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Naive Lie Theory [Hardcover]

John Stillwell

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Book Description

Aug. 14 2008 0387782141 978-0387782140 2008

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.

This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history.

John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).


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From the reviews:

“An excellent read. In just 200 pages the author explains what Lie groups and algebras actually are. … An undergraduate who has taken the calculus series, had a course in linear algebra that discusses matrices, has some knowledge of complex variables and some understanding of group theory should easily follow the material to this point. … the best book to get you going.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)

“There are several aspects of Stillwell’s book that I particularly appreciate. He keeps the sections very short and straightforward, with a few exercises at the end of each to cement understanding. The theory is built up in small bites. He develops an intuition for what is happening by starting with very simple examples and building toward more complicated groups. … In short, if you want to teach an undergraduate course on Lie theory, I recommend Stillwell.”(David Bressoud, The UMAP Journal, Vol. 31 (4), 2010)

"Lie theory, basically the study of continuous symmetry, certainly occupies a central position in modern mathematics … . In Naive Lie Theory, Stillwell (Univ. of San Franciso) concentrates on the simplest examples and stops short of representation theory … . Summing Up: Recommended. Upper-division undergraduates and graduate students." (D. V. Feldman, Choice, Vol. 46 (9), May, 2009)

"This book provides an introduction to Lie groups and Lie algebras suitable for undergraduates having no more background than calculus and linear algebra. … Each chapter concludes with a lively and informative account of the history behind the mathematics in it. The author writes in a clear and engaging style … . The book is a welcome addition to the literature in representation theory." (William M. McGovern, Mathematical Reviews, Issue 2009 g)

"This is a beautifully clear exposition of the main points of Lie theory, aimed at undergraduates who have … calculus and linear algebra. … The book is well equipped with examples … . The book has a very strong geometric flavor, both in the use of rotation groups and in the connection between Lie algebras and Lie groups." (Allen Stenger, The Mathematical Association of America, October, 2008)

From the Back Cover

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.

This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history.

John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).


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Amazon.com: 4.5 out of 5 stars  11 reviews
51 of 53 people found the following review helpful
5.0 out of 5 stars An excellent introduction to Lie Theory May 1 2009
By Hedin Peter - Published on Amazon.com
Format:Hardcover
I'm no expert in Lie groups or Lie algebras, I didn't read any of that stuff in my M. Sc. Eng. Phys. so i decided to try Professor Stilwells book as an introduction to the subject. I am very glad that I bought this book. What Prof. Stilwell promises in the foreword is true - you can read and understand this book with a background of only calculus and linear algebra. The book introduces a lot of advanced concepts, but in a very clear and logic way - there is no problem for an undergraduate to comprehend the material. I guess the book is meant to be a school text book - it was a little hard for me to try to self-study some of the excercises, because there are no solutions provided. I like that every chapter starts with a preview to give an orientation of what will be presented in the chapter. Every chapter also ends with a discussion, which gives historical aspects of the presented theory, and some suggestions for further litterature on the various subjects. This is nice - it gives a wider perspective to the subject. I think this book is a very good stepping-stone on the reader's way from undergrad math to graduate topics, and I hope there will be more books of this kind.
37 of 39 people found the following review helpful
5.0 out of 5 stars Excellent read June 19 2009
By Rick Martinelli - Published on Amazon.com
Format:Hardcover
An excellent read. In just 200 pages the author explains what Lie groups and algebras actually are. Most books on Lie theory are aimed at professional mathematicians, so begin with lots of topological and algebraic preliminaries and finally define a Lie group as a group that is also a manifold, or something similar. Stillwell begins with an example of the simplest Lie group, SO(2), as a group of rotations in the circle, then proceeds methodically to the next example SU(2), the first non-commutative Lie group. In short order all the other classical groups are discussed and, in chapter 5, the concepts of tangent space and Lie algebra are made clear through more examples. An undergraduate who has taken the calculus series, had a course in linear algebra that discusses matrices, has some knowledge of complex variables and some understanding of group theory should easily follow the material to this point. Topology, usually a graduate topic, is introduced later while showing which Lie groups are simply-connected, and how this is used to distinguish between similar Lie groups.

