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

From almost (para)-complex structures to affine structures on Lie groups

Let

On first integrals of the geodesic flow on Heisenberg nilmanifolds

G=Hltimes K

Hamiltonian systems related to invariant Metrics

G=H⋉K denote a semidirect product Lie group with Lie algebra