Book Description

Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right.

Customer Reviews (1)

Wonderful Introduction
As a senior math major I decided I wished to learn Lie Algebra, and based on my experience with the SUMS Number Theory book, I chose this one. I am taking a reading course on Lie Algebras which consists of me reading and doing the problems and asked the teacher if i have issues. This book is perfect. In about two weeks I have gotten through the first 11 chapters, and done about 80% of the problems. In addition to being very clear and simple, it is very complete. Often my adviser will ask if the book covered a particular concept, it has yet to fail. It also provides some nice examples to relate to. Unfortunately it does have several typos and not a complete solution guide, these things kept it from the 5. I especially recommend this book for self-study. ... Read more

Book Description

Customer Reviews (6)

Lie groups, examples and exercises
An excellent overview of Lie Groups and Algebras. Gilmore, as he notes himself, has concentrated on producing a self contained course for physicists. The mathematical treatment is generally detailed and shows most steps. He notes the omission of various topics in physics and mathematics, but refers the reader to specialized texts in his comprehensive bibliography. My course of Lie Groups at university was focused on mathematical applications and differential equations and this text by Gilmore provides a satisfying broader appreciation of Lie Groups and Algebras in their own right and their applications to fields and problems I wasn't previously aware of. I'm especially pleased with the many exercises which I find a great help in developing greater understanding and testing my grasp of the text.

This book becomes my reference on group theory in physics
I've waited many years to find a book like this.
It may take me many years to master everything in it,
but at least with this book I have a chance to try.
I contrast this text to books and papers by Gell-Mann, Richard Feynman,
and Steven Weinberg and these great men come off second best
when it comes to exposition of the relationships between groups.
I have found what appear to be factor of two difference
between the examples and the tables for A(n)
but those once corrected seem to leave this the complete
reference on group theory for physics that I've been looking for for a long time.
I congratulate Robert Gilmore for his well written book.

Rave Review
I haven't read this whole book cover to cover, because of time constraints.However, I can say that it is extremely clear in it's exposition.The material is very well chosen for use by physicists.I have read pure math books on this topic, and while they can be more sophisticated and thorough, they are rarely as straight forward, nor do they cover the breadth of material in this book.

In sum I would have to agree with what I was told: "this is the book on Lie Algebra for a physicist".

A classic
Gilmore's treatment of Lie groups and Lie algebras is written in the mathematical languague which theoretical physicists should be comfortable with. The notation is very clear, the discussion is nearly flawless and the physical relevance is not omitted, which is for example done in more mathematically oriented books. Very thorough, very readable and cheap!

An excellent treatment of the subject written in "mathematics."
I find Professor Gilmore's book a top-flight exposition of a "not so easy" subject.As a mathematician, I am very comfortable with the degree of mathematical rigor.It is not sloppy.I hope physicists won't be put off by the fact that this book is written in "mathematics."We are in a time when the distinction between pure mathematics and theoretical physics is rapidly blurring, if not already extinct.The physicist must become accustomed to the protocal of pure mathematics, as the mathematician must understand the needs of theoretical physics and be motivated by them.Many will disagree.

As a Ph.D candidate in mathematics a "few" years ago, on creating a proof (hopefully elegant), I used to joke with my major professor about what a disaster it would be if the physicists at our university found some way to put our reselt to practical use.
Imagine the horror of our lemma on ideal theory in a C* Algebra
being so desecrated.Oh, the shame!Now I say "a pox on such elitism and snobbery."

The work under discussion is very well motivated (the preface is an excellent historical summary of how this mathematics became so necessary for progress in physics, and how theoretical physics has motivated mathematical research.The copious "no frills" illustrations are particularly valuable to the reader, particularly if she is not accustomed to just accepting chains of head-to-tail syllogisms as sufficient.

Two thumbs up!

... Read more

Book Description
This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations.A good knowledge of linear algebra (including eigenvalues, bilinear forms, Euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra.The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are more demanding.This text grew out of lectures which the author gave at the N.S.F. Advanced Science Seminar on Algebraic Groups at Bowdoin College in 1968. ... Read more

Customer Reviews (4)

a good text
I must admit, my progress through this book can be measured in lines. It's not that it's confusing, but that it's pretty dense. The proofs are structured in such a way as to leave teasing amount of details to the reader, and the text measures understanding as much as the exercises. It is that which makes reading this book worthwhile.

From an academic point of view, the material in this book is very standard. The content of the first four chapters is closely paralleled by an introductory graduate level course in Lie Algebra and Representation Theory at MIT (although the instructor did not explicitly declare this as class text.) In many ways, this book is my ticket out of attending lectures, and it has done a great job so far.

I must admit that it can be frustrating at times to work out the statements of the proofs, but it only makes the understanding just that much more pleasant and adds the perfect amount of emotion to an otherwise black/white text.

dense and uninviting
This is a typical mathematical monograph
which means it is densely written with
almost no examples.It's too bad since
that makes decoding the text much more
timeconsuming.

There is a lot here for such a short book
This book is a pretty good introduction to the theory of Lie algebras and their representations, and its importance cannot be overstated, due to the myriads of applications of Lie algebras to physics, engineering, and computer graphics. The subject can be abstract, and may at first seem to have minimal applicability to beginners, but after one gets accustomed to thinking in terms of the representations of Lie algebras, the resulting matrix operations seem perfectly natural (and this is usually the approach taken by physicists). The book is aimed at an audience of mathematicians, and there is a lot of material covered, in spite of the size of the book. Readers who desire an historical approach should probably supplement their reading with other sources. Readers are expected to have a strong background in linear and abstract algebra, and the book as a textbook is geared toward graduate students in mathematics. Only semisimple Lie algebras over algebraically closed fields are considered, so readers interested in Lie algebras over prime characteristic or infinite-dimensional Lie algebras (such as arise in high energy physics), will have to look elsewhere. Physicists can profit from the reading of this book but close attention to detail will be required.

