Theory of transformation groups I : General properties of continuous transformation groups ; a contemporary approach and translation

Revalues the purity and the incredible beauty of Lie’s architectural Theorie der Transformations gruppen Makes available Lie’s classification theorems that are not reproved in any modern textbook Offers to researchers the domain of PDEs' symmetries and the classification of differential equations in several (in)dependent variables This modern translation of Sophus Lie's and Friedrich Engel's “Theorie der Transformationsgruppen Band I” will allow readers to discover the striking conceptual clarity and remarkably systematic organizational thought of the original German text. Volume I presents a comprehensive introduction to the theory and is mainly directed towards the generalization of ideas drawn from the study of examples. The major part of the present volume offers an extremely clear translation of the lucid original. The first four chapters provide not only a translation, but also a contemporary approach, which will help present day readers to familiarize themselves with the concepts at the heart of the subject. The editor's main objective was to encourage a renewed interest in the detailed classification of Lie algebras in dimensions 1, 2 and 3, and to offer access to Sophus Lie's monumental Galois theory of continuous transformation groups, established at the end of the 19th Century. Lie groups are widespread in mathematics, playing a role in representation theory, algebraic geometry, Galois theory, the theory of partial differential equations, and also in physics, for example in general relativity. This volume is of interest to researchers in Lie theory and exterior differential systems and also to historians of mathematics. The prerequisites are a basic knowledge of differential calculus, ordinary differential equations and differential geometry. Content Level » Research Keywords » 17B45,17B56,17B66,17B70,22F30,12H05,14P05,14P15 - 22E05,22E10,22E60,2203,1A05,1A55,17B30,17B40, - classifications of Lie Algebras - complete systems of PDEs - continuous transformation groups - general projective group - infinitesimal transformations - local holomorphic vector fields Related subjects » Algebra - History of Mathematical Science Front Matter Pages i-xv Modern Presentation Front Matter Pages 1-1 Book Chapter Pages 3-12 Three Principles of Thought Governing the Theory of Lie Book Chapter Pages 13-22 Local Transformation Equations and Essential Parameters Book Chapter Pages 23-60 Fundamental Differential Equations for Finite Continuous Transformation Groups Book Chapter Pages 61-91 One-Term Groups and Ordinary Differential Equations English Translation Front Matter Pages 93-93 Book Chapter Pages 95-110 Complete Systems of Partial Differential Equations Book Chapter Pages 111-122 New Interpretation of the Solutions of a Complete System Book Chapter Pages 123-149 Determination of All Systems of Equations Which Admit Given Infinitesimal Transformations Book Chapter Pages 151-159 Complete Systems Which Admit All Transformations of a One-term Group Book Chapter Pages 161-186 Characteristic Relationships Between the Infinitesimal Transformations of a Group Book Chapter Pages 187-197 Systems of Partial Differential Equations the General Solution of Which Depends Only Upon a Finite Number of Arbitrary Constants Book Chapter Pages 199-216 The Defining Equations for the Infinitesimal Transformations of a Group Book Chapter Pages 217-223 Determination of All Subgroups of an rr -term Group Book Chapter Pages 225-234 Transitivity, Invariants, Primitivity Book Chapter Pages 235-255 Determination of All Systems of Equations Which Admit a Given rr -term Group Book Chapter Pages 257-281 Invariant Families of Infinitesimal Transformations Book Chapter Pages 283-299 The Adjoint Group Book Chapter Pages 301-321 Composition and Isomorphism Book Chapter Pages 323-339 Finite Groups, the Transformations of Which Form Discrete Continuous Families Book Chapter Pages 341-377 Theory of the Similarity [Aehnlichkeit] of rr -term Groups Book Chapter Pages 379-410 Groups, the Transformations of Which Are Interchangeable with All Transformations of a Given Group English Translation Front Matter Pages 93-93 Book Chapter Pages 411-438 The Group of Parameters Book Chapter Pages 439-465 The Determination of All rr -term Groups Book Chapter Pages 467-501 Invariant Families of Manifolds Book Chapter Pages 503-525 Systatic and Asystatic Transformation Groups Book Chapter Pages 527-557 Differential Invariants Book Chapter Pages 559-580 The General Projective Group Book Chapter Pages 581-598 Linear Homogeneous Groups Book Chapter Pages 599-618 Approach [Ansatz] Towards the Determination of All Finite Continuous Groups of nn -times Extended Space Book Chapter Pages 619-634 Characteristic Properties of the Groups Which Are Equivalent to Certain Projective Groups Back Matter Pages 635-643 "Esta traducción moderna de la "Theorie der Transformationsgruppen Band I" de Sophus Lie y Friedrich Engel permitirá a los lectores descubrir la sorprendente claridad conceptual y el pensamiento organizativo notablemente sistemático del texto original alemán. El volumen I presenta una amplia introducción a la teoría y se dirige principalmente a la generalización de las ideas extraídas del estudio de los ejemplos. La mayor parte del presente volumen ofrece una traducción extremadamente clara del lúcido original. Los cuatro primeros capítulos ofrecen no sólo una traducción, sino también un enfoque contemporáneo, que ayudará a los lectores actuales a familiarizarse con los conceptos que constituyen el núcleo del tema. El principal objetivo del editor era fomentar un interés renovado por la clasificación detallada de las álgebras de Lie en dimensiones 1, 2 y 3, y ofrecer acceso a la monumental teoría de Galois de Sophus Lie sobre los grupos de transformación continua, establecida a finales del siglo XIX. Los grupos de Lie están muy extendidos en las matemáticas, ya que desempeñan un papel en la teoría de la representación, la geometría algebraica, la teoría de Galois, la teoría de las ecuaciones diferenciales parciales, y también en la física, por ejemplo en la relatividad general. Este volumen es de interés para los investigadores de la teoría de Lie y de los sistemas diferenciales exteriores, así como para los historiadores de las matemáticas. Los requisitos previos son un conocimiento básico del cálculo diferencial, las ecuaciones diferenciales ordinarias y la geometría diferencial". Editor "This modern translation of Sophus Lie's and Friedrich Engel's ĺlTheorie der Transformationsgruppen Band Iĺl will allow readers to discover the striking conceptual clarity and remarkably systematic organizational thought of the original German text. Volume I presents a comprehensive introduction to the theory and is mainly directed towards the generalization of ideas drawn from the study of examples. The major part of the present volume offers an extremely clear translation of the lucid original. The first four chapters provide not only a translation, but also a contemporary approach, which will help present day readers to familiarize themselves with the concepts at the heart of the subject. The editor's main objective was to encourage a renewed interest in the detailed classification of Lie algebras in dimensions 1, 2 and 3, and to offer access to Sophus Lie's monumental Galois theory of continuous transformation groups, established at the end of the 19th Century. Lie groups are widespread in mathematics, playing a role in representation theory, algebraic geometry, Galois theory, the theory of partial differential equations, and also in physics, for example in general relativity. This volume℗lis of interest to researchers in Lie theory and exterior differential systems and also to historians of mathematics. The prerequisites are a basic knowledge of differential calculus, ordinary differential equations and differential geometry."--Back cover