Algebraic Structures in Integrability

Relationships of the theory of integrable systems with various branches of mathematics are extremely deep and diverse. On the other hand, the most fundamental exactly integrable systems often have applications in theoretical physics. Therefore, many mathematicians and physicists are interested in integrable models.The book is intelligible to graduate and PhD students and can serve as an introduction to separate sections of the theory of classical integrable systems for scientists with algebraic inclinations. For the young, the book can serve as a starting point in the study of various aspects of integrability, while professional algebraists will be able to use some examples of algebraic structures, which appear in the theory of integrable systems, for wide-ranging generalizations.The statements are formulated in the simplest possible form. However, some ways of generalization are indicated. In the proofs, only essential points are mentioned, while for technical details, references are provided. The focus is on carefully selected examples. In addition, the book proposes many unsolved problems of various levels of complexity. A deeper understanding of every chapter of the book may require the study of more rigorous and specialized literature.<b>Contents:</b> <ul><li>Foreword</li><li>Preface</li><li>Introduction</li><li><b><i>Lax Representations for Integrable Systems:</i></b><ul><li>Lax Pairs and Decomposition of Lie Algebras</li></ul></li><li><b><i>Algebraic Structures in Theory of Bi-Hamiltonian Systems:</i></b><ul><li>Bi-Hamiltonian Formalism</li></ul></li><li><b><i>Symmetry Approach to Integrability:</i></b><ul><li>Basic Concepts of Symmetry Approach</li><li>Integrable Hyperbolic Equations of Liouville Type</li><li>Integrable Nonabelian Equations</li><li>Integrable Evolution Systems and Nonassociative Algebras </li><li>Integrable Vector Evolution Equations</li><li>Appendices</li></ul></li><li>Bibliography</li><li>Index</li></ul><br><b>Readership:</b> Graduate.Integrable Systems;Lax Pairs;Symmetries;Nonabelian Differential Equations0<b>Key Features:</b><ul><li>For scientists who prefer to begin the study of scientific theories with a consideration of simple but substantial examples</li><li>For scientists who prefer to first understand the approximate content of the theory and solve several simple problems</li><li>Discusses the results and algebraic structures associated with most important concepts of the theory of classical integrable systems</li></ul>