Customer Reviews

Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

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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.

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 roots 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 root 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.

This is an excellent book on a difficult subject.

When learning Group Theory from the viewpoint of physics, one can miss out completely on some of the important mathematical aspects.
Halls book solved that problem for me. But, I can imagine that it also works in the reverse;
If one studies Group Theory from a pure mathematical viewpoint, one can miss out on a multitude of computational techniques and some important results.

The paramount example of Halls book is the handling of the representations of the group SU(3).
To gain even more insight into that group one can use Halls book together with Quantum Mechanics: Symmetries.
There one can see "Groups, Algebras and their Representaions in Action", especially SU(3),
in numerous solved excercises and problems displaying a multitude of relevant computational techniques.

The two books begin at about the same point (groups, algebras, representations, the exponential map),
and end at about the same point (classification of the classical groups).
Halls book provides the correct mathematical setting and Greiners book the solved examples.

The two books together add up to a lot of value.
The pure math student can easily ignore the physics in Greiners book and pick up some new things in representation theory,
such as Cartans criterion for irreducibility, derivations of dimension formulas for representations, etc.
Meanwhile, the pure physics student should probably avoid trying to learn Group Theory from physics books (including Greiners).
There is a lot of confusion in the physics books as to what is what. Groups, algebras, representations and invariant subspaces are constantly mixed up.

In conclusion, one benifits from a math book, and a large collection of examples. Halls book and Greiners book work surprisingly well together.

Brian Hall's book is a welcome addition to the material available for the study of Lie Groups. This book in particular provides a good basis for the study of Lie Groups without getting caught up in the study of Manifold Theory. The book is easy to access, requiring only a basic background in Modern and Linear Algebra and has many applications pertaining both to mathematics and physics.

This review is for the Kindle edition. I have the hard copy, which I like, and downloaded a sample Kindle version, which I do not like. The Kindle edition occasionally breaks words up; sometimes, more than once per word. While not a deal breaker, editing should be better. The big problems are equations and italics. The equation characters, particularly exponents, are sometimes difficult to identify because they are not fully rendered, and the issue is not with the Kindle. I have a Kindle DX, and expanding the font size does not correct the problem. Missing segments remain missing. Italics have the same problem along slants.

[EDITED 2012: This review used to contain information about the (unreadable) Kindle edition of this book. I removed that information after the publisher, wisely, stopped offering the Kindle edition.]

As a graduate student, I love this book. It requires surprisingly little familiarity with topology and algebra; I could have taken this course in my first year without being taxed by prerequisites. Its focus on specific examples, such as SU(2) and SO(3), match well with the situations in which I have previously encountered Lie groups outside of the course. The extensive discussion of examples also helps me to structure the big picture in my head, in that I feel more confident asserting why we take an interest in, for example, the connectedness of a Lie group. Hall's discussion of the behavior of, and topological properties of, the most commonly encountered Lie groups is superb. The writing clarifies which details I ought to work out on my own as I read (which is another reason I would have been able to take this course in my first year, when my skill at actively reading and engaging with upper-level textbooks was still budding).

I don't know if this book is useful as a reference text for individuals who have a research interest in algebraic topology. I doubt it's extensive enough. However, as a learning text for physicists and mathematicians whose main research interest lies in analysis or PDE, this book has proven very satisfactory. Future authors of textbooks might also skim through it to discover an excellent example of how to connect with a competent student audience and guide them into an unfamiliar topic without obscuring its purpose.

Also worth recommending here, specifically as a companion volume: the text Linearity, Symmetry, and Prediction in the Hydrogen Atom (Undergraduate Texts in Mathematics) by Stephanie Frank Singer. I think an excellent and interesting first-year graduate course in mathematical physics could be constructed from Hall's and Singer's books.

I am a graduate student at UC Berkeley who used this book for an introductory course in Lie theory.

I found that Hall's book focuses too much on examples, often allowing the reader to lose sight of the underlying algebra. In the midst of hacking and slashing one's way through matrix computations, one fails to gain a deeper, and more valuable intuition for what Lie groups and Lie algebras are.

My classmates discovered and took a liking to Fulton and Harris. I wound up using Humphreys. Humphreys suffers the opposite failing, of being overly sparse and direct, but it succeeds in conveying an appreciation for the elegance of Lie algebras.

It doesn't take a lot of intelligence to figure
out how to present lie algebras and lie groups
if you are going to take the matrix route.
Namely, you give lots of concrete examples
(requiring nothing more than calculus as
background) and then just state what the general
case is. In this book, the author uselessly drags
the uninitiated through swamps of archaic notation
(save that for the real thing) and incomplete
proofs (where invariably the hard parts are just quoted)
so that you have to wonder what in the world is the point
of committing this mess to paper. It is ironic that the
very same publisher already has better books out on exactly
the same topics. Finally, if this really were an introduction
you wouldn't have to add 'elementary' to the title - so let's
call a spade a spade and leave the spin to the politicians.