37 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
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
Classification of irreducible weight modules over -algebra W(2,2)
We show that the support of an irreducible weight module over the -algebra
, which has an infinite dimensional weight space, coincides with the
weight lattice and that all nontrivial weight spaces of such a module are
infinite dimensional. As a corollary, we obtain that every irreducible weight
module over the the -algebra , having a nontrivial finite
dimensional weight space, is a Harish-Chandra module (and hence is either an
irreducible highest or lowest weight module or an irreducible module of the
intermediate series).Comment: 10 page
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
Lattice algebras and quantum groups
We represent Feigin's construction [22] of lattice W algebras and give some
simple results: lattice Virasoro and algebras. For simplest case
we introduce whole quantum group on this lattice. We
find simplest two-dimensional module as well as exchange relations and define
lattice Virasoro algebra as algebra of invariants of . Another
generalization is connected with lattice integrals of motion as the invariants
of quantum affine group . We show that Volkov's scheme leads
to the system of difference equations for the function from non-commutative
variables.Comment: 13 pages, misprints have been correcte
Geometry of W-algebras from the affine Lie algebra point of view
To classify the classical field theories with W-symmetry one has to classify
the symplectic leaves of the corresponding W-algebra, which are the
intersection of the defining constraint and the coadjoint orbit of the affine
Lie algebra if the W-algebra in question is obtained by reducing a WZNW model.
The fields that survive the reduction will obey non-linear Poisson bracket (or
commutator) relations in general. For example the Toda models are well-known
theories which possess such a non-linear W-symmetry and many features of these
models can only be understood if one investigates the reduction procedure. In
this paper we analyze the SL(n,R) case from which the so-called W_n-algebras
can be obtained. One advantage of the reduction viewpoint is that it gives a
constructive way to classify the symplectic leaves of the W-algebra which we
had done in the n=2 case which will correspond to the coadjoint orbits of the
Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov
algebra. Our method in principle is capable of constructing explicit
representatives on each leaf. Another attractive feature of this approach is
the fact that the global nature of the W-transformations can be explicitly
described. The reduction method also enables one to determine the ``classical
highest weight (h. w.) states'' which are the stable minima of the energy on a
W-leaf. These are important as only to those leaves can a highest weight
representation space of the W-algebra be associated which contains a
``classical h. w. state''.Comment: 17 pages, LaTeX, revised 1. and 7. chapter