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Lie Algebras

Nathan Jacobson
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
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Back in its day (that being 1962, the year it was first published by Wiley Interscience), this book was regarded as being one of the most accessible introductions to Lie algebras at the early graduate level. Of course, honesty compels me to point out that, back then, the bar was not set particularly high; in fact this book may well have been the only real textbook-level introduction to the subject.

Jacobson has, of course, a well-deserved reputation as a top-tier mathematician, but he is not generally known as an author of student-friendly books. The succinctness of his exposition has been commented on before in this column (see the review of his Basic Algebra I) and is certainly evident in this book as well. It is not an easy read by any means. Since its publication other textbooks have appeared that cover the purely algebraic theory of Lie algebras, notably Humphreys’ Introduction to Lie Algebras and Representation Theory, Erdmann and Wildon’s Introduction to Lie Algebras, and Henderson’s Representations of Lie Algebras. These are much more accessible to graduate students (and perhaps even very good undergraduates). In addition, books have appeared (such as Stillwell’s Naive Lie Theory) that discuss, at an undergraduate level, the connections between Lie algebras and Lie groups. Nevertheless this book remains, I think, a classic in the field.

Although it starts at the beginning with the definition of a Lie algebra, the book winds up covering a great deal of material, some of it quite advanced (and still, to this day, not easily found in other textbooks). A quick (non-exhaustive) summary of the contents follows.

The first three chapters introduce Lie algebras and discuss some of the basic examples of them, as well as the standard theorems about, and kinds of, Lie algebras (solvable, nilpotent, semi-simple, etc.) Some fairly sophisticated ideas, such as cohomology groups, the theorems of Levi and Mal’cev, and Weyl’s theorem on complete reducibility, are discussed as well. These first two topics are not covered in Humphreys’ text; he states in the preface of his book that he views them as being “better suited to a second course”.

In chapter IV, the classification theory of simple Lie algebras is discussed. Most textbook treatments do this for Lie algebras over an algebraically closed field of characteristic 0; Jacobson retains the characteristic 0 hypothesis but instead of working over algebraically closed fields, he deals more generally with “split” Lie algebras — i.e., those where a Cartan subalgebra splits into root spaces. (This splitting always occurs when the field is algebraically closed.)

Chapters V and VI develop some more general theory of Lie algebras. Chapter V introduces the universal enveloping algebra and proves the Poincaré-Birkhoff-Witt theorem. In addition, this chapter introduces the concept of a restricted Lie algebra over a field of characteristic p; these are Lie algebras that are equipped with a “p-power mapping” satisfying certain properties. First introduced by Jacobson in a paper in 1937, these Lie algebras are easier to handle than are general algebras over a field of characteristic p; they played a major role in the classification of simple Lie algebras over such fields. They are rarely mentioned in introductory texts on the subject.

Chapter VI is devoted to the twin theorems of Ado and Iwasawa. These theorems essentially say that any finite-dimensional Lie algebra is isomorphic to a subalgebra of the Lie algebra of n x n matrices (with the bracket operation being the ordinary matrix commutator AB – BA). These results are true in both characteristic 0 (Ado) and characteristic p (Iwasawa); they are also not covered in Humphreys’ book.

Chapters VII and VIII discuss topics in the representation theory of semisimple Lie algebras over a field of characteristic 0, specifically the characterization of the irreducible representations of a such a Lie algebra in terms of weights (with respect to a Cartan subalgebra) and Weyl’s character formula.

Chapter IX establishes (via a detour through algebraic geometry) the conjugacy of any two Cartan subalgebras of a semi-simple Lie algebra over an algebraically closed field of characteristic 0, and also determines the automorphism groups of the simple Lie algebras over these fields. The final chapter extends some of these results to Lie algebras over arbitrary fields of characteristic 0. This last chapter is algebraically quite demanding, and requires as a prerequisite a good knowledge of both Galois theory and the Weddeburn theory of associative algebras.

In some respects, this book does show its age. Jacobson’s notation, especially his use of Fraktur letters such as \(\mathfrak{L}\) and \(\mathfrak{M}\) seems old-fashioned and, to my eye anyway, unpleasant. In addition, as I noted earlier, there have been improvements in the development of Lie algebra theory. More elementary proofs of old results (such as the conjugacy of Cartan subalgebras) have appeared, and new approaches to the subject (such as an axiomatic approach to root systems) tend to focus attention on what is important, and improve overall clarity.

For these reasons, and given Jacobson’s concise writing style, I would not use this book as a text or recommend it to a student wanting to know what Lie algebras are all about. (I’d go with the previously mentioned books by Stillwell and Erdmann and Wilder, instead.)

Nevertheless, this book retains an undeniable appeal. Part of this appeal, to me, may be sentimental, because I studied from it in my younger days. Part of the appeal also lies in the fact, mentioned earlier, that there are things here that are still not easily found in the textbook literature. And then there is also simple appreciation for a job well-done. I think it is worthwhile to remember that blazing a trail is often harder than clearing up the trail later on, and there is no denying that this book was a trail-blazer.

Mark Hunacek ( teaches mathematics at Iowa State University. 

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