Theory of Continuous Groups

by Charles Loewner

Overview

Professor of Mathematics at Stanford University from 1950 until his death in 1968, Charles Loewner occasionally taught as a Visiting Professor at the University of California at Berkeley. After his 1955 course at Berkeley on continuous groups, Loewner's lectures were reproduced in the form of mimeographed notes. The professor had intended to develop these notes into a book, but the project was still in formative stages at the time of his death. The 1971 edition compiles edited and updated versions of Professor Loewner's original fourteen lectures, making them available in permanent form.
Professor Loewner's interest in continuous groups—particularly with respect to applications in geometry and analysis—began with his study of Sophus Lie's three-volume work on transformation groups. He was able to reconstruct a coherent development of the subject by synthesizing Lie's numerous illustrative examples, many of which appeared only as footnotes. The examples contained in this book—primarily geometric in character—reflect the professor's unique view and treatment of continuous groups.

Product Details

ISBN-13:	9780486462929
Publisher:	Dover Publications
Publication date:	02/04/2008
Series:	Dover Books on Mathematics
Pages:	126
Product dimensions:	6.12(w) x 9.25(h) x (d)

About the Author

Charles Loewner was a renowned Professor of Mathematics at Stanford University.

Table of Contents

Preface
Lecture I: Transformation Groups; Similarity
Lecture II: Representations of Groups; Combinations of Representations; Similarity and Reducibility
Lecture III: Representations of Cyclic Groups; Representations of Finite Abelian Groups; Representations of Finite Groups
Lecture IV: Representations of Finite Groups (cont.); Characters
Lecture V: Representations of Finite Groups (conc.); Introduction to Differentiable Manifolds; Tensor Calculus on a Manifold
Lecture VI: Quantities, Vectors, and Tensors; Generation of Quantities by Differentiation; Commutator of Two Contravariant Vector Fields; Hurwitz Integration on a Group Manifold
Lecture VII: Hurwitz Integration on a Group Manifold (cont.); Representation of Compact Groups; Existence of Representations
Lecture VIII: Representation of Compact Groups (cont.); Characters; Examples
Lecture IX: Lie Groups; Infinitesimal Transformations on a Manifold
Lecture X: Infinitesimal Transformations of a Group; Examples; Geometry on the Group Space
Lecture XI: Parallelism; First Fundamental Theorem of Lie Groups; Mayer-Lie Systems
Lecture XII: The Sufficiency Proof; First Fundamental Theorem; Converse; Second Fundamental Theorem; Converse
Lecture XIII: Converse of the Second Fundamental Theorem (cont.); Concept of Group Germ
Lecture XIV: Converse of the Third Fundamental Theorem; The Helmholtz-Lie Problem
Index