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

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概要 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 organ...izational 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.続きを見る
目次 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>.
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本文を見る Full text available from Springer Mathematics and Statistics eBooks 2015 English/International

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登録日 2020.06.27
更新日 2020.06.28