The first chapter covers the basic definitions of Lie algebras and the algebraic properties of Lie algebras. No historical motivation is given, such as the connection of the theory with Lie groups, and Lie algebras are defined as vector spaces over fields, and not in the general setting of modules over a commutative ring. The four classical Lie algebras are defined, namely the special linear, symplectic, and orthogonal algebras. The physicist reader should pay attention to the (short) discussion on Lie algebras of derivations, given its connection to the adjoint representation and its importance in applications. The important notions of solvability and nilpotency are covered in fairly good detail. Engel's theorem, which essentially says that if all elements of a Lie algebra are nilpotent under the 'bracket", then the Lie algebra itself is nilpotent, is proven.

The second chapter gives more into the structure of semisimple Lie algebras with the first result being the solution of the "eigenvalue" problem for solvable subalgebras of gl(V), where V is finite-dimensional. Cartan's criterion, giving conditions for the solvability of a Lie algebra, is proven, along with the criterion of semisimplicity using the Killing form. The representation theory of Lie algebras is begun in this chapter, with proof of Weyl's theorem. This theorem is essentially a generalization to Lie algebras of a similar result from elementary linear algebra, namely the Jordan decomposition of matrices. Again, physicist readers should pay close attention to the details of the discussion on root space decompositions.

This is followed in chapter 3 by an in-depth treatment of root systems, wherein a positive-definite symmetric bilinear form is chosen on a fixed Euclidean space. These root systems enable a more transparent approach to the representation theory of Lie algebras. The theory of weights along with the Weyl group, allow a description of the representation theory that depends only on the root system. In addition, one can prove that two semisimple Lie algebras with the same root system are isomorphic, as is done in the next chapter. More precisely, it is shown that a semisimple Lie algebra and a maximal toral subalgebra is determined up to isomorphism by its root system. These maximal toral subalgebras are conjugate under the automorphisms of the Lie algebra. The author further shows that for an arbitary Lie algebra that is true, if one replaces the maximal toral subalgebra by a Cartan subalgebra. The proofs given do not use algebraic geometry, and so they are more accessible to beginning students.

In chapter 5, the author introduces the universal enveloping algebra, and proves the Poincare-Birkhoff-Witt theorem. The goal of the author is to find a presentation of a semisimple Lie algebra over a field of characteristic 0 by generators and relations which depend only on the root system. This will show that a semisimple Lie algebra is completely determined by its root system (even if it is infinite dimensional).

Chapter 6 is very demanding, and will require a lot of time to get through for the newcomer to the representation theory of Lie algebras. Weight spaces and maximal vectors are introduced in the context of modules over semisimple Lie algebras L. Finite dimensional irreducible L-modules are studied by first considering L-modules generated by a maximal vector. It is shown that if two standard cyclic modules of highest weight are irreducible, then they are isomorphic. The existence of a finite dimensional irreducible standard cyclic module is shown. Freudenthal's formula, which gives a formula for the multiplicity of an element of an irreducible L-module of heighest weight, is proven. A consideration of characters on infinite-dimensional modules leads to a proof of Weyl's formulas on characters of finite dimensional modules.

The last chapter of the book considers Chevelley algebras and groups. Their introduction is done in the context of constructing irreducible integral representations of semisimple Lie algebras.

Excellent Introduction to Lie Algebras
Humphreys' book on Lie algebras is rightly considered the standard text.Very thorough, covering the essential classical algebras, basic results on nilpotent and solvable Lie algebras, classification, etc. up to andincluding representations.Don't let the relatively small number of pagesfool you; the book is quite dense, and so even covering the first 30 pagesis a nice accomplishment for a student.Small caveat, the notation mightbe a bit confusing until you get used to it, but this is a common problemdue to having both a Lie and a matrix product floating around, and is not afault of the text.There is also a nice selection of exercises, between 5and 10 per section.

Highly recommended; every mathematician should knowthe basics of Lie algebras. ... Read more

Book Description

Customer Reviews (4)

A small book with a big Kernal
I bought the book for Dynkin diagrams,
Cartan matrices and a better understanding of group theory
as it applies toLie Algebras.
I got that so I'm satisfied with the book.
What I would like is a better coverage of the Standard theory and
ideas of symmetry breaking.
I also miss the connection to Weyl gauge theory and
the differential geometry involved.
Picky , Picky , Picky...
I think that Robert N. Cahn has done a very good job with this book
for price and content, but I can also see why
Europe is ahead of the USA in physics,
since it is not what is in the book,
but what is left out that troubles me.

