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Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory

Ibrahim Assem, Daniel Simson, and Andrzej Skowroński
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts 65
[Reviewed by
Gizem Karaali
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The first point that I should make about this book is about its audience. Though published in the series of London Mathematical Society Student Texts, Elements of the Representation Theory of Associative Algebras is by no means geared toward advanced undergraduate or beginning graduate students. In fact, its authors explain at the very beginning that it is mainly aimed at the graduate student beginning research in representation theory. This should be interpreted not as a word of discouragement, but rather as an indication that the reader should have a substantial background in graduate level algebra. Moreover, looking at the contents of the book, one immediately notices that some previous exposure to basic category theory and homological algebra would be necessary for a comfortable ride. Quite a few facts and basic results about categories, functors and general homological algebra can be found in the detailed appendix, and the first chapter is on algebras and modules. However a reader who has never been exposed to this material will have a very difficult time surviving most of the text.

Having said that, I would very heartily recommend this text to anyone wishing to learn about representation theory of associative algebras. As long as the reader has the prerequisite algebra background and the motivation to read a tome of this size (over 400 pages of pure mathematical prose!), this book will prove a most welcoming introduction to the theory.

The material here is in textbook form for the first time and what a delight it is to read! Through the words of the authors, the expansive and ever-growing literature in the field is transformed into a great textbook which finally makes this beautiful subject accessible to beginners. I, for one, would have benefited immensely if this book had been published during my graduate school years. Even though much (if not all) of its content is well-known by active researchers in the field, it is rather difficult for a fresh graduate student to access all this information by going through the original research papers in a reasonable period of time. There are plenty of well-chosen examples and exercises throughout the text, making it even more suitable to the classroom or for self-study.

This text will be a welcome addition to mathematical libraries, private or institutional, and not only as a textbook for graduate students who wish to specialize in representation theory. Ideas and techniques of representation theory have proved relevant to developments in many different fields. Now those who were not trained as representation theorists, (but of course only those who have the algebraic prerequisites), will find in this book a very efficient way to introduce themselves to this vital mathematical theory. In fact even before I got my hands on my copy of the text, I had already met mathematicians working in different fields who had been studying it and applying ideas found there to their fields to get novel results.

After the introductory chapter on algebras and modules, the book has eight other chapters, and these all have very distinct flavors. In a book written by three authors I was very happy to find no glaring marks of this collaboration; the voices of the authors seems to have blended very well into one another and the book reads very smoothly. The variance in flavors is due mainly to the nature of the subject under study. The representation theory of associative algebras is basically the study of modules of algebras. However there have been many independent directions that researchers in the field have followed. The last few decades have seen incredible advances in many of these directions, but, most interestingly, the various approaches of different schools have finally been woven into a tapestry of mathematics at its peak of elegance. This book tells this story as it should be told.

In Chapter 2 the authors first introduce then study in detail the theory of quivers, or directed graphs, as a tool for describing and understanding associative algebras and their representations. This part of the theory is relatively basic and a reader with a solid graduate algebra course under his belt will travel problem-free through the terrain. In Chapter 3, the ideas developed are put into use in a deeper study of modules. Then, in Chapