788 research outputs found
Locally conformal symplectic nilmanifolds with no locally conformal K\"ahler metrics
We obtain an example of a compact locally conformal symplectic nilmanifold
which admits no locally conformal K\"ahler metrics. This gives a new positive
answer to a question raised by L. Ornea and M. Verbitsky.Comment: 7 pages, no figures. Comments are welcome
Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations
Given a Lie-Poisson completely integrable bi-Hamiltonian system on
, we present a method which allows us to construct, under certain
conditions, a completely integrable bi-Hamiltonian deformation of the initial
Lie-Poisson system on a non-abelian Poisson-Lie group of dimension
, where is the deformation parameter. Moreover, we
show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on
that underly the dynamics of the deformed system and by making use of
the group law on , one may obtain two completely integrable Hamiltonian
systems on . By construction, both systems admit
reduction, via the multiplication in , to the deformed bi-Hamiltonian
system in . The previous approach is applied to two relevant
Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler
top systems.Comment: 23 pages, 2 figures. Revised versio
Reduction of Poisson-Nijenhuis Lie algebroids to symplectic-Nijenhuis Lie algebroids with nondegenerate Nijenhuis tensor
We show how to reduce, under certain regularities conditions, a
Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with
nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi
for the reduction of Poisson-Nijenhuis manifolds. The choice of the more
general framework of Lie algebroids is motivated by the geometrical study of
some reduced bi-Hamiltonian systems. An explicit example of reduction of a
Poisson-Nijenhuis Lie algebroid is also provided.Comment: 35 pages, final version to appear in J. Phys. A: Math. Theo
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