A practical guide to Lie algebras and representations
The objective of this book is to provide a readable synthesis of the theory of (complex) semisimple Lie algebras and their representations which are usually needed in physics. There is no attempt to develop the theory formally, as done in usual textbooks on Lie algebras, but to present the material motivated by the rotation group SU(2), and also SU(3). The book is divided into sixteen sections. The first ten give a brief overview of the classification of semisimple algebras and their representations. For the proofs the reader is referred to the book of Jacobson [Lie algebras, Wiley 1962]. The purpose of this presentation is to introduce the concepts like Killing form, weights, root system, etc, using the examples of the two groups cited above, and then give the general description. Technical results are kept to a minimum, which causes a couple of omissions which are however used in later chapters [this is the case for the decomposition of any positive root as a sum of simple roots with integer positive coefficients]. The eleventh chapter introduces the Casimir operators of Lie algebras (more precisely the quadratic Casimir operator) and Freudenthal's formula for the dimension of weights spaces. In chapter 12 the Weyl group of a root system is discussed (but without commenting the Weyl chambers). Chapter XIII presents Weyl's formula for the dimension of irreducible representations, and illustrated with examples like sl(3) or the exceptional algebra of rank two. Chapter XIV begins with topics usually encountered in physical applications, like the decomposition of the tensor product of two irreducible representations. This and later chapters are strongly influenced by Dynkin's original work. In particular the theorem for the second highest representation is developed in detail. The last two chapters are devoted to the analysis of subalgebras of semisimple Lie algebras and the branching rules (i.e., decomposition of representations with respect to a certain subalgebra). The method based on the extension of the Dynkin diagrams is carefully developed, and the question of maximality of the subalgebra (regular or not) discussed. Here an extremely important observation is made, namely the existence of some little mistakes in the Dynkin's method (concerning the maximality of certain subalgebras in the exceptional case). This is pointed out with explicit exhibition of examples. The last chapter gives an insight into the branching rules, by the development of carefully chosen examples and the presentation of some results (without proof) due also to Dynkin.
Resuming, this book provides a quick introduction to the techniques and features of (finite dimensional) Lie algebras appearing in physical theories (e.g. the interacting boson model) without being forced to digest a formal mathematical development. Inspite of few points where the reader can get puzzled (due to the use of noncommented general properties), the text achieves its purpose and constitutes a valuable reference for physicists.

A pleasant read
Not only are Lie algebras interesting and important from a mathematical standpoint, an in-depth understanding of them is essential if one is to fully comprehend the physical theories of elementary particle interactions. All of these theories, from quantum field theories to string theories, to the current research on D-branes and M-theories, are dependent on the theory of Lie groups and Lie algebras. Because of its relaxed informal style, this book would be a good choice for the physics graduate student who intends to specialize in high energy physics. Those interested in mathematical rigor would probably want to select another text. Because of space restrictions, only the first thirteen chapters will be reviewed here.

In chapter 1 the author begins the study of SU(2), the group of unitary 2 x 2 matrices of determinant 1. He does this by first considering the matrix representations of infinitesimal rotations in 3-dimenensional space. "Exponentiating" these matrices gives the finite rotational matrices. He then shows that the consideration of products of finite rotations involves knowledge of the commutators of the infinitesimal rotations. Viewing these commutators abstractly motivates the definition of a Lie algebra. He then shows that the rotation matrices form a (3-dimensional) 'representation' of the Lie algebra. Higher-dimensional representations he shows can be obtained by analogies to what is done in quantum mechanics, via the addition of angular momentum and are parametrized by spin (denoted j). The representation of smallest dimension is given by j = 1/2 and corresponds to SU(2). He is careful to point out that the rotations in 3 dimensions and SU(2) have the same Lie algebra but are not the same group.

The constructions in chapter 1, particularly the concept of "exponentiating", are central to the understanding of Lie algebras in general. This is readily apparent in the next chapter wherein he studies the Lie algebra of SU(3), the 3x3 unitary matrices of determinant 1. SU(3) has to rank as one of the most important groups in elementary particle physics. The (abstract) Lie algebra corresponding to the commutation relations of this group have various representations, the 8-dimensional, or "adjoint" representation being one of great interest. The author finds the famous 'Cartan subalgebra' of the Lie algebra, shows that it 2-dimensional and Abelian, and how eigenvectors of the adjoint operator can form a basis for the Lie algebra, as long as this operator corrresponds to an element of the Cartan subalgebra. Further, he shows that the eigenvalues of this operator depend linearly on this element, and then defines functionals on the Cartan subalgebra, called the roots, and they form the dual space to the Lie algebra. Dual spaces are familiar to physicists in the Dirac bra-ket formalism.

The geometry of Lie algebras is very well understood and is formulated in terms of the roots of the algebra and a kind of scalar product (except is not positive definite) for the Lie algebra called the 'Killing form'. The Killing form is defined on the root space, and gives a correspondence between the Cartan subalgebra and its dual. The author then shows how to use the Killing form to obtain a scalar product on the root space, and this scalar product illustrates more clearly the symmetry of the Lie algebra. The property of being semisimple is then defined abstractly by the author, namely a Lie algebra with no Abelian ideals. He states, but does not prove entirely, that the Killing form is non-degenerate if and only if the Lie algebra is semisimple.

The treatment becomes more abstract in chapter 4, wherein the author studies the structure of simple Lie algebras, since every semisimple algebra can be written as the sum of simple Lie algebras. The author shows how to obtain the Cartan subalgebra in general, motivating his procedures with what is done for SU(3). He also proves the invariance of the Lie algebra and shows that it is the only invariant bilinear form on a simple Lie algebra. After a detour on properties of representations in chapter 5, wherein he constructs some useful relations for adjoint representations, the author uses these to again study the structure of simple Lie algebras in chapters 6 and 7. This involves the notion of positive and negative roots, and simple roots, and from the latter the author constructs the 'Cartan matrix', which summarizes all of the properties of the simple Lie algebra to which it corresponds. The author shows how the contents of the Cartan matrix can be summarized in terms of 'Dynkin diagrams'.

These considerations allow an explicit characterization of the 'classical' Lie algebras: SU(n), SO(n), and Sp(2n) in chapter 8. The Dynkin diagrams of these Lie algebras are constructed. Then in chapter 9, the author considers the 'exceptional' Lie algebras, which are the last of the simple Lie algebras (5 in all). Their Dynkin diagrams are also constructed explicitly.

