Advertisement

© 2000

Lie Groups

Benefits

  • The authors are leading specialists and excellent expositors, who have worked a long time on this book project

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. J. J. Duistermaat, J. A. C. Kolk
    Pages 1-92
  3. J. J. Duistermaat, J. A. C. Kolk
    Pages 93-130
  4. J. J. Duistermaat, J. A. C. Kolk
    Pages 131-207
  5. J. J. Duistermaat, J. A. C. Kolk
    Pages 209-328
  6. Back Matter
    Pages 339-344

About this book

Introduction

This book is devoted to an exposition of the theory of finite-dimensional Lie groups and Lie algebras, which is a beautiful and central topic in modern mathematics. At the end of the nineteenth century this theory came to life in the works of Sophus Lie. It had its origins in Lie's idea of applying Galois theory to differential equations and in Klein's "Erlanger Programm" of treat­ ing symmetry groups as the fundamental objects in geometry. Lie's approach to many problems of analysis and geometry was mainly local, that is, valid in local coordinate systems only. At the beginning of the twentieth century E. Cartan and Weyl began a systematic treatment of the global aspects of Lie's theory. Since then this theory has ramified tremendously and now, as the twentieth century is coming to a close, its concepts and methods pervade mathematics and theoretical physics. Despite the plethora of books devoted to Lie groups and Lie algebras we feel there is justification for a text that puts emphasis on Lie's principal idea, namely, geometry treated by a blend of algebra and analysis. Lie groups are geometrical objects whose structure can be described conveniently in terms of group actions and fiber bundles. Therefore our point of view is mainly differential geometrical. We have made no attempt to discuss systematically the theory of infinite-dimensional Lie groups and Lie algebras, which is cur­ rently an active area of research. We now give a short description of the contents of each chapter.

Keywords

Group actions algebra algebras groups Matrix Representations of algebras Representations of groups Vector space

Authors and affiliations

  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands

About the authors

Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was (co-)author of eleven books.

Duistermaat was affiliated to the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.

 

Johan Kolk published about harmonic analysis on semisimple Lie groups, the theory of distributions, and classical analysis. Jointly with Duistermaat he has written four books: besides the present one, on Lie groups, and on multidimensional real analysis. Until his retirement in 2009, he was affiliated to the Mathematical Institute of Utrecht University. For more information, see http://www.staff.science.uu.nl/~kolk0101/

Bibliographic information

  • Book Title Lie Groups
  • Authors J.J. Duistermaat
    Johan A.C. Kolk
  • Series Title Universitext
  • DOI https://doi.org/10.1007/978-3-642-56936-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 2000
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-15293-4
  • eBook ISBN 978-3-642-56936-4
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages VIII, 344
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Group Theory and Generalizations
    Topological Groups, Lie Groups
  • Buy this book on publisher's site

Reviews

From the reviews:

"This one is worth to read and to keep on your shelf! It presents the theory of Lie groups not only in the usual way of Lie algebraic treatment, but also from the global point of view. … Every chapter ends with very useful notes on the origins and connections of the chapter’s subject. References are given separately in each chapter. ... It is highly recommended to advanced undergraduate and graduated students in mathematics and physics." (Árpád Kurusa, Acta Scientiarum Mathematicarum, Vol. 75, 2009)