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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 (17)

mystery and math
This is a juvenile mystery written with math clues.The heroine is a math whiz in school and applies mathematical principles to most everything.It uses Venn Diagrams, Parallel Lines, and the Additive Property of Equality to solve a possible murder.It gives young readers an intriguing approach to math.

More gimmick than tale.
The idea is an interesting one: create a story based on math concepts. Much of the time, though, it seemed like the story was being twisted around so that a math concept could be inserted. Worse, the author couldn't resist waxing didactic about other things, like what the Bill of Rights is.

I agree with the reviewer who said that Tess isn't particularly likeable. She's unkind to others, she's full of herself, and she snidely explains very simple math concepts to her parents (like what a negative number is and the difference between a line segment and a line). For some reason, her parents have never heard these things before and are amazed.

Some of the math concepts introduced in the book are very simple (the meaning of > and < is taught in the primary grades). Others are very complex. The book will not really explain much about math concepts for struggling students, though it is an interesting idea.

I also agree with those who said that the plot elements were all pinched off at the end with quick and not-very-satisfying solutions. And these solutions are achieved by other people, not the protagonist.

One last peeve. As a teacher myself, I'm always annoyed by authors who are not teachers telling us what they think "good teaching" is. Invariably they seem to think a good teacher is one who imparts buckets of facts all the time (for example, the history teacher in the book rattling off facts about the Bill of Rights). This is not true. A good teacher is generally one who enables her students to *interact* with facts and concepts and to construct their own knowledge.

Inappropriate for 9 year olds!
My daughter wanted this book, so I bought it for her for her birthday.I was very disappointed when a few pages into it she asked me what Suicide was!That topic is not appropriate and way too depressing for a nine year old!

Reading AND Math are cool in Middle School
As the School-wide Language Arts Chair of a large middle school, I am always looking for books that will hook our kids.Secrets, Lies, and Algebra offers an interesting look into the 8th grade year of all students who are trying to find themselves in a world of mazes of friendships and family relationships. Most of the plot was believable in the middle school world. I particularly liked the "math" reasoning at the basis of the central plot.For those students who LOVE math and worry about reading a novel, this book offers a way to look at the universe in mathematical terms.Although I felt that the last few chapters left some gaps that were intended to be filled in by a sequel, the novel moved well, stuck to its format of offering each problem as an algebraic equation, and resolved the mysteries of the book cleanly and appropriately.I will certainly offer this title as an example of "reading across the curriculum" to all of the math teachers in my building. In short, the reader + the book = an interesting story line.

A fun read
I bought this for my daughter.Okay - I had to force it on her.She began laughing by the end of the first chapter and said she learned more about Algebra from the book than from her own school.Ms. Lichtman has done the unthinkable - used algebra to describe the angst of eighth grade and all the ensuing politics involved.

I read it and it is laugh-out-loud funny.The illustrations bring the points home.

A must have for anyone looking for a quick, fun read (and a great way to sneak in some math that is otherwise uncomprehensible in some academic environments!)

Bravo.This book is a winner! ... Read more

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Customer Reviews (8)

Lie groups, not just for particle physics
Having read, and loved, Lie Groups for Pedestrians, I picked up this book to further my knowledge of this wonderful subject.
I am not a particle physicist nor am I mathematician, I am a spectroscopist and had read some about Lie groups and their applications to spectroscopy. However to read and digest the material that was contained in the books and articles I was coming across, it was clear that I needed to know more about Lie groups and algebras. This book was exactly what I needed. It gave very clear and concise definitions (if you have had an introduction to group theory) of what Lie groups and algebras are and the tools that are needed to use them.
The exercises at the end of the sections were a real joy for me. Working problems is the best way to learn a subject like this, and they helped to clarify what the preceding chapter had talked about. The writing is anything but dry and an easy read.
To start this book I would recommend that if you are a scientist you have taken, and understood, a good introductory course to QM and group theory; if you are a mathematician that you have taken and understood a good abstract algebra course. Do not do yourself a disservice by trying to digest this book without the proper background. You will most likely turn yourself off from a very beautiful and exciting area.
This book is not for someone who has taken an intro to physics course and wants to know about all the riddles of the universe. They will be lost, frustrated and otherwise flummoxed by this book.

Not fair on non-physicist mathematicians
Couldn't get into this, I gave up in the first chapter after failing to understand how he was applying his Kronecker product to his vectors. He just failed to explain his notation adequately. *And* there were mistakes in that first bit I read up to then.

I appreciate that physicists and mathematicians use different language, and I also appreciate that this was an advanced work, i.e. postgrad plus, but it would have been nice to have seen a glossary of terms and a little more background.

This may be a competent and erudite work, but unfortunately impenetrable without unspecified previous knowledge, and that's not the way these books ought to be.

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". ... Read more

Product Description
This book addresses Lie groups, Lie algebras, and representation theory. In order to keep the prerequisites to a minimum, theauthor restricts attention to matrix Lie groups and Lie algebras.This approach keeps the discussion concrete, allows the reader toget to the heart of the subject quickly, and covers all of themost interesting examples. The book also introduces theoften-intimidating machinery of roots and the Weyl group in agradual way, using examples and representation theory asmotivation.