The author returns to representation theory in chapter 10, wherein he introduces the concept of a 'weight'. These come in sequences with successive weights differing by the roots of the Lie algebra. A finite dimensional irreducible representation has a highest weight, and each greatest weight is specified by a set of non-negative integers called 'Dynkin coefficients'. He then shows how to classify representations as 'fundamental' or 'basic', the later being ones where the Dynkin coefficients are all zero except for one entry.

In complete analogy with the theory of angular momenta in quantum mechanics, the author illustrates the role of Casimir operators in chapter 11. Freudenthal's recursion formula, which gives the dimension of the weight space, is used to derive Weyl's formula for the dimension of an irreducible representation in chapter 13. The reader can see clearly the power of the 'Weyl group' in exploiting the symmetries of representations.

A nice little summary of the theory
Very well written account of the theory, with almost all the necessary proofs to get familiar with the it. It's inspired by Jacobson's book, however a lot easier to read. It's out of print, but there is an onlinecopy. ... Read more

Book Description

In the eighth grade, 1 math whiz < 1 popular boy, according to Tess's calculations. That is, until she has to factor in a few more variables, like:
1 stolen test (x),
3 cheaters (y),
and 2 best friends (z) who can't keep a secret.
Oh, and she can't forget the winter dance (d)!

Customer Reviews (7)

You don't need numbers to learn math
This book was great fun to read whether you like math or not.It takes mathematical concepts and shows how they apply in real life situations without the use of numbers.For example, should you provide police with information about a crime that may have been committed? Tess is trying to decide whether to tell someone about what she believes is important evidence in the death of her mother's friend, but her mother insists that the family should remain silent because they have no proof.Using the concept of axioms,the information should be reported."In math, if something is an axiom, it means you don't need proof for it to be true."The proof may be left to the courts, like a theorem, which does need proof. Finally, a fun work of math fiction!

Engrossing from Page One
Even though I am way past my teenage years, I still loved reading the story of Tess and her friends, their mystery, and discoveries of real math. Lichtman's characters are distinct and vivid; her plot line is creative and original.I even learned about some math principles.My twenty-something daughter and 12 year old nephew both really enjoyed reading it as well. My daughter said it would make a fun movie.

Math is cool
What a great concept to make math cool.I loved the imagery and the way Tess thinks through her world AND that she's a girl (sorry for the run-on).Page turner even for a 36 year old kid.

Algrebra comes alive
This book is brilliant.If you love math, you'll be amazed at how the author uses the theoretical to describe human relationships and events.If you hate math, this book will give you a new appreciation for the elusive concepts you never thought you'd ever need again.If you're afraid of math, this book will make it accessible in a way it never has been before and it's all accomplished through the backdrop of a great story.Lichtman captures the voice and social culture of adolescence and so seamlessly weaves math concepts throughout that you don't even realize you're learning algebra.This book should be required reading for every teen taking algebra and for every parent who might be asked to help with homework.

Great book!
This innovative book captures the essence of middle school life and engages the mathematical and creative interests of young readers.I read this book with a small group of 11 to 15 year olds and they loved the combination of mystery, math and tales of teenage life.It's a great educational tool... and a great read! ... Read more

Book Description
An exciting new edition of a classic text

Howard Georgi is the co-inventor (with Sheldon Glashow) of the SU(5) theory. This extensively revised and updated edition of his classic text makes the theory of Lie groups accessible to graduate students, while offering a perspective on the way in which knowledge of such groups can provide an insight into the development of unified theories of strong, weak, and electromagnetic interactions. ... Read more

Customer Reviews (11)

Simple and easy to read
Its an excellent book for Physics students. It is not very mathematical and explains the concepts in very simple terms.

Group Theory Supplement
Anyone who has taken a course on the Standard Model or wants to apply group theory to physics knows that there is not one definitive resource on the subject, but Georgi comes close. Any physicist in need of group theory should find a place for Georgi on their shelf as a resource. The material is well-presented and logical, while not going too overboard in presenting material.

classical
very well written text about the algebra of standard model,
but not for beginers,a very solid background in particle physics
and symmetry methods for physics is required

A good *first* start
This book is good for what it is, namely, something to get your feet wet.When learning the basics of particle physics, e.g. as an undergrad or a beginning experimentalist, this is the quickest way to get a feel for the standard model gauge group.
However, this is *not* a complete text on group theory in particle physics (and therefore, little of what you need for supersymmetric field theories and string theories).So in addition to this book, you'd need something else with an introduction to the other things you need for your particular interest.Try Gilmore's "Applications of Lie algebras...", which I believe is out of print (in libraries).Also, Cornwell's abridged "Group theory in physics" is good (though if you can find the older set of three volumes, that may be more suited to your desires).
I don't suggest many of the other books on group theory for particles/fields/strings.There are tidbits of group theory you can pick up in the particular text you are working with, e.g. "Quantum theory of Fields" by Weinberg if you are learning quantum field theory.
For mathematical physics in general, I strongly suggest "Gauge fields, knots, and gravity" (John Baez), "Differential Geometry for physicists" (Chris Isham), and "Mathematical Physics" (Geroch).

What do you need more?
I'd say that, at least, the Georgi's book is too underestimated here.

