Groups and Manifolds

The underlying theme of this textbook can be summed up in one word: symmetry. In its most elementary mathematical formulation, symmetry is represented by finite groups (e.g., dihedral groups); in more advanced formulations, mathematical concepts such as manifolds, Lie groups, Lie algebras and symmetric spaces play a role. This book, starting from a point near scratch, covers all these topics.

The first three chapters are mostly about groups. (I say “mostly” because one section discusses Lie algebras.) The discussion in these chapters begins at a very elementary level with the definition of a group and lots of very standard examples. By the end of chapter 3, however, the exposition has moved beyond what is commonly covered in introductory undergraduate abstract algebra courses and has discussed group actions and the rudiments of group representation theory. A fourth chapter provides a fairly detailed look at classical matrix groups and crystallographic groups.

Chapter 5 introduces manifolds and Lie groups; because a major tool for studying a Lie group is the associated Lie algebra, this chapter is followed by four others on Lie algebras. Quite a lot is covered here, likely more than is typically discussed in a full semester course on Lie algebras. The topics include the standard introductory material on Lie algebras, the classification theory of semisimple Lie algebras via root systems and Dynkin diagrams, Lie algebra cohomology (including an excursion into the fundamental group, homology and cohomology), and representation theory. In addition, much more attention is paid to the five exceptional Lie algebras than is customary in textbooks;

After these chapters, the authors return to differential geometry, and spend two chapters covering such topics as Riemannian manifolds, fiber bundles, and coset manifolds and symmetric spaces. This is then followed by a chapter on applications of all of the preceding to physics, discussing, for example, molecular vibration, quantum theory of the hydrogen atom, and the anti de Sitter space. Readers without a strong background in physics will likely find this chapter very difficult.

While the book is intended as a text, exercises are provided only for the first three chapters; these are located in the back of the book, along with solutions. The authors explain why exercises are not provided for later chapters as follows: “For Lie groups and manifolds we rather rely on a collection of explicitly worked out examples of increasing complexity.”

The authors of this book are physicists, and they are writing for physics students. Fortunately for those of us who have atrociously weak physics backgrounds, the discussions throughout are very mathematical. With the exception of the last chapter, background experience in physics is generally not needed. Occasional references to objects of interest to physicists, like the Galilean Lie group and associated Lie algebra, and magnetic monopoles and the Hopf fibration, are made, however, as illustrative examples.

There is no denying, though, that a book written by physicists just reads somewhat differently than one written by mathematicians. As Sir Michael Atiyah (paraphrasing George Bernard Shaw — or Oscar Wilde; internet sources differ) said in his speech to the International Congress in Mathematical Physics in 2000, mathematicians and physicists are two people separated by a common language.

This difference manifests itself in this book in several ways. There is an intangible difference, for example, in the way things are written and expressed, and the authors are not afraid to skip some proofs. But there are also more tangible differences. For example, superscripts abound, and along with them there is frequent use of the dreadful Einstein Summation Convention.

In addition, things are also not always stated as precisely as they should be by mathematical standards. In the definition of an eigenvalue, for example, the authors omit the requirement that the associated eigenvector must be nonzero; under their definition, every \(\lambda\) will always be an eigenvalue of any matrix. Also, the authors define a maximal ideal of a Lie algebra \(L\) to be an ideal \(H\) that is not contained in any other ideal except \(H\); they should, of course, say “except \(H\) or \(L\)”. For still another (more serious) example, the derived algebra of a Lie algebra \(L\) is defined to be the set of all elements of the form \([x,y]\) where \(x,y\in L\), rather than the space spanned by all such elements. As defined by the authors, the derived algebra of a Lie algebra is not necessarily even a subspace, let alone the ideal that they claim it obviously is.

Some terminology is nonstandard, too. For example, what I would call the stabilizer of an element of a set that is acted on by a group is here called the “stability subgroup”. Fair enough; that’s no big deal. But while I define the character \(\chi\) of a representation \(R\) of a group \(G\) to be the function with domain \(G\) defined by \(\chi(g)=\mathrm{Tr}(R(g))\), the authors define it to be a vector: they take an ordered set of representatives from the various conjugacy classes of \(G\), evaluate the trace of \(R(g)\) for each \(g\) in the set, and then string the various values together to form a vector. In other words, their idea of a character (of an irreducible representation, anyway) is basically a row in the character table of the group. I don’t know whether this definition is standard in the physics community or is unique to the authors.

The physicsy style of writing may not be a huge defect in the textbook, but something else certainly is: there is no index at all. Perhaps one day in the future, print books will no longer exist and everybody will be reading word-searchable e-books, but until that day arrives, I think it is inexcusable to publish a mathematics text without an index. Unfortunately, this seems to be happening more and more, and I have now reached the point where I automatically refuse to adopt any book as a text that lacks an index.

The concerns expressed above notwithstanding, the sheer number of topics covered in this book does give it considerable value as a reference for graduate students (and faculty) in both physics and mathematics. For all such readers, however, the lack of an index is a serious impediment.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.