Algebraic Integrability, Painlevé Geometry and Lie Algebras (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 47)

From the reviews of the first edition: "The aim of this book is to explain ‘how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations’. ... One of the main advantages of this book is that the authors ... succeeded to present the material in a self-contained manner with numerous examples. As a result it can be also used as a reference book for many subjects in mathematics. In summary ... a very good book which covers many interesting subjects in modern mathematical physics." (Vladimir Mangazeev, The Australian Mathematical Society Gazette, Vol. 33 (4), 2006) "This is an extensive volume devoted to the integrability of nonlinear Hamiltonian differential equations. The book is designed as a teaching textbook and aims at a wide readership of mathematicians and physicists, graduate students and professionals. ... The book provides many useful tools and techniques in the field of completely integrable systems. It is a valuable source for graduate students and researchers who like to enter the integrability theory or to learn fascinating aspects of integrable geometry of nonlinear differential equations." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1083, 2006)

This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic.

This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic Front Matter....Pages I-XII Introduction....Pages 1-4 Front Matter....Pages 5-5 Lie Algebras....Pages 7-39 Poisson Manifolds....Pages 41-66 Integrable Systems on Poisson Manifolds....Pages 67-104 Front Matter....Pages 105-105 The Geometry of Abelian Varieties....Pages 107-152 A.c.i. Systems....Pages 153-197 Weight Homogeneous A.c.i. Systems....Pages 199-261 Front Matter....Pages 263-263 Integrable Geodesic Flow on SO(4)....Pages 265-360 Periodic Toda Lattices Associated to Cartan Matrices....Pages 361-418 Integrable Spinning Tops....Pages 419-468 Back Matter....Pages 469-483 Introduction Part I: Liouville Integrable Systems; Lie Algebras; Poisson Manifolds; Integrable Systems on Poisson Manifolds Part II: Algebraic Completely Integrable Systems; The Geometry of Abelian Varieties; A.c.i. Systems; Weight Homogeneous A.c.i. Systems Part III: Examples; Integrable Geodesic Flow on SO(4); Periodic Toda Lattices Associated to Cartan Matrices; Integrable Spinning Tops References Index.