Theory of Transformation Groups I : General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation / by Sophus Lie ; edited by Joël Merker.

Format:

Book

Author/Creator:

Lie, Sophus., Author.

Contributor:

Merker, Joël., Editor.

Language:

English

Subjects (All):

Topological groups.

Lie groups.

Geometry, Projective.

Mathematics.

History.

Topological Groups, Lie Groups.

Projective Geometry.

History of Mathematical Sciences.

Local Subjects:

Topological Groups, Lie Groups.

Projective Geometry.

History of Mathematical Sciences.

Physical Description:

1 online resource (640 p.)

Edition:

1st ed. 2015.

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2015.

Language Note:

English

Summary:

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.

Contents:

Part I Modern Presentation

1 Three Principles of Thought Governing the Theory of Lie

2 Local Transformation Equations and Essential Parameters

3 Fundamental Differential Equations for Finite Continuous Transformation Groups

4 One-Term Groups and Ordinary Differential.- Part II English Translation

5 Complete Systems of Partial Differential Equations

7 Determination of All Systems of Equations Which Admit Given Infinitesimal Transformations

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

9 Characteristic Relationships Between the Infinitesimal Transformations of a Group

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

11 The Defining Equations for the Infinitesimal Transformations of a Group

12 Determination of All Subgroups of an r-term Group

13 Transitivity, Invariants

14 Determination of All Systems of Equations Which Admit a Given r-term Group

15 Invariant Families of Infinitesimal Transformations

16 The Adjoint Group

17 Composition and Isomorphism

18 Finite Groups, the Transformations of Which Form Discrete Continuous Families

19 Theory of the Similarity [AEHNLICHKEIT] of r-term Groups

20 Groups, the Transformations of Which Are Interchangeable With All Transformations of a Given Group

21 The Group of Parameters

22 The Determination of All r-term Groups

23 Invariant Families of Manifolds.- 24 Systatic and Asystatic Transformation Groups

25 Differential Invariants

26 The General Projective Group

27 Linear Homogeneous Groups

28 Approach [ANSATZ] towards the Determination of All Finite Continuous Groups of the n-times Extended Space

29 Characteristic Properties of the Groups Which are Equivalent to Certain Projective Groups

Glossary of significantly used words

Index./p>.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

3-662-46211-7