340 research outputs found
Integrable Systems and Factorization Problems
The present lectures were prepared for the Faro International Summer School
on Factorization and Integrable Systems in September 2000. They were intended
for participants with the background in Analysis and Operator Theory but
without special knowledge of Geometry and Lie Groups. In order to make the main
ideas reasonably clear, I tried to use only matrix algebras such as
and its natural subalgebras; Lie groups used are either GL(n)
and its subgroups, or loop groups consisting of matrix-valued functions on the
circle (possibly admitting an extension to parts of the Riemann sphere). I hope
this makes the environment sufficiently easy to live in for an analyst. The
main goal is to explain how the factorization problems (typically, the matrix
Riemann problem) generate the entire small world of Integrable Systems along
with the geometry of the phase space, Hamiltonian structure, Lax
representations, integrals of motion and explicit solutions. The key tool will
be the \emph{% classical r-matrix} (an object whose other guise is the
well-known Hilbert transform). I do not give technical details, unless they may
be exposed in a few lines; on the other hand, all motivations are given in full
scale whenever possible.Comment: LaTeX 2.09, 69 pages. Introductory lectures on Integrable systems,
Classical r-matrices and Factorization problem
Drinfeld-Sokolov reduction for difference operators and deformations of W-algebras. II. General Semisimple Case
The paper is the sequel to q-alg/9704011. We extend the Drinfeld-Sokolov
reduction procedure to q-difference operators associated with arbitrary
semisimple Lie algebras. This leads to a new elliptic deformation of the Lie
bialgebra structure on the associated loop algebra. The related classical
r-matrix is explicitly described in terms of the Coxeter transformation. We
also present a cross-section theorem for q-gauge transformations which
generalizes a theorem due to R.Steinberg.Comment: 19 pp., AMS-LaTeX. The paper replaces a temporarily withdrawn text;
the first part (written by E. Frenkel, N. Reshetikhin, and M. A.
Semenov-Tian-Shansky) is available as q-alg/970401
Classical and Quantum Nonultralocal Systems on the Lattice
We classify nonultralocal Poisson brackets for 1-dimensional lattice systems
and describe the corresponding regularizations of the Poisson bracket relations
for the monodromy matrix . A nonultralocal quantum algebras on the lattices for
these systems are constructed.For some class of such algebras an
ultralocalization procedure is proposed.The technique of the modified
Bethe-Anzatz for these algebras is developed.This technique is applied to the
nonlinear sigma model problem.Comment: 33 pp. Latex. The file is resubmitted since it was spoiled during
transmissio
Renormalization programme for effective theories
We summarize our latest developments in perturbative treating the effective
theories of strong interactions. We discuss the principles of constructing the
mathematically correct expressions for the S-matrix elements at a given loop
order and briefly review the renormalization procedure. This talk shall provide
the philosophical basement as well as serve as an introduction for the material
presented at this conference by A. Vereshagin and K. Semenov-Tian-Shansky.Comment: 6 pages, talk given at HSQCD 2004, Russia, May 2004, to be published
in Proceeding
Classification of All Poisson-Lie Structures on an Infinite-Dimensional Jet Group
A local classification of all Poisson-Lie structures on an
infinite-dimensional group of formal power series is given. All
Lie bialgebra structures on the Lie algebra {\Cal G}_{\infty} of
are also classified.Comment: 11 pages, AmSTeX fil
Bootstrap and the physical values of resonance parameters
This is the 6th paper in the series developing the formalism to manage the
effective scattering theory of strong interactions. Relying on the theoretical
scheme suggested in our previous publications we concentrate here on the
practical aspect and apply our technique to the elastic pion-nucleon scattering
amplitude. We test numerically the pion-nucleon spectrum sum rules that follow
from the tree level bootstrap constraints. We show how these constraints can be
used to estimate the tensor and vector coupling constants. At last, we
demonstrate that the tree-level low energy expansion coefficients computed in
the framework of our approach show nice agreement with known experimental data.
These results allow us to claim that the extended perturbation scheme is quite
reasonable from the computational point of view.Comment: 41 pages, 7 figure
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