The authors of this problem book are all associated with the International School for Scientific Computing at the University of Johannesburg in South Africa. It is clearly based on a course taught there. There is little indication of what the intended audience is; given the style of the book and the content of the “Applications” chapter, one assumes it would largely consist of students with a background in physics and computation.
There are four chapters, one for each of the four topics listed in the title. Each chapter is preceded by a very brief summary of basic ideas. (The one on groups is two pages long; lie algebras get four.) Then come problems, each followed by a solution. Most of the problems are computational, some messily so. In fact, each of the chapters ends with a section giving problems to be solved using a computer algebra system, with examples and solutions given in Maxima.
The section on groups is fairly typical of the book. The first thirty or so problems are straightforward exercises that either ask the reader to check something (that this is a group, that this group is abelian, that conjugation is an automorphism) or to prove a standard result in elementary group theory. Many standard examples of small finite groups appear, but often in an unusual form. Sn is the group of n×n permutation matrices, for example. The “Pauli spin matrices” show up a lot, as does the Kronecker product of matrices and other explicit constructions. Group representations are also treated. Problem 59, for example, gives an explicit degree 3 representation of the cyclic group with three elements. The problem asks the reader to show that this is indeed a representation and to decide whether it is reducible. The solution checks all products explicitly and then decomposes each matrix in the representation into an explicit direct sum. Curiously, the direct sum of two matrices is defined only in Problem 60.
The remaining sections are in a similar spirit, with lots of explicit computations involving small matrix groups. To a pure mathematician, the “applications” look very strange. There is stuff on quantum computing and some differential equations derived from physics, but most of the “applications” are messy computations whose point is not made explicit. For example, Problem 85 in the applications section asks the reader to do some calculations in su(1,1) using an explicit basis, after which comes stuff on the “Bose annihilation and creation operators” and the vacuum state. I can follow the computations but have no idea what the point is. Problem 59 even asks explicitly for a discussion of the results of a computation; alas, the solution does not include such a discussion.
Here and there I saw solutions that would not pass muster in my classes. Problem 81 in the chapter on groups asks for a proof that if H = SL2(R) and g is a matrix with determinant 2, then the coset gH is the set of all matrices with determinant 2. The proof checks that the determinant of any matrix in gH is in fact 2, but does not check the reverse inclusion.
This book contains many problems of a sort that don’t come naturally to my mind; I suspect most mathematicians would have a similar reaction. As such, it might serve as a useful complement to a more standard textbook.
Fernando Q. Gouvêa is currently trying to teach some of his Colby College students about Lie theory.