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
Author(s): Sophus Lie; Joel Merker (ed., transl.)
Publisher: Springer
Year: 2015
Language: English
Pages: XV, 643
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