I agree that this book lacks some notions and concepts which are usually dealt with in the matmatical literature, but not on logical clearity. Every book has its own way. For example the later parts of Green, Schwarz and Witten are also a mere sketches but it sufficiently pinpoints every important steps. A physically inclined reader(?), soon realize that it is filled with (and you may feel the leakage of) the master's intuition. You can see what mathematics going on beneath the physics. It is a well-framed series of informal lectures which reveals some space-between-lines secret. ... Read more

Book Description
This book addresses Lie groups, Lie algebras, and representation theory. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame. ... Read more

Customer Reviews (5)

Excellent introduction into the theory of Lie Groups
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.

Horrible
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.

Companion book suggestion
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, 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.

A refreshingly clear introductory text on Lie groups
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.

AT LAST, LIE GROUPS & ALGEBRAS I CAN UNDERSTAND!!
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. ... Read more

Book Description
This book is a collection of a series of lectures given by Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. These lectures focus on the idea of a highest weight representation, which goes through four different incarnations.

The first is the canonical commutation relations of the infinite-dimensional Heisenberg Algebra (= ocillator algebra). The second is the highest weight representations of the Lie algebra gl¥ of infinite matrices, along with their applications to the theory of soliton equations, discovered by Sato and Date, Jimbo, Kashiwara and Miwa. The third is the unitary highest weight representations of the current (= affine Kac-Moody) algebras. These algebras appear in the lectures twice, in the reduction theory of soliton equations (KP ® KdV) and in the Sugawara construction as the main tool in the study of the fourth incarnation of the main idea, the theory of the highest weight representations of the Virasoro algebra.

This book should be very useful for both mathematicians and physicists. To mathematicians, it illustrates the interaction of the key ideas of the representation theory of infinite-dimensional Lie algebras; and to physicists, this theory is turning into an important component of such domains of theoretical physics as soliton theory, theory of two-dimensional statistical models, and string theory. ... Read more

Book Description
This is an introduction to Lie algebras and their applications in physics. The first three chapters show how Lie algebras arise naturally from symmetries of physical systems and illustrate through examples much of their general structure. Chapters 4 to 13 give a detailed introduction to Lie algebras and their representations, covering the Cartan-Weyl basis, simple and affine Lie algebras, real forms and Lie groups, the Weyl group, automorphisms, loop algebras and highest weight representations. Chapters 14 to 22 cover specific further topics, such as Verma modules, Casimirs, tensor products and Clebsch-Gordan coefficients, invariant tensors, subalgebras and branching rules, Young tableaux, spinors, Clifford algebras and supersymmetry, representations on function spaces, and Hopf algebras and representation rings. A detailed reference list is provided, and many exercises and examples throughout the book illustrate the use of Lie algebras in real physical problems. The text is written at a level accessible to graduate students, but will also provide a comprehensive reference for researchers. ... Read more

Customer Reviews (1)

Mixed feelings
Lie groups and Lie algebras permeate most parts of theoretical physics. Every student in physics should have some basic notions of the subject as it sometimes tends to have unsuspected applications.

The first three chapters of this book include exemples and motivation for the more formal aspect of the Lie theory. Those are also meant to set the notation used later throughout the book. Topics covered should be well-known from a senior undergraduate student with a good background in quantum mechanics (harmonic oscillator, the rotation group) and particle physics (mostly the "zoological" part of it : classification of particles, the eightfold way and so on).
From chapter 4 on, the Maths definitely take the most prominent part of the stage. Chapter 4 is a reminder of basic notions in algebra, as covered in an undergraduate course in algebra and classical groups.
Chapter 5, on representation, should not be a challenge to the physicist.
The core of the subject is presented in chapter 6, where the idea of the Cartan-Weyl basis is given a nice presentation. This chapter is a little bit more demanding. Some statements are not proved. However, a committed student in physics, should be able to devise proofs for him/herself.
Chapter 7 is particularly enjoyable, dealing with Dynkin diagrams and the classification of finite simple Lie algebras, and introducing infinite dimensional ones. The way Kac-Moody algebras appear, through relaxing the axioms of the Chevalley-Serre construction should be appreciated. Also, physical exemples are to the point.
However, beginning with chapter 12, the wrongs of this book become somewhat annoying. For instance, in chapter 12, the authors of this book freely speak of Verma modules, highest weight representations, while these concepts are to be introduced and properly developped in later chapters. I found this chaffing from an introductory book. From chapter 12, it seems that the reader is to gently follow and accept the statements made by the author, without encountering much proof or hint to this all.
Things come more acceptable in later chapters only, where invariant tensors and other things more familiar from a physicist with no previous acquaintance to Lie algebras, are exposed.

All in all, a good book for some parts of it but whose value could have surely been enhanced by adopting a more pedagogical presentations. Some proofs to key facts in the more "exotic subjects", would have been welcome, too. All the more, that some chapters of this book did not require much work from the authors, as it seems that they were taken from Dr. Fuchs "Affine Lie algebras".
Hopefully, welcome additions will be added to a further edition.
Beginners or readers with a casual interest in Lie algebras should better learn it from another source.
... Read more

Book Description

Customer Reviews (1)

A Gem from The Past
I recently noticed that my early edition of this book could not be found, so, I ordered another copy. It is just as good as my recollections some fourty and more years later told me. There is really no more to be said, despite more recent work, and the discoveries by physicists between 1965 and 1995. I think we all hope that Lie Algebras will be just as useful in interpretation of results soon to be forthcoming from the European Super Collider, and we hope from an even bigger paticle accelerator built somewhere in the United States. ... Read more

Book Description

This book, designed for advanced graduate students and post-graduate researchers, provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, a concise exposition is given of the basic concepts of Lie algebras, their representations and their invariants. The second part contains a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book contains many examples that help to elucidate the abstract algebraic definitions. It provides a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators and the dimensions of the representations of all classical Lie algebras.