The text is divided into two parts. The first covers Lie groupsand Lie algebras and the relationship between them, along withbasic representation theory. The second part covers the theory ofsemisimple Lie groups and Lie algebras, beginning with a detailedanalysis of the representations of SU(3). The author illustratesthe general theory with numerous images pertaining to Liealgebras of rank two and rank three, including images of rootsystems, lattices of dominant integral weights, and weightdiagrams. This book is sure to become a standard textbook forgraduate students in mathematics and physics with little or noprior exposure to Lie theory.

Brian Hall is an Associate Professor of Mathematics at theUniversity of Notre Dame. ... Read more

Customer Reviews (8)

Abominable formatting
This review is only relevant to the Kindle edition, read on a well-functioning latest-edition (K3) Kindle e-reader.

Do not buy the Kindle edition of this book.

The default font used for the book is a struggle to read in normal font, and very nearly unreadable in italics. This is because portions of letters are missing, such as the lower half of most lowercase e's and the right half of many lowercase h's. The normal options to adjust the font are disabled, except for resizing. The Kindle version inserts spaces in the mi ddl e of wo rds, as demonst rate d in thi s rev ie w. To reduce the occurrence of this problem, it is necessary to shrink the font to its smallest size.

Entire lines in some paragraphs appear with drastically altered font size, even though in the original hardcover, these lines are normal and unaltered (all being part of the same paragraph).

This is a terrible, terrible, terrible conversion to Kindle format. Do not under any circumstances purchase this Kindle edition. The author ought to be infuriated, and the publisher ought to be ashamed. I intend to return the Kindle edition, if possible, and order a hard copy instead.

...

I should mention that, 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 are interested 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.

I would give the paper copy five stars, but Amazon does not allow me to distinguish between editions (I tried). Please bear this ratings dichotomy in mind if you wish to let my review influence your purchase.

Distracting focus on examples
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.

Lie Groups on Kindle
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.

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. ... Read more

Product Description
Definitive treatment covers split semi-simple Lie algebras, universal enveloping algebras, classification of irreducible modules, automorphisms, simple Lie algebras over an arbitrary field, and more. Classic handbook for researchers and students; useable in graduate courses or for self-study.
... Read more

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

Product 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 (5)

It's so good l, I covet to have written it myself!
A quite complex combo-area, and he's done more than an adequate job of it. Hurrah!

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

Product Description
This is the first textbook treatment of work leading to the landmark 1979 Kazhdan-Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The setting is the module category $\mathscr {O}$ introduced by Bernstein-Gelfand-Gelfand, which includes all highest weight modules for $\mathfrak{g}$ such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of $\mathfrak{g}$. Basic techniques in category $\mathscr {O}$ such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan-Lusztig Conjecture (due to Beilinson-Bernstein and Brylinski-Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: $D$-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel. ... Read more

Product 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.

Based on a lecture course given to fourth-year undergraduates, this book provides an elementary introduction to Lie algebras. It starts with basic concepts. A section on low-dimensional Lie algebras provides readers with experience of some useful examples. This is followed by a discussion of solvable Lie algebras and a strategy towards a classification of finite-dimensional complex Lie algebras. The next chapters cover Engel's theorem, Lie's theorem and Cartan's criteria and introduce some representation theory. The root-space decomposition of a semisimple Lie algebra is discussed, and the classical Lie algebras studied in detail. The authors also classify root systems, and give an outline of Serre's construction of complex semisimple Lie algebras. An overview of further directions then concludes the book and shows the high degree to which Lie algebras influence present-day mathematics.

The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions.

Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

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

Product 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 (2)

An encyclopedia
This book is really dense, and not recommended for the first encounter with Lie algebras. However, it contains exactly the things one ends up remembering after a year's course, so it's great for review, reminder, or a reference book. The exposition, as usual for Serre, is very clear.

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

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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

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This is the third, substantially revised edition of this important monograph. The book is concerned with Kac-Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems. ... Read more

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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

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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

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A comprehensive and modern account of the structure and classification of Lie groups and finite-dimensional Lie algebras, by internationally known specialists in the field. This Encyclopaedia volume will be immensely useful to graduate students in differential geometry, algebra and theoretical physics.

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Now in paperback, this book provides a self-contained introduction to the cohomology theory of Lie groups and algebras and to some of its applications in physics. No previous knowledge of the mathematical theory is assumed beyond some notions of Cartan calculus and differential geometry (which are nevertheless reviewed in the book in detail). The examples, of current interest, are intended to clarify certain mathematical aspects and to show their usefulness in physical problems. The topics treated include the differential geometry of Lie groups, fiber bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and algebras, some applications in supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite Lie algebras, the cohomological descent in mechanics and in gauge theories and anomalies. This book will be of interest to graduate students and researchers in theoretical physics and applied mathematics. ... Read more

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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

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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

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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 classictext makes the theory of Lie groups accessible to graduate students,while offering a perspective on the way in which knowledge of suchgroups can provide an insight into the development of unified theoriesof strong, weak, and electromagnetic interactions. ... Read more

Customer Reviews (13)

i'll probably try to read it again, someday.
it's here if you want it.
you'll have to work for it.
this is not written by a mathematician, so fellow math phDs be ready.
i didn't make it too far, but i wasn't so beaten down that i might not crack this one open again.
do the world a favor, please:read this book and write a slightly more understandable version of it (with more pictures :)).

