85 research outputs found
On the structure of the B\"acklund transformations for the relativistic lattices
The B\"acklund transformations for the relativistic lattices of the Toda type
and their discrete analogues can be obtained as the composition of two duality
transformations. The condition of invariance under this composition allows to
distinguish effectively the integrable cases. Iterations of the B\"acklund
transformations can be described in the terms of nonrelativistic lattices of
the Toda type. Several multifield generalizations are presented
Higher Dimensional Classical W-Algebras
Classical -algebras in higher dimensions are constructed. This is achieved
by generalizing the classical Gel'fand-Dickey brackets to the commutative limit
of the ring of classical pseudodifferential operators in arbitrary dimension.
These -algebras are the Poisson structures associated with a higher
dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless
KP-hierarchy). The two dimensional case is worked out explicitly and it is
shown that the role of Diff is taken by the algebra of generators of
local diffeomorphisms in two dimensions.Comment: 22 pages, Plain TeX, KUL-TF-92/19, US-FT/6-9
Poisson-Lie group of pseudodifferential symbols
We introduce a Lie bialgebra structure on the central extension of the Lie
algebra of differential operators on the line and the circle (with scalar or
matrix coefficients). This defines a Poisson--Lie structure on the dual group
of pseudodifferential symbols of an arbitrary real (or complex) order. We show
that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP
Poisson structures are naturally realized as restrictions of this Poisson
structure to submanifolds of this ``universal'' Poisson--Lie group.
Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in
physical terminology) can be viewed as subspaces of the quotient (or Poisson
reduction) of this Poisson--Lie group by the dressing action of the group of
functions.
Finally, we define an infinite set of functions in involution on the
Poisson--Lie group that give the standard families of Hamiltonians when
restricted to the submanifolds mentioned above. The Poisson structure and
Hamiltonians on the whole group interpolate between the Poisson structures and
Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical
meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes
to 0.Comment: 64 pages, no figure
Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction
The matrix version of the -KdV hierarchy has been recently
treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian
symmetry reduction applied to a Poisson submanifold in the dual of the Lie
algebra . Here a
series of extensions of this matrix Gelfand-Dickey system is derived by means
of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra
using the natural
embedding for any positive integer. The
hierarchies obtained admit a description in terms of a matrix
pseudo-differential operator comprising an -KdV type positive part and a
non-trivial negative part. This system has been investigated previously in the
case as a constrained KP system. In this paper the previous results are
considerably extended and a systematic study is presented on the basis of the
Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson
brackets and makes clear the conformal (-algebra) structures related to
the KdV type hierarchies. Discrete reductions and modified versions of the
extended -KdV hierarchies are also discussed.Comment: 60 pages, plain TE
A One-Parameter Family of Hamiltonian Structures for the KP Hierarchy and a Continuous Deformation of the Nonlinear \W_{\rm KP} Algebra
The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson
structures obtained from a generalized Adler map in the space of formal
pseudodifferential symbols with noninteger powers. The resulting \W-algebra
is a one-parameter deformation of \W_{\rm KP} admitting a central extension
for generic values of the parameter, reducing naturally to \W_n for special
values of the parameter, and contracting to the centrally extended
\W_{1+\infty}, \W_\infty and further truncations. In the classical limit,
all algebras in the one-parameter family are equivalent and isomorphic to
\w_{\rm KP}. The reduction induced by setting the spin-one field to zero
yields a one-parameter deformation of \widehat{\W}_\infty which contracts to
a new nonlinear algebra of the \W_\infty-type.Comment: 31 pages, compressed uuencoded .dvi file, BONN-HE-92/20, US-FT-7/92,
KUL-TF-92/20. [version just replaced was truncated by some mailer
Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
We construct affinization of the algebra of ``complex size''
matrices, that contains the algebras for integral values of the
parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra
results in the quadratic Gelfand--Dickey structure on the
Poisson--Lie group of all pseudodifferential operators of fractional order.
This construction is extended to the simultaneous deformation of orthogonal and
simplectic algebras that produces self-adjoint operators, and it has a
counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure
Quantum and Classical Integrable Systems
The key concept discussed in these lectures is the relation between the
Hamiltonians of a quantum integrable system and the Casimir elements in the
underlying hidden symmetry algebra. (In typical applications the latter is
either the universal enveloping algebra of an affine Lie algebra, or its
q-deformation.) A similar relation also holds in the classical case. We discuss
different guises of this very important relation and its implication for the
description of the spectrum and the eigenfunctions of the quantum system.
Parallels between the classical and the quantum cases are thoroughly discussed.Comment: 59 pages, LaTeX2.09 with AMS symbols. Lectures at the CIMPA Winter
School on Nonlinear Systems, Pondicherry, January 199
Non-Local Matrix Generalizations of W-Algebras
There is a standard way to define two symplectic (hamiltonian) structures,
the first and second Gelfand-Dikii brackets, on the space of ordinary linear
differential operators of order , . In this paper, I consider in detail the case where the are
-matrix-valued functions, with particular emphasis on the (more
interesting) second Gelfand-Dikii bracket. Of particular interest is the
reduction to the symplectic submanifold . This reduction gives rise to
matrix generalizations of (the classical version of) the {\it non-linear}
-algebras, called -algebras. The non-commutativity of the
matrices leads to {\it non-local} terms in these -algebras. I show
that these algebras contain a conformal Virasoro subalgebra and that
combinations of the can be formed that are -matrices of
conformally primary fields of spin , in analogy with the scalar case .
In general however, the -algebras have a much richer structure than
the -algebras as can be seen on the examples of the {\it non-linear} and
{\it non-local} Poisson brackets of any two matrix elements of or
which I work out explicitly for all and . A matrix Miura transformation
is derived, mapping these complicated second Gelfand-Dikii brackets of the
to a set of much simpler Poisson brackets, providing the analogue of the
free-field realization of the -algebras.Comment: 43 pages, a reference and a remark on the conformal properties for
adde
Schwinger Terms and Cohomology of Pseudodifferential Operators
We study the cohomology of the Schwinger term arising in second quantization
of the class of observables belonging to the restricted general linear algebra.
We prove that, for all pseudodifferential operators in 3+1 dimensions of this
type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a
modified version of the Radul cocycle arising in non-commutative differential
geometry. In the process we also show how the ordinary Radul cocycle for any
pair of pseudodifferential operators in any dimension can be written as the
phase space integral of the star commutator of their symbols projected to the
appropriate asymptotic component.Comment: 19 pages, plain te
Higher spin quaternion waves in the Klein-Gordon theory
Electromagnetic interactions are discussed in the context of the Klein-Gordon
fermion equation. The Mott scattering amplitude is derived in leading order
perturbation theory and the result of the Dirac theory is reproduced except for
an overall factor of sixteen. The discrepancy is not resolved as the study
points into another direction. The vertex structures involved in the scattering
calculations indicate the relevance of a modified Klein-Gordon equation, which
takes into account the number of polarization states of the considered quantum
field. In this equation the d'Alembertian is acting on quaternion-like plane
waves, which can be generalized to representations of arbitrary spin. The
method provides the same relation between mass and spin that has been found
previously by Majorana, Gelfand, and Yaglom in infinite spin theories
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