Factorization and Integrable Systems

In September 2000 a Summer School on "Factorization and Integrable Systems" was held at the University of Algarve in Portugal. The main aim of the school was to review the modern factorization theory and its application to classical and quantum integrable systems. The program consisted of a number of short courses given by leading experts in the field. The lecture notes of the courses have been specially prepared for publication in this volume. The book consists of four contributions. I. Gohberg, M.A. Kaashoek and I.M. Spitkovsky present an extensive review of the factorization theory of matrix functions relative to a curve, with emphasis on the developments of the last 20-25 years. The group-theoretical approach to classical integrable systems is reviewed by M.A. Semenov-Tian-Shansky. P.P. Kulish surveyed the quantum inverse scattering method using the isotropic Heisenberg spin chain as the main example.

Preface

vii

An Overview of Matrix Factorization Theory and Operator Applications

I. Gohberg

M.A. Kaashoek

I.M. Spitkovsky

Introduction

1

(2)

General theorems

3

(21)

Factorization in decomposing Banach algebras

24

(14)

Generalized Lp factorization

38

(19)

State space method

57

(46)

References

95

(8)

Matrix Riemann-Hilbert Problems Related to Branched Coverings of CP1

Dmitry Korotkin

Introduction

103

(4)

Riemann-Hilbert problem with quasi-permutation monodromies and algebraic curves

107

(4)

Riemann surfaces. Rauch variational formulas

111

(5)

Solution of Riemann-Hilbert problems with quasi-permutation monodromies and Szego kernel

116

(3)

Isomonodromic tau-function and Cauchy-Riemann determinants

119

(12)

Acknowledgements

127

(1)

References

127

(4)

Quantum Groups and Integrable Models

Peter P. Kulish

Introduction

131

(2)

Algebraic Bethe Ansatz and QISM

133

(11)

Yang-Baxter equation

144

(4)

Thermodynamic limits

148

(4)

Conclusion

152

(3)

Acknowledgements

152

(1)

References

153

(2)

Integrable Systems and Factorization Problems

M.A. Semenov-Tian-Shansky

Introduction

155

(3)

A few preliminaries: Poisson brackets, coadjoint orbits, etc.

158

(4)

Classical r-matrices and Lax equations

162

(5)

Classical Yang-Baxter identity

167

(3)

A finite-dimensional example

170

(4)

Loop algebras and the Riemann problem

174

(5)

More examples

179

(5)

Zero curvature equations

184

(18)

Difference equations and Poisson-Lie groups

202

(17)

Acknowledgements

217

(1)

References

217

(2)

Programme of the Summer School on Factorization and Integrable Systems

219

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Factorization and Integrable Systems

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Table of Contents

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