Introduction to Lie Algebras and Representation Theory (Graduate Texts in Mathematics, 9)

The text on the subject - read it and work through the examples and you'll go far in life.

Classical book about Lie algebras and Representaion Theory.

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.

Moves at an advanced pace, but doesn't skip any major steps in any arguments. The exercises make you think about the material.

Professor Humphreys has accomplished clarification and teaching of this very core area of modern mathematics. He gives instructive examples and exercises.

Its beyond me how anyone could give this five stars. Its a typical mathematical monograph. That means its dense and impenetrable without the expert help of a good lecturer. Luckily the lecturer in charge of my unit is very good and after he explains something I can read this text and understand it. But its extremely hard to penetrate this text without help. These mathematical monograph texts are full of statements such as "Now it is obvious that............." and this would be true ifa) you are already an expert on Lie groupsorb) you are Evariste Galois or Srinivasan RamanujanI miss the days where I had textbooks like Anton's "Contemporary linear algebra" which is full of colour diagrams, in depth explanations and all round good vibes. This textbook has all the information you need, but its pretty hard to penetrate.

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 and including representations. Don't let the relatively small number of pages fool you; the book is quite dense, and so even covering the first 30 pages is a nice accomplishment for a student. Small caveat, the notation might be a bit confusing until you get used to it, but this is a common problem due to having both a Lie and a matrix product floating around, and is not a fault of the text. There is also a nice selection of exercises, between 5 and 10 per section.Highly recommended; every mathematician should know the basics of Lie algebras.

A quite complex combo-area, and he's done more than an adequate job of it. Hurrah!

前のレビューの方がおっしゃる通り、この本はリー代数の理論を知るには今でも欠かせないものだと思います。 中盤からの理論やその証明がごつごつしてて人によってはとっつきにくいですが、逆に言えばリー代数というものを打ち立ててきたのに先人(Lie, Engel, Killing, Weyl, Cartan...)がどれだけ苦労されてきたかを感じることができると思います。佐武先生の「リー環の話」はこの本と筋道が非常に似ていてより分かりやすく書かれているので併読するのもいいかも。線形代数と代数学の基本的な知識だけで読めるのもポイントであると思います。ある程度読み終えたら、リー代数の(普遍包絡代数の)q-変形である量子群、リー群(特に連結であるもの)や簡約代数群の理論、半単純リー代数の一般化であるKac-Moody代数とそれから得られるアフィンリー代数、リー代数のZ_{2}-次数付き版であるリー超代数など進路が豊富にありますので、これらを勉強したいならばこの本によるリー代数は知っておいて損はないです。

古典になりつつある本書であるが、リー代数についての重要な１冊と言える。半単純リー代数について、ルート系やウェイトなど基本的な事柄は一通り記述されている。また、一意性や既約性をワイル群をもちいて慎重に証明されている。数学専攻の人たちには講義などで学習済みの事かもしれないが、数学専攻以外の理工系の方々にも、リー代数を学んだときに出てくるワイル群の重要性・有用性を認識させてくれるものである。さらには、無限次元リー代数や量子群など発展しつつある時に書かれているので、有限次元リー代数に限らない証明が時折なされているのが興味深い。本書は７０年代に書かれているので用語や記号が現在と異なるところが多々あるがあとがきに現在の記法との違いを説明しているので問題ないと思う。ページ数の少ない本であるが、リー代数を知る上で重要な本であると、再度言えるのではないかと思う。