3,979 research outputs found
Lie algebroid structures on a class of affine bundles
We introduce the notion of a Lie algebroid structure on an affine bundle
whose base manifold is fibred over the real numbers. It is argued that this is
the framework which one needs for coming to a time-dependent generalization of
the theory of Lagrangian systems on Lie algebroids. An extensive discussion is
given of a way one can think of forms acting on sections of the affine bundle.
It is further shown that the affine Lie algebroid structure gives rise to a
coboundary operator on such forms. The concept of admissible curves and
dynamical systems whose integral curves are admissible, brings an associated
affine bundle into the picture, on which one can define in a natural way a
prolongation of the original affine Lie algebroid structure.Comment: 28 page
Construction of completely integrable systems by Poisson mappings
Pulling back sets of functions in involution by Poisson mappings and adding
Casimir functions during the process allows to construct completely integrable
systems. Some examples are investigated in detail.Comment: AmsTeX, 9 page
Completely integrable systems: a generalization
We present a slight generalization of the notion of completely integrable
systems to get them being integrable by quadratures. We use this generalization
to integrate dynamical systems on double Lie groups.Comment: Latex, 15 page
Generalized n-Poisson brackets on a symplectic manifold
On a symplectic manifold a family of generalized Poisson brackets associated
with powers of the symplectic form is studied. The extreme cases are related to
the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can
be obtained in a similar way.Comment: Latex, 10 pages, to appear in Mod. Phys. Lett.
Modular classes of skew algebroid relations
Skew algebroid is a natural generalization of the concept of Lie algebroid.
In this paper, for a skew algebroid E, its modular class mod(E) is defined in
the classical as well as in the supergeometric formulation. It is proved that
there is a homogeneous nowhere-vanishing 1-density on E* which is invariant
with respect to all Hamiltonian vector fields if and only if E is modular, i.e.
mod(E)=0. Further, relative modular class of a subalgebroid is introduced and
studied together with its application to holonomy, as well as modular class of
a skew algebroid relation. These notions provide, in particular, a unified
approach to the concepts of a modular class of a Lie algebroid morphism and
that of a Poisson map.Comment: 20 page
The Structure of Conserved Charges in Open Spin Chains
We study the local conserved charges in integrable spin chains of the XYZ
type with nontrivial boundary conditions. The general structure of these
charges consists of a bulk part, whose density is identical to that of a
periodic chain, and a boundary part. In contrast with the periodic case, only
charges corresponding to interactions of even number of spins exist for the
open chain. Hence, there are half as many charges in the open case as in the
closed case. For the open spin-1/2 XY chain, we derive the explicit expressions
of all the charges. For the open spin-1/2 XXX chain, several lowest order
charges are presented and a general method of obtaining the boundary terms is
indicated. In contrast with the closed case, the XXX charges cannot be
described in terms of a Catalan tree pattern.Comment: 22 pages, harvmac.tex (minor clarifications and reference corrections
added
Modular classes of Poisson-Nijenhuis Lie algebroids
The modular vector field of a Poisson-Nijenhuis Lie algebroid is defined
and we prove that, in case of non-degeneracy, this vector field defines a
hierarchy of bi-Hamiltonian -vector fields. This hierarchy covers an
integrable hierarchy on the base manifold, which may not have a
Poisson-Nijenhuis structure.Comment: To appear in Letters in Mathematical Physic
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