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 $\frak{gl}(n)$ 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

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

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

Correcting low-frequency noise with continuous measurement

Low-frequency noise presents a serious source of decoherence in solid-state qubits. When combined with a continuous weak measurement of the eigenstates, the low-frequency noise induces a second-order relaxation between the qubit states. Here we show that the relaxation provides a unique approach to calibrate the low-frequency noise in the time-domain. By encoding one qubit with two physical qubits that are alternatively calibrated, quantum logic gates with high fidelity can be performed.Comment: 10 pages, 3 figures, submitte

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 $G_{\infty}$ of formal power series is given. All Lie bialgebra structures on the Lie algebra {\Cal G}_{\infty} of $G_{\infty}$ are also classified.Comment: 11 pages, AmSTeX fil

Entanglement and quantum phase transitions

We examine several well known quantum spin models and categorize behavior of pairwise entanglement at quantum phase transitions. A unified picture on the connection between the entanglement and quantum phase transition is given.Comment: 4 pages, 3 figure