The material was clearly discussed and I found only a couple of typos. But I also found the use of the word vector and matrix for the same object in the same paragraph somewhat dis-quieting. Lastly, I would have liked to have seen some mention of Lie theory connections with modern physics.
19 of 19 people found the following review helpful
5.0 out of 5 stars Spectacular introduction to Lie groups and algebras June 12 2010
By Joshua E. Hill - Published on Amazon.com
Format:Paperback|Verified Purchase
Let me start by stating my point of view: I'm a math grad student, so I'm not really the nominal audience for the book (the book is targeted toward undergraduates). Having said that, I found this book to be wonderfully conversational in tone, amusing, very honest (if there is slogging to be done in a proof, the author says so, and if the author leaves something out he tells you why), and very useful in gaining an intuitive feel for the material. The prerequisites for this book are very modest: if you've seen linear algebra and calculus, then you could give it a go. Some sort of exposure to abstract algebra of some sort would be useful, but may not be required. Some intuition for manifolds is is similarly useful, but certainly not required.

Even with these modest prerequisites, the author manages to do much with Lie Theory. This is a jewel of a book, much like its spiritual predecessor, Halmos's Naive Set Theory (Undergraduate Texts in Mathematics).

So, this book is accessible, well written and useful. What more could you ask for in an introduction?
8 of 9 people found the following review helpful
5.0 out of 5 stars Review of Naive Lie Theory Oct. 4 2010
By Ray Bagley - Published on Amazon.com
Format:Hardcover|Verified Purchase
This review is on the textbook Naive Lie Theory by John Stillwell. Recently I purchased this book with hopes of having a study reference to the more elementary parts in preparation for more advanced study of Lie Theory and other theoretical math that involves these ideas. I have not yet finished the book. This book is well written with clear and accurate developments and good examples. There are well placed exercises. One is tempted to try various things, to explore variations based on the readings. I find this exciting the way the book let's me explore ideas. The Author lets you know about the more advanced parts of Lie Theory he is not going to cover so you have an idea what to study later to complete the picture. He decides to use simpler concepts of matrix processes and linear algebra with the understanding that this will allow you to do quite a bit. It is a nice start using the unit circle on the complex plane as an elementary first example. A clear context is given why certain inventions and discoveries were made. I am a mathematician, computer scientist, mathematical physicist, and Formal Languages.
3 of 3 people found the following review helpful
5.0 out of 5 stars Best introduction to Lie theory May 27 2012
By Emizco - Published on Amazon.com
Format:Paperback|Verified Purchase
It is not often that I buy a math textbook, read it cover to cover, and long for more. Stillwell is an exceptional writer. What differentiates this textbook from others is (1) the historical background material that seamlessly mixes with the equations, and (2) a clear motivation and exposition of important concepts.
For readers with physics background: in my opinion Stillwell is the David Griffiths of math. Stillwell does not cover indefinite groups (Lorentz groups) nor does he cover representations. But it is still the best book to get you going.
I found this textbook more interesting than Tapp's Matrix Groups for Undergraduates (Student Mathematical Library,). I think Tapp's book is somewhat more elementary. I could not read Kosmann-Schwarzbach's Groups and Symmetries: From Finite Groups to Lie Groups (Universitext) (translated) beyond chapter 1, the text was concise but encryptic. Georgi's Lie Algebras In Particle Physics: from Isospin To Unified Theories (Frontiers in Physics) was more advanced and kind of dry. Lipkin's Lie Groups for Pedestrians (Dover Books on Physics) was more advanced also. After finishing Stillwell's book I would recommend Hall's Lie Groups, Lie Algebras, and Representations: An Elementary Introduction.

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