Covers the material very well
The Dover books Semi-Simple Lie Algebras and Their Representations (Dover Books on Mathematics) and Lie Groups, Lie Algebras, and Some of Their Applications cover the topic, but maybe not as well and the leave out a good coverage of
angles in groups and Young's combinatorial tableaux.
The relationship of Dynkin diagrams to SU(n)( A_n) and SO(n) ( D_n and B_n) groups is well covered. I liked the coverage of generalized Gell -Mann
groups as well. The explanation of the relationship of SU(3) to SU(6) was also helpful. In general except for the price this is one of the better books on the market on this subject.

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 ... Read more

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This classic graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Lie theory, in its own right, has become regarded as a classical branch of mathematics. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras. ... Read more

Customer Reviews (2)

very, very good
this is a wickedly good book.it's concise (yeah!) and it's well written.it misses out on lots of stuff (spin representations, etc..). but once you read this book you will have the formalism down pat, and then everything else becomes easy.

if you put in the hours to read this book cover to cover -- like sitting down for 3 days straight 8 hours a day, then will learn the stuff.if you don't persevere and get overwhelmed with the stuff that is not clear at the beginning, then you will probably chuck it out the window.

lie groups and lie algebras in 200 pages done in an elegant way that doesn't look like lecture notes cobbled together is pretty impressive.

An absolute must for beginners.
I used this book as the primary text for an introductory course on Lie groups and Lie algebras. There are several aspects of the book which distinguish it from every other book on the same topic, making it an indespensable resource for the beginning student.

First, the book is, as its title indicates, an introduction, and a fairly brief one at that. It is not intended to be comprehensive in scope or in depth, rather to gently introduce some fairly complex ideas in the most basic way possible. This is the primary reason it is so useful to start with: The author knows just how much detail is necessary and skips cumbersome and unenlightening proofs. For example, he doesn't prove Serre's theorem or finish the proof of the PBW theorem, but rather refers to other books for these. In contrast to other books on the subject, the student doesn't have to sift the important points from the nitty-gritty details. Every section is important and worth reading. I particularly appreciate that the sections on Lie groups don't require that the reader is an expert in differential geometry and reviews all essential prerequisites.

Of particular value is the excellent collection of exercises. The majority of these are not particularly difficult, but most are enormously worthwhile. Having done lots of exercises from other books, including Knapp (Lie groups beyond an introduction), Hall, Humphreys, and others, I can safely say these are among the best, reaching both an optimal level of difficulty and a fair balance between computation and theory. (Note: Hall's book has great exercises too and are goodfor those who want more practice with computations). One of the main problems I have with problems in most math books is that they often feel unrelated to the material of the book and don't help to understand the material. Kirillov's exercise practically all require verifying simple details from the book or proving small parts of theorems and are all worth doing.

Finally, the book outlines many more advanced directions in Lie theory and gives appropriate references. Overall, the book has the feel of a rigorous exposition without scaring away the student with 800 pages of technical details. Of course there are simpler texts (like Hall) which just focus on matrix Lie groups and more sophisticated (Knapp) which contain everything in this book and a LOT more, but I'd say for a first read, this book is the most suitable.
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This is an introduction to the theory of affine Lie algebras and to the theory of quantum groups. It is unique in discussing these two subjects in a unified manner, which is made possible by discussing their respective applications in conformal field theory.The description of affine algebras covers the classification problem, the connection with loop algebras, and representation theory including modular properties.The necessary background from the theory of semisimple Lie algebras is also provided.The discussion of quantum groups concentrates on deformed enveloping algebras and their representation theory, but other aspects such as R-matrices and matrix quantum groups are also dealt with. ... Read more

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Affine Lie Algebras and Quantum Groups: An Introduction
A very good book, that takes every steps you need to know in Affine Lie Algebras and Quantum Groups. ... Read more

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Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the 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.

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This book is a detailed reference on Lie algebras and Lie superalgebras presented in the form of a dictionary. It is intended to be useful to mathematical and theoretical physicists, from the level of the graduate student upwards. The Dictionary will serve as the reference of choice for practitioners and students alike.

Key Features:
* Compiles and presents material currently scattered throughout numerous textbooks and specialist journal articles
* Dictionary format provides an easy to use reference on the essential topics concerning Lie algebras and Lie superalgebras
* Covers the structure of Lie algebras and Lie superalgebras and their finite dimensional representation theory
* Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras ... Read more