Customer Reviews (1)

A welcomed unified approach of Lie algebras applied to Physics
This book is not a manual in the usual sense, but a compilation of facts concerning Lie algebras that continuously appear in physical problems. The material covered is the result of various seminars given by the author during many years, and synthetize the main facts that should be known to any physicist.

The material is divided into 12 chapters of variable length. The first two present the main theory of semisimple Lie algebras, enumerating the key results from root theory and Dynkin-Coxeter diagrams to classify the complex simple algebras. The real forms for the classical algebras are given in table form, without going into its detailed obtainment. It should also be taken into account that for Dynkin diagrams, the author does not distinguish betweenpositively and negatively oriented angles, thus the angles between roots in equation (2.8) are reduced to five (unoriented) angles instead of the usual eight (oriented) angles.

Chapter three compiles the most important facts about Lie algebras of Lie groups, mainly focused on matrix groups. Important techniques like the exponential map and the covering of groups are nicely illustrated with the classical unitary algebra su(2) and the Lorentz group (in one dimension). I personally miss some comment on the left invariant vector fields or 1-forms (Maurer-Cartan equations), of importance in many applications to cosmology.

The fourth chapter is devoted to representation theory. Although the Weyl decomposition theorem is not included, it is assumed that any representationdecomposes as a direct sum of irreducible modules (valid for semisimple Lie algebras). The fundamental representations are discussed for the classical algebras (symplectic, unitary and orthogonal), and for the latter, the spinor representations are also given. The dimension formulae are given, and the tensor products (Clebsch-Gordan problem) is developed by means of Young tableaux. This is applied to the branching rules of representations with respect to some chain and the missing label problem, illustrated by examples that are typical in the interacting boson model.

In chapter five, Casimir operators of Lie algebras are defined and obtained for the classical Lie algebras. Here the author uses the Perelomov-Popov approach of operators that can be identified with symmetric elements in the universal enveloping algebra. At the beginning of the chapter it is said that the number of Casimir operators equals the rank of the algebra. Again, this is only valid for semisimple Lie algebras, and generally false for arbitrary Lie algebras. The eigenvalue problem is presented using important examples, and the results resumed in a table at the end of the chapter.

The previous chapter is a nice motivation for tensor operators in general, which comprise essential techniques like the coupling and recoupling coefficients, how to determine them and their symmetries (much of this material was originally developed by Racah in his Princeton lectures of 1951). This chapter is of great importance for applications.

Chapters 8 and 9 are devoted to another technique of great relevance, the realizations of Lie algebras by means of creation and annihilation operators, divided into boson and fermion operators, according to commutation or anticommutation relations. Here the unitary case is exploited, and many subalgebra chains are analyzed with respect to these realizations. Of special interest are the sections concerning the L-S and j-j couplings used in spectroscopy of light nuclei and shell models, and where original examples have been used.

Chapter 9 presents another possibility for realizing Lie algebras, namely by differential operators. Although a short chapter, important topics like the Casimir operators as differential operators or the Laplace-Beltrami form is presented. In chapter 10, the classical matrix realizations (in fact representations by linear operators) are briefly recalled, and the classical interpretation of the Casimir operators is recovered (without using the Schur lemma).

The two last chapters deal with quite more specific topics, like dynamic symmetries, studied in both fermionic and bosonic systems, in the unitary algebras u(6) and u(4), in order to obtain mass and energy level diagrams. For the part of degeneracy algebras, the problems illustrated are the isotropic harmonic oscillator, the Coulomb problem and the Teller-Pöschl and Morse potentials. In all these problems the reader is referred to original articles to complete the information presented.

The chapters of the book do not develop the theory systematically, but rather focus on a type of problem or technique which is developed using the main Lie algebras appearing (mainly) in spectroscopy, atomic, nuclear and molecular physics, as well as quantum mechanics. No proofs are given, which prevents the reader from being distracted from the main objective of the lectures. To fill the gaps, the reader is led, at many places, to consult either original references or more formal books.

The book is written in an informal style, which simplifies its reading and makes it a suitable consultation work. The profusion of examples (many of them actually coming from original references) explains quite well the topics studied, and gives a concrete idea how to apply the techniques. It is a very welcomed addition to the literature that contains much topics treated for the first time in textbook form.

There are few misprints and mistakes in the text, which can however confuse the reader having no previous knowledge on Lie algebras. For example, on page 7, the definition of semidirect sum is confusing and wrong. It is actually not necessary that one of the algebras is an ideal in the other, it suffices that one of them acts by derivations on the other. The definition given in the book is incompatible with example 13 on the same page. Another confusing point is subsection 1.13. Here by derivations the author means the derived series of an algebra, determining whether it is solvable or not. Derivations are linear maps satisfying the Leibniz rule, and are completely independent on the solvable character. The notation is very confusing, since the derived subalgebra (commutator ideal) is denoted in the same manner as the Lie algebra of derivations (which is actually a linear Lie algebra).

Inspite of these minor details, the book will certainly be of great use for students or specialists that want to refresh their knowledge on Lie algebras applied to physics. The list of references is quite complete and provides a deeper insight into the problems where these structures appear. However, there are also some surprising absences in the references, such as the books ofJ. F. Cornwell or H. Lipkin, in my opinion two classicals on group theory in physics. Among the original articles, I miss for example the relevant review article by R. Slansky [Phys. Rep. 79 (1981), 1-128], although it is clear that giving a complete reference list is impossible.

