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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
Brian C. Hall
Publisher: Springer Verlag (2003)
Details: 351 pages, Hardcover
Series: Graduate Texts in Mathematics
Topics: Lie Groups
MAA Review[Reviewed by Gizem Karaali, on 01/02/2006]
This book is a great find for those who want to learn about Lie groups or Lie algebras and basics of their representation theory. It is a well-written text which introduces all the basic notions of the theory with many examples and several colored illustrations. The author, Brian Hall of the University of Notre Dame, writes as if he is talking to his best students; without losing rigor and attention to details, he provides many informal explanations, several examples and counterexamples to definitions, discussions and warnings about different conventions, and so on.
But one may ask: Why choose this book among so many others with similar titles? What is so special about this particular one?
The idea is simple: Lie groups and Lie algebras are relevant and useful to many mathematicians (and physicists) with diverse backgrounds. However, at least until recently, these mathematical structures have not been included in the standard curricula of most undergraduate or graduate programs in mathematics. Probably the main reason for this has been the perception that there just are too many prerequisites to even get one's feet wet in the subject and only those few who have been motivated to study the subject in all its details have ventured to test the waters. (How is that for mixed metaphors?) I would go further and dare to say that this is mainly due to the nature of most of the standard texts in the topic, which either choose to focus only on Lie algebras and provide a deep algebraic theory with very little attempt at explaining the geometric connotations (à la Humphreys' Introduction to Lie Algebras and Their Representations), or require a solid background on basic manifold theory and lose a lot of otherwise enthusiastic readers from the beginning (à la Varadarajan's Lie Groups, Lie Algebras, and Their Representations).
However, according to Brian Hall, (and after reading his text, this reviewer most definitely agrees), there is a way to make almost all the basics of Lie theory and representation theory accessible to a mathematically mature audience who may not have either the prerequisites in manifold theory required to feel comfortable with Varadarajan's text or the interest in a purely algebraic approach like Humphreys'. Hall restricts himself to matrix Lie groups and matrix Lie algebras, which are the main finite dimensional examples. However it is worth noting that he still ends up developing all the theory that one would come across in a more standard text, like the representation theory of semisimple Lie algebras, and in particular the theory of roots and weights.
By restricting to the matrix case, though admittedly losing some generality, the text gains immensely in accessibility. The formal prerequisites reduce now to a solid background in linear algebra (though a detailed appendix covers some of the more advanced topics that are relevant), and a knowledge of some facts about groups and about the convergence of sequences of real or complex numbers. This basically amounts to the usual basic undergraduate courses, and one can even contemplate using the book in a course for an advanced undergraduate audience. However, except for a rather ambitious and highly motivated group of undergraduates, this text could prove to be rather difficult at the undergraduate level. The mathematical maturity required from the reader makes it much more appropriate for beginning graduate students in mathematics or physics. It would also make a great read for mathematicians who want to learn about the subject.
Gizem Karaali teaches at the University of California in Santa Barbara.