Lie Groups: A Problem-Oriented Introduction via Matrix Groups, by Harriet Pollatsek, the Mathematical Association of America, Washington, D. C., 2009. 164 pp., $63.95 (hardbound). ISBN 9780883857595.
Five out of five stars
A measure of how complex Lie groups can be is that it is part of problem number five in the famous set of twenty-three stated by David Hilbert in 1900.
“Under what conditions is a topological group also a Lie group?”
In this book, there is a restriction to the matrix Lie groups, which places them within the skill set of even sophomores, provided that they have a few courses behind them. Specifically calculus and linear algebra. This book contains some abstract algebra, but it is presented when needed, so it can be understood within the flow of the material.
The tactic used can be summed up by the phrase, “a little bit of explanatory text, a list of problems, . . . repeat.” The count of problems is stated as 200, which, given that there are 139 pages of the main content including other material such as suggestions for further reading, works out to about two problems per page. Hints to the solutions to some of the problems are given in an appendix.
If you are interested in developing a special topics or independent study course that is over material not normally covered in an undergraduate curriculum, this is a text that will allow you to cover Lie groups. It is very understandable and readable, the kind of book that a prepared student will be able to read on their own.