Lie Groups

This second edition of a successful graduate textbook and reference is now divided in four parts. Part I, rather short, focuses on two properties of compact Lie groups: Schur orthogonality (thee versions of it) and the Peter-Weyl theorem. This part also sets the level of the book: It is not an introductory textbook and assumes from the reader more than some acquaintance with analysis, differential equations, differential topology and geometry. For example, in first two pages of the first chapter the author just recalls the existence and basic properties of the Haar measure on locally compact groups. On the other hand, in chapter three we find a discussion of compact operators on normed spaces, including proofs of the spectral theorem for compact operators on Hilbert spaces and the Arzelà-Ascoli theorem.

Part II of the book, chapters 5 to 23, are entirely devoted the theory of compact Lie groups, covering all basic facts and methods. This part starts with the all-important example of closed matrix groups, their Lie algebras, exponential maps and one-parameter subgroups. Following this introduction, the general notions of manifolds and Lie groups are introduced, together with the corresponding generalized notions of exponential maps via left-invariant vector fields. Next, after reviewing some needed facts from multilinear algebra, comes the universal enveloping algebra and its main properties, and a chapter on extension of scalars.

All of these preliminaries pave the way for the introduction of (complex) representations, starting with the all-important example of SL(2,C) and its Lie algebra. Here is where we find, in a concrete setting, the first appearance of concepts that will play an important role later on, e.g., weight spaces. (I must call attention to the non-standard notation for the symmetric power of vector space, on page 59, which, if you skip Chapter 9 on multilinear algebra, could catch you off-guard when it reappears in Chapter 12, page 71ff.) The highlight of the first half of Part II is the correspondence between Lie algebra homomorphisms and Lie group homomorphisms, a consequence of the local Frobenius theorem.

The second half of Part II develops the representation theoretical aspects of the theory: Maximal tori and geodesics, root systems, Weyl groups, weights, the Weyl integration formula and the Weyl character formula. The last chapter turns towards a more topological side, focusing on the fundamental group of Lie groups. Part II of the book, about 160 pages, could be used for a one-term course on compact Lie groups and their representations.

Part III of the book is devoted to non-compact Lie groups. This part starts, in a natural way, first considering the complexification of (connected) Lie groups, a universal construction that allows the introduction of the class of reductive groups that can then be studied by means of Weyl’s unitary trick. Indeed, a large portion of Part III studies semi-simple, or more generally complex reductive, groups, where the algebraic part of the theory is more developed. Moreover, some of algebraic geometry techniques here introduced can be applied to algebraic groups over non-algebraically closed fields such as number fields, p-adic fields or finite fields, important for their arithmetic significance. Thus, in the chapters of Part III we are introduced to the Iwasawa and Bruhat decompositions, the Weyl group and the affine Weyl group as Coxeter groups, Weyl chambers and alcoves, Demazure characters and the Bruhat order. Later on, we find symmetric spaces, embeddings of non-compact symmetric spaces in its compact dual, and Cartan’s classification. The last chapter of Part III deals with the spin representation using Clifford algebras and, at some point, following an approach pioneered by R. Howe as an alternative to Chevalley’s classical approach.

Part IV of the book, its last one, is mostly devoted to the Frobenius-Schur duality in several guises. Various and important applications of this philosophy are included, ranging from the combinatorics of branching rules for representations of the symmetric group to recent results of Keating and Snaith on the distribution of values of Riemann’s zeta function in terms of random matrix theory. The last chapters of the book introduce Hecke algebras as deformations of Coxeter groups. Just mentioning Hecke algebras opens a deep well of rich interactions with several important branches of the mathematical tree: From modular representations of groups of Lie type to the theory of automorphic forms. Here the author takes to a tour to some of the vistas associated to this philosophy, opening up the vast panorama lying ahead.

Lie theory is an important and massive field that interacts with many of the main parts of mathematics. It is usually assumed that most mathematicians must know Lie theory up to a certain point, depending on their main interests. There are even attempts, with various degrees of success, to introduce the basic, concrete Lie groups, say over the real field, to the undergraduate curriculum. I think I may safely assume that if you are reading this review, we agree on the importance of Lie theory. Perhaps then I may be allowed to add that the book under review is the one every one of us must have on its desk or night table. I might be biased, but its algebraic-geometry approach to some topics just adds to its appeal.

Lastly, I must add, especially for the younger generations, that in some parts of the book and when deemed appropriate, the author mentions the well-known and ever-evolving Sage software to do some calculations. An appendix gives a quick overview of some Sage commands to illustrate the power of this software.

To sum up, this is a well-organized book with clear and well-established goals, taking the interested reader to the frontiers of today’s research.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.