106 research outputs found
On Gelfand-Zetlin modules
summary:[For the entire collection see Zbl 0742.00067.]\par Let {\germ g}\sb k be the Lie algebra {\germ gl}(k,\mathcal{C}), and let U\sb k be the universal enveloping algebra for {\germ g}\sb k. Let Z\sb k be the center of U\sb k. The authors consider the chain of Lie algebras {\germ g}\sb n\supset {\germ g}\sb{n-1}\supset\dots\supset {\germ g}\sb 1. Then Z=\langle Z\sb k\mid k=1,2,\dots n\rangle is an associative algebra which is called the Gel'fand-Zetlin subalgebra of U\sb n. A {\germ g}\sb n module is called a -module if V=\sum\sb x\oplus V(x), where the summation is over the space of characters of and V(x)=\{v\in V\mid(a-x(a))\sp mv=0, m\in\mathcal{Z}\sb +, . The authors describe several properties of - modules. For example, they prove that if for some and the module is simple, then is a -module. Indecomposable - modules are also described. The authors give three conjectures on - modules and
On the Classification of Automorphic Lie Algebras
It is shown that the problem of reduction can be formulated in a uniform way
using the theory of invariants. This provides a powerful tool of analysis and
it opens the road to new applications of these algebras, beyond the context of
integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras
associated to the icosahedral group I, the octahedral group O, the tetrahedral
group T, and the dihedral group Dn are isomorphic. The proof is based on
techniques from classical invariant theory and makes use of Clebsch-Gordan
decomposition and transvectants, Molien functions and the trace-form. This
result provides a complete classification of sl2-Automorphic Lie Algebras
associated to finite groups when the group representations are chosen to be the
same and it is a crucial step towards the complete classification of
Automorphic Lie Algebras.Comment: 29 pages, 1 diagram, 9 tables, standard LaTeX2e, submitted for
publicatio
Discrete Lagrangian systems on the Virasoro group and Camassa-Holm family
We show that the continuous limit of a wide natural class of the
right-invariant discrete Lagrangian systems on the Virasoro group gives the
family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and
Korteweg-de Vries equations. This family has been recently derived by Khesin
and Misiolek as Euler equations on the Virasoro algebra for
-metrics. Our result demonstrates a universal nature of
these equations.Comment: 6 pages, no figures, AMS-LaTeX. Version 2: minor changes. Version 3:
minor change
Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy
The lattice Gel'fand-Dikii hierarchy was introduced by Nijhoff, Papageorgiou,
Capel and Quispel in 1992 as the family of partial difference equations
generalizing to higher rank the lattice Korteweg-de Vries systems, and includes
in particular the lattice Boussinesq system. We present a Lagrangian for the
generic member of the lattice Gel'fand-Dikii hierarchy, and show that it can be
considered as a Lagrangian 2-form when embedded in a higher dimensional
lattice, obeying a closure relation. Thus the multiform structure proposed in
arXiv:0903.4086v2 [nlin.SI] is extended to a multi-component system.Comment: 12 page
Cohomology of groups of diffeomorphims related to the modules of differential operators on a smooth manifold
Let be a manifold and be the cotangent bundle. We introduce a
1-cocycle on the group of diffeomorphisms of with values in the space of
linear differential operators acting on When is the
-dimensional sphere, , we use this 1-cocycle to compute the
first-cohomology group of the group of diffeomorphisms of , with
coefficients in the space of linear differential operators acting on
contravariant tensor fields.Comment: arxiv version is already officia
Cohomology of the Lie Superalgebra of Contact Vector Fields on and Deformations of the Superspace of Symbols
Following Feigin and Fuchs, we compute the first cohomology of the Lie
superalgebra of contact vector fields on the (1,1)-dimensional
real superspace with coefficients in the superspace of linear differential
operators acting on the superspaces of weighted densities. We also compute the
same, but -relative, cohomology. We explicitly give
1-cocycles spanning these cohomology. We classify generic formal
-trivial deformations of the -module
structure on the superspaces of symbols of differential operators. We prove
that any generic formal -trivial deformation of this
-module is equivalent to a polynomial one of degree .
This work is the simplest superization of a result by Bouarroudj [On
(2)-relative cohomology of the Lie algebra of vector fields and
differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127].
Further superizations correspond to -relative cohomology
of the Lie superalgebras of contact vector fields on -dimensional
superspace
A spinor-like representation of the contact superconformal algebra K'(4)
In this work we construct an embedding of a nontrivial central extension of
the contact superconformal algebra K'(4) into the Lie superalgebra of
pseudodifferential symbols on the supercircle S^{1|2}. Associated with this
embedding is a one-parameter family of spinor-like tiny irreducible
representations of K'(4) realized just on 4 fields instead of the usual 16.Comment: 19 pages, TeX. Corrections to the references in the paper to be
published in J. Math. Phys. v 42, no 1, 200
Magnetic hydrodynamics with asymmetric stress tensor
In this paper we study equations of magnetic hydrodynamics with a stress
tensor. We interpret this system as the generalized Euler equation associated
with an abelian extension of the Lie algebra of vector fields with a
non-trivial 2-cocycle. We use the Lie algebra approach to prove the energy
conservation law and the conservation of cross-helicity
Contact complete integrability
Complete integrability in a symplectic setting means the existence of a
Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we
describe complete integrability in a contact set-up as a more subtle structure:
a flag of two foliations, Legendrian and co-Legendrian, and a
holonomy-invariant transverse measure of the former in the latter. This turns
out to be equivalent to the existence of a canonical
structure on the leaves of the co-Legendrian foliation. Further, the above
structure implies the existence of contact fields preserving a special
contact 1-form, thus providing the geometric framework and establishing
equivalence with previously known definitions of contact integrability. We also
show that contact completely integrable systems are solvable in quadratures. We
present an example of contact complete integrability: the billiard system
inside an ellipsoid in pseudo-Euclidean space, restricted to the space of
oriented null geodesics. We describe a surprising acceleration mechanism for
closed light-like billiard trajectories
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