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  • Textbook
  • © 2002

Matrix Groups

An Introduction to Lie Group Theory

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  • Only introduction to Lie group theory aimed at the undergraduate

  • Discusses applications in mathematics and physics

  • Provides a self-contained introduction to Lie groups and serves as a foundation for a more abstract course in Lie theory and differential geometry

  • Uses solved examples and exercises to build the reader's confidence in a subject that can be difficult to tackle

Part of the book series: Springer Undergraduate Mathematics Series (SUMS)

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  • ISBN: 978-1-4471-0183-3
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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xi
  2. Basic Ideas and Examples

    1. Front Matter

      Pages 1-1
    2. Real and Complex Matrix Groups

      • Andrew Baker
      Pages 3-43
    3. Tangent Spaces and Lie Algebras

      • Andrew Baker
      Pages 67-97
    4. Clifford Algebras and Spinor Groups

      • Andrew Baker
      Pages 129-156
    5. Lorentz Groups

      • Andrew Baker
      Pages 157-178
  3. Matrix Groups as Lie Groups

    1. Front Matter

      Pages 179-179
    2. Lie Groups

      • Andrew Baker
      Pages 181-209
    3. Homogeneous Spaces

      • Andrew Baker
      Pages 211-233
    4. Connectivity of Matrix Groups

      • Andrew Baker
      Pages 235-247
  4. Compact Connected Lie Groups and their Classification

    1. Front Matter

      Pages 249-249
    2. Semi-simple Factorisation

      • Andrew Baker
      Pages 267-288
  5. Back Matter

    Pages 303-330

About this book

Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.
The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.
Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.

Keywords

  • Group theory
  • Lie group
  • Lie groups
  • Matrix
  • Matrix groups
  • algebra
  • differential geometry
  • linear algebra
  • matrix theory

Reviews

From the reviews of the first edition:

MATHEMATICAL REVIEWS

"This excellent book gives an easy introduction to the theory of Lie groups and Lie algebras by restricting the material to real and complex matrix groups. This provides the reader not only with a wealth of examples, but it also makes the key concepts much more concrete. This combination makes the material in this book more easily accessible for the readers with a limited background…The book is very easy to read and suitable for an elementary course in Lie theory aimed at advanced undergraduates or beginning graduate students…To summarize, this is a well-written book, which is highly suited as an introductory text for beginning graduate students without much background in differential geometry or for advanced undergraduates. It is a welcome addition to the literature in Lie theory."

"This book is an introduction to Lie group theory with focus on the matrix case. … This book can be recommended to students, making Lie group theory more accessible to them." (A. Akutowicz, Zentralblatt MATH, Vol. 1009, 2003)

Authors and Affiliations

  • Department of Mathematics, University of Glasgow, Glasgow, UK

    Andrew Baker

Bibliographic Information

Buying options

eBook
USD 34.99
Price excludes VAT (USA)
  • ISBN: 978-1-4471-0183-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 44.99
Price excludes VAT (USA)