Resuming, the book by Iachello constitutes an excellent reference for those interested in the practical application and techniques of Lie algebras to physics, and that try to avoid the often embarrassing theoretical works. It should also be mentioned that much of the material is divided into hundreds of original articles, and therefore a unified presentation will be of great use for the physical community. ... Read more

Book Description
Three of the leading figures in the field have composed this excellent introduction to the theory of Lie groups and Lie algebras. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part, Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie groups. And in the final part, Ian Macdonald gives an introduction to special linear groups.Graduate students requiring an introduction to the theory of Lie groups and their applications should look no further than this book. ... Read more

Customer Reviews (1)

A very recommanded book on the subject
This book is an introduction to the subject of Lie groups and Lie algebras. This is a general overview on the subject for students with no background on the subject. There are almost no proofs and this is not a text book. Nevertheless for the mature reader (say someone toward the end of his graduate studies) this is an amazing general introduction. It is clear that a lot of effort was used in order to give the reader as much knowledge as possible with minimal technical details as possible. Intuitive arguments are used when ever possible and motivation is there all along the way. The writing is very elegant and it is clear that the writers are masters in their field. So, if you want an overview on the subject and you do not need to go into technical details look no further then this book. ... Read more

Book Description
This book reproduces J-P. Serre's 1964 Harvard lectures. The aim is to introduce the reader to the "Lie dictionary": Lie algebras and Lie groups. Special features of the presentation are its emphasis on formal groups (in the Lie group part) and the use of analytic manifolds on p-adic fields. Some knowledge of algebra and calculus is required of the reader, but the text is easily accessible to graduate students, and to mathematicians at large. ... Read more

Customer Reviews (1)

Nice and complete, but very Bourbaki-looking.
The same problem with all Bourbaki authors: They treat the subject in a very concise, abstract, and authoritative way, but present almost no motivation to introduce the subject, and they are not so used to giveextensive and accurate references. Of course Serre is a leading expert inthe field, but he (nobody) cannot be regarded as the inventor of thetheory, so the absence of such a bibliography is not justifiable.

Thecontents of the book are: Lie algebras, filtered groups and lie algebras,universal algebra of a Lie algebra, free Lie algebras, nilpotent andsolvable Lie algebras, semisimple Lie algebras, representations of sl_n,complete fields, analytic functions, analytic manifolds, analytic groups,Lie theory. Includes excercises.

Useful for graduate students and workingmathematicians, along with a "lighter" reference.

Please checkmy other reviews (just click on my name above). ... Read more

Book Description

The theory of Lie algebras and algebraic groups has been an area of active research for the last 50 years. It intervenes in many different areas of mathematics: for example invariant theory, Poisson geometry, harmonic analysis, mathematical physics. The aim of this book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the final chapters. All the prerequisites on commutative algebra and algebraic geometry are included.

Book Description
These notes, already well known in their original French edition, give the basic theory of semisimple Lie algebras over the complex numbers including the basic classification theorem. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and representation theory. The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out. The last chapter discusses the connection between Lie algebras and Lie groups, and is intended to guide the reader towards further study. ... Read more

Customer Reviews (1)

One of the most valuable expositions in Lie-theory
This book is intended as a short concise overview of the theory of complex semisimple Lie algebras. Inspite of its small volume, this text is far from being of easy lecture, since it assumes the knowledge of some basic facts concerning Lie algebras, as well as associative algebras. Indeed the first chapters are a résumé, without proofs, of some basic theorems of Lie algebras. This concerns solvable and nilpotent Lie algebras, as well as some generic results on semisimple algebras (results that do not involve Cartan subalgebras).
The proper exposition begins with the third chapter, dealing with Cartan subalgebras. Two fundamental facts are exposed in this chapter: existence and conjugacy of these subalgebras. The existence is proved by exhibiting the classical construction by means of regular elements, i.e., elements of the algebra whose annihilator is of minimal dimension. The conjugacy of Cartan subalgebras, which enables us to define the numeric invariant called rank, is developed in analogous way to the book of Chevalley [Théorie des Groupes de Lie, 1951]. Chapter four is devoted to the study of the complex simple Lie algebra of rank one, sl(2,C). This algebra plays the key role in the study of semisimple algebras and their representations, which justifies a separated treatment. The irreducible representations of sl(2,C) are obtained.
The root theory is introduced in the following chapter. Here the first innovation is made, namely, developing the root systems before dealing with the Cartan decomposition. In particular, no inner product has been used yet. Root systems are defined over a real vector space V, and the Weyl group is defined as the group generated by certain involutions associated to the roots [one will observe observe the similarity of this definition and the theory of Coxeter groups]. The inner product on V is obtained as an inner product which is invariant under the Weyl group. Bases of roots and their elementary properties are developed, and how to go from a basis to another by emans of the Weyl group [it is supposed that the root system is reduced, for nonreduced systems see for example the sixth chapter of Bourbaki: Algèbres de Lie, Hermann 1967]. Then it follows the notion of Cartan matrix (obtained from the inner product previously defined), and the associated Dynkin diagram. All admissible Dynkin diagrams are enumerated, and their corresponding root systems enumerated. Chapter six begin with the classical Weyl theorems, and the Cartan decomposition of a semisimple Lie algebra is obtained. From this the root system associated to the algebra follows naturally. The core of the chapter is the existence and uniqueness proofs of semisimple Lie algebras corresponding to a root system. As an appendix, a theorem showing how to construct semisimple Lie algebras from root systems by means of generators and relations [that is, using presentations]. This result is of extreme importance, and constitutes one of the germs that lead to the notion of Kac-Moody algebras in 1968. The next chapter is a standard treatment of representation theory of semisimple Lie algebras. The existence of dominant weights is shown, from which the (canonical) bijection between dominant integral forms and the finite dimensional irreducible modules follows. The Weyl character formula is also presented, but without proof.
The final chapter presents some important results of compact groups, intended to facilitate the lecture of more advanced texts [like the book of Pontryaguin, for example].
Resuming, an excellent text that presents a good insight to the theory of complex semiimple algebras. It should however be said that probably this book is not convenient for a first contact with these structures, due to its comprised presentation and the complete absence of exercises or many examples [this applies at least to the original french edition]. Actually, some acquaitance with Lie theory is implictly supposed through the text. ... Read more

Book Description
The third, substantially revised edition of a monograph concerned with Kac-Moody algebras, a particular class of infinite-dimensional Lie albegras, and their representations, based on courses given over a number of years at MIT and in Paris. Suitable for graduate courses. ... Read more

Book Description
This is the softcover reprint of the English translation of 1975 (available from Springer since 1989) of the first 3 chapters of Bourbaki's 'Groupes et algèbres de Lie'. The first chapter describes the theory of Lie algebras, their derivations, their representations and their enveloping algebras. In Ch. 2, free Lie algebras are introduced in order to discuss the exponential, logarithmic and the Hausdorff series. Ch. 3 deals with the theory of Lie groups over R and C and ultrametric fields. It describes the connections between their local and global properties, and the properties of their Lie algebras. It is one of the very best references on this subject. ... Read more

Customer Reviews (1)

great as references, and..
...even better when supplemented by other texts (be wary though, as the notation of Bourbaki is not universally accepted).
I'd buy themsimply for their sparkling clarity and careful attention to rigour (esp. Commalg, Lie theory and integration).
These arewonderfully crafted, masterful expositions, and shouldnot, by any means, be overlooked by the (aspiring) pure mathematician.

Cheers,
A. ... Read more

Book Description
From reviews of the first edition: "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and text book in the future." -- Newsletter of the EMS"This is a fundamental book and none, beginner or expert, could afford to ignore it. Some results are really difficult to be found in other monographs, while others are for the first time included in a book." -- Mathematica"Each chapter begins with an excellent summary of the content and ends with an exercise section... This is really an outstanding book, well written and beautifully produced. It is both a graduate text and a monograph, so it can be recommended to graduate students as well as to specialists." -- Publicationes MathematicaeLie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups.Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond. ... Read more

Customer Reviews (1)

Review of Knapp's "Lie groups: beyond an introduction."
The short version:this is a superbly written and conceived book;if I had to learn this material (the basic theory of
structure and representation of Lie algebras and groups,
especially semimsimple ones) from a single book, this is
the one I'd choose, among those I've seen.If you know the
basics of abstract algebra and some very basic concepts from
topology and manifolds, and you want to learn this material,
use this book.It would be a good reference, too, as it is
easy to find things in it, and takes a fairly modern, sophisticated approach (without sacrificing motivation and
intuition).

The long version, if you want more convincing or details:

I have used several books recently in learning the structure and
representation theory of Lie algebras and groups (especially Humphreys' Introduction to Lie algebras and representation theory, Fulton
and Harris' "Representation Theory," Varadarajan's "Lie groups,
Lie algebras, and their representations.")Although I came to Knapp's book with a decent background from the others, I think it's the best pedagogically, for someone with a modicum of mathematical sophistication and some basics like abstract
algebra and an idea of what a smooth manifold is), and a smattering of Lie theory.Some examples of the book's strength:
Elementary but potentially confusing concepts (like complexification, real forms, field extensions)
are explained thoroughly but in a sophisticated way, rather
than viewed as obvious.Carefully chosen examples motivate and
clarify the general theory;consequently even though the book
is completely rigorous, and carefully delineates lemmas, proofs,
remarks, definitions, and the like, it seems less dry then some
others (e.g. Varadarajan, from my point of view).But the point
of the examples, and their relation to the general theory, is
made clear, so they do not provide an overload of detail or b
obscure the main structure.Thought is always given to the
reader's understanding, not just to logical correctness, though
the author also takes the point of view, with which I concur,
that logical clarity and sufficient detail are essential
to understanding.Relations between ideas, alternative
proofs, and the structure of the theory to come are discussed
thoroughly, but such discussion is clearly demarcated from
the main structure of the argument, so that the latter is never
obscured.This is a fantastic book, and exactly what I was
looking for.Whether you are learning the material for the
first time, or want to review it or refer to, it is a superb
source. ... Read more

Book Description

This is the English translation of Bourbaki's text Groupes et Algèbres de Lie, Chapters 7 to 9. It completes the previously published translations of Chapters 1 to 3 (3-540-64242-0) and 4 to 6 (3-540-42650-7) by covering the structure and representation theory of semi-simple Lie algebras and compact Lie groups. Chapter 7 deals with Cartan subalgebras of Lie algebras, regular elements and conjugacy theorems. Chapter 8 begins with the structure of split semi-simple Lie algebras and their root systems. It goes on to describe the finite-dimensional modules for such algebras, including the character formula of Hermann Weyl. It concludes with the theory of Chevalley orders. Chapter 9 is devoted to the theory of compact Lie groups, beginning with a discussion of their maximal tori, root systems and Weyl groups. It goes on to describe the representation theory of compact Lie groups, including the application of integration to establish Weyl's formula in this context. The chapter concludes with a discussion of the actions of compact Lie groups on manifolds. The nine chapters together form the most comprehensive text available on the theory of Lie groups and Lie algebras.