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Theory of Transformation Groups I

General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation

  • Book
  • © 2015

Overview

Authors:
  1. Sophus Lie
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Editors:
  1. Joël Merker

    1. Laboratoire de Mathématiques, Université Paris-Sud 11 Faculté des Sciences d'Orsay, Orsay, France

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  • 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

  • Includes supplementary material: sn.pub/extras

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Table of contents (29 chapters)

  1. Front Matter

    Pages i-xv
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  3. English Translation

    1. Front Matter

      Pages 93-93
      Download chapter PDF

    2. Complete Systems of Partial Differential Equations

      • Sophus Lie
      Pages 95-110

    3. New Interpretation of the Solutions of a Complete System

      • Sophus Lie
      Pages 111-122

    4. Determination of All Systems of Equations Which Admit Given Infinitesimal Transformations

      • Sophus Lie
      Pages 123-149

    5. Complete Systems Which Admit All Transformations of a One-term Group

      • Sophus Lie
      Pages 151-159

    6. Characteristic Relationships Between the Infinitesimal Transformations of a Group

      • Sophus Lie
      Pages 161-186

    7. Systems of Partial Differential Equations the General Solution of Which Depends Only Upon a Finite Number of Arbitrary Constants

      • Sophus Lie
      Pages 187-197

    8. The Defining Equations for the Infinitesimal Transformations of a Group

      • Sophus Lie
      Pages 199-216

    9. Determination of All Subgroups of an \(r\)-term Group

      • Sophus Lie
      Pages 217-223

    10. Transitivity, Invariants, Primitivity

      • Sophus Lie
      Pages 225-234

    11. Determination of All Systems of Equations Which Admit a Given \(r\)-term Group

      • Sophus Lie
      Pages 235-255

    12. Invariant Families of Infinitesimal Transformations

      • Sophus Lie
      Pages 257-281

    13. The Adjoint Group

      • Sophus Lie
      Pages 283-299

    14. Composition and Isomorphism

      • Sophus Lie
      Pages 301-321
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Keywords

About this book

This modern translation of Sophus Lie's and Friedrich Engel's “Theorie der Transformationsgruppen 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.

Editors and Affiliations

  • Laboratoire de Mathématiques, Université Paris-Sud 11 Faculté des Sciences d'Orsay, Orsay, France

    Joël Merker

About the editor

Professor Joël Merker studied Mathematics and Philosophy at the Ecole Normale Supérieure in Paris where he received his Ph. D. in Mathematics (1996), followed by his habilitation in Mathematics (2006) and Ph. D. in Philosophy (2012). He was a CNRS researcher (1997-2010) and is currently Professor of Mathematics at Paris-Sud-Orsay University.

Bibliographic Information

  • Book Title: Theory of Transformation Groups I

  • Book Subtitle: General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation

  • Authors: Sophus Lie

  • Editors: Joël Merker

  • DOI: https://doi.org/10.1007/978-3-662-46211-9

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2015

  • Hardcover ISBN: 978-3-662-46210-2Published: 27 March 2015

  • Softcover ISBN: 978-3-662-51266-1Published: 09 October 2016

  • eBook ISBN: 978-3-662-46211-9Published: 12 March 2015

  • Edition Number: 1

  • Number of Pages: XV, 643

  • Number of Illustrations: 7 illustrations in colour

  • Topics: Topological Groups, Lie Groups, Projective Geometry, History of Mathematical Sciences

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