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Naive Lie Theory
Publisher: Springer (2008)
Details: 215 pages, Hardcover
Series: Undergraduate Texts in Mathematics
Topics: Lie Groups
MAA Review[Reviewed by Allen Stenger, on 10/06/2008]
This is a beautifully clear exposition of the main points of Lie theory, aimed at undergraduates who have studied calculus and linear algebra. The book is modeled after (and named in homage to) Halmos's Naive Set Theory. The key simplification is that it deals only with matrix groups.
The book is well equipped with examples, and it always ties the matrix groups back to concrete examples, especially the complex numbers and the quaternions. 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.
The book's most conspicuous weakness is that it treats Lie theory in isolation. It doesn't give any clue where the subject came from or what it is used for today. Each chapter ends with a very informative "Discussion" section, but what is discussed is the portions of Lie theory that we couldn't get to in this book. There's no mention of differential equations, Klein's Erlangen Program, or representation theory.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.