Introduction to Lie Algebras and Representation Theory

Graduate Texts in Mathematics #9

by James E. Humphreys

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 admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor­ porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.

Genres: Mathematics, Academic, Algebra, Textbooks

186 pages, Hardcover

First published January 23, 1973

Reviews

[J · Rating 4 out of 5]

This is a great introduction to Lie Algebras and Representation Theory as the title claims, and probably the best textbook currently in existence (although perhaps combining this book with Fulton & Harris would make it even better). The book is, however, very dense -- partly due to the fact that the subject is somewhat computational (as are most of the classification theorems), but also many proofs in the book are far from optimal and can be simplified greatly. As a result, it takes longer to read than it actually should.

[Jorge Ortiz · Rating 4 out of 5]

Nice but difficult book. The topic treated is a complex one, but I think he gives some intuition in introducing concepts (at least in the first two chapters). Such as why Engels or Lies theorem, why sl(2,C) is so important, or why the adjoint rep, and many more. Maybe the universal enveloping algebra or the Verma modules should be read somewhere else. From section 4 everything becomes tedious. But all in all, is a really abstract topic what we are dealing with, and the book relies on rigor

[NoName66 · Rating 3 out of 5]

finished chapter 1 - 4 and some parts of chapter 5. The proofs in this book are not easy to read, but they deserve some time to digest.

[Oliver Waite · Rating 5 out of 5]

Remarkably readable and a good reference.

[Mikhail Ignatev · Rating 5 out of 5]

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МНОЖЕСТВО упражнений позволят любому читателю стать вполне familiar с основами теории алгебр Ли.