Lie Groups, Lie Algebras, and Representations

Rating:
Summary: A refreshingly clear introductory text on Lie groups
Review: I rarely have time or feel strongly enough about a text to write a review. However, with Hall's book, I feel compelled. After struggling with the rather compact sixth chapter of Wulf Rossman's book on representations of Lie groups and algebras during a course on representation theory (the first five chapters were assumed), I turned to this one, and boy, am I ever glad I did.

The main and overriding strength of this book is the willingness of the author to guide the reader in digesting definitions and proofs. This comes in the form of numerous examples and counterexamples to point the reader in the right direction after a definition. And Hall constantly reminds readers of particular relevant terms in the course of applying them, which I found very effective in reinforcing concepts, and which allowed me to focus on the task at hand rather than spending time sifting through previous chapters, often losing sight of the main point of the argument.

Another strong point is the approach taken to introducing weights and roots of particular representations. I have found this a very difficult subject (as I guess a lot of students do) and Rossman's book was not helping much. As the previous reviewer noted, this book starts out (chapters four and five) with detailed treatments of the representations of su(2) and su(3) via the complexifications sl(2; C) and sl(3; C) and introduces weights in these contexts as pairs of simultaneous eigenvalues of the basis elements of the Cartan subalgebra.This requires only a background in linear algebra to digest and really hits home the point of these constructs in the whole scheme of things. After these examples under the belt, the reader is then able to take in the general definition of a weight as a linear functional in chapter six. Representations of general semisimple Lie algebras are covered in chapter seven.

Throughout it all, Hall's style is very clear and his proofs are complete and illuminating. If you have had courses in linear and modern algebra, you should be fine with this one. Very well suited for self study. I can't recommend this book highly enough.

Rating:
Summary: AT LAST, LIE GROUPS & ALGEBRAS I CAN UNDERSTAND!!
Review: This book focuses on matrix Lie groups and Lie algebras, and their relations and representations. This makes things a bit simpler, and not much is lost, because most of the interesting Lie groups & algebras are (isomorphic to)groups & algebras of matrices.
I believe that most mathematicians are more concerned with impressing their colleagues with their subtlety and erudition than they are in making a clear, simple and comprehensible presentation. This is mitigated by the publisher's insistence that the first 10 pages be clear to a mid-level undergraduate so the book will sell. So I usually get stuck at page 10 in those books.
This book is clear (to me) at least to page 168 (as far as I have progressed). There are even appendices on finite groups and key aspects of linear algebra. After introducing the classical groups and their algebras and the exponential map relating one to the other, the author introduces representations. He then details the representations of sl(2,C) and sl(3,C) (a.k.a. the complexifications of su(2) and su(3), respectively). By going through the details on these [with their Cartan subalgebras, weights, roots, Weyl groups, etc.], the general theory that follows is more palatable than it might otherwise be. Little rigor is sacrificed (if I am qualified to judge that - probably not). A few proofs are left out, but not many.

Another virtue of this book is that there are very few mistakes. I have trouble distinguishing an author's typos from my thinkos, so this is a particularly impotant feature of this book.
I very highly recoommend this book to anyone who does not already know the subject; it would be a perfect first book on this area. This book is really written with the student in mind. As a "shade - tree" mathematician, I need all the help I can get in understanding this difficult subject. Hall has done the best job I have seen at making the theory accessible without sacrificing rigor.

Rating:
Summary: AT LAST, LIE GROUPS & ALGEBRAS I CAN UNDERSTAND!!
Review: This book focuses on matrix Lie groups and Lie algebras, and their relations and representations. This makes things a bit simpler, and not much is lost, because most of the interesting Lie groups & algebras are (isomorphic to)groups & algebras of matrices.
I believe that most mathematicians are more concerned with impressing their colleagues with their subtlety and erudition than they are in making a clear, simple and comprehensible presentation. This is mitigated by the publisher's insistence that the first 10 pages be clear to a mid-level undergraduate so the book will sell. So I usually get stuck at page 10 in those books.
This book is clear (to me) at least to page 168 (as far as I have progressed). There are even appendices on finite groups and key aspects of linear algebra. After introducing the classical groups and their algebras and the exponential map relating one to the other, the author introduces representations. He then details the representations of sl(2,C) and sl(3,C) (a.k.a. the complexifications of su(2) and su(3), respectively). By going through the details on these [with their Cartan subalgebras, weights, roots, Weyl groups, etc.], the general theory that follows is more palatable than it might otherwise be. Little rigor is sacrificed (if I am qualified to judge that - probably not). A few proofs are left out, but not many.

Another virtue of this book is that there are very few mistakes. I have trouble distinguishing an author's typos from my thinkos, so this is a particularly impotant feature of this book.
I very highly recoommend this book to anyone who does not already know the subject; it would be a perfect first book on this area. This book is really written with the student in mind. As a "shade - tree" mathematician, I need all the help I can get in understanding this difficult subject. Hall has done the best job I have seen at making the theory accessible without sacrificing rigor.