Customer Reviews

Naive Lie Theory (Undergraduate Texts in Mathematics)

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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.

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.

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?

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.

This book is clear and neither what I already knew nor over my head. I wish the answers to the exercises were available.

I was very disapointed with the way the author treated the subject.

The thing is, when you talk about a subject such as Lie Theory, you know it's a graduate level subject. In my opinion, it's totally unrealistic to try and teach (in a formal manner) the subject at an undergraduate level, and that's exactly what this book is about. The author assumes you have no knowledge of topology, differential calculus or group theory whatsoever, and tries to teach you the bare minimum so you can understand the basics ideas of Lie Theory.

In my opinion, the book should be titled "Introduction to what you must know in order to start wanting to learn Lie Theory". I recognize now that it was very unrealistic of me to want to learn the subject as an undergraduate (it's not a coincidence that it's thought at grad school). I think one must be really familiar with topology and differential calculus (maybe even differential geometry) in order to grasp the interesting stuff about Lie Theory.

To end with an analogy, it would be like writing a book about measure theory but aimed at high-school students. Sure, they could probably understand the concept of sigma-algebra and measure, but that would be it! You would be stuck with showing them trivial examples, therefore making no point whatsoever. It would be like : "You know how you measure the surface of a square, guess what, you can do it through a fancy thing called the 'Lebesgue-Measure' and thus obtaining the same result that with your high-school method."

Anyway, in my opinion, this book is pretty pointless.

A very good text for graduate students with a little or no knowledge of Quantum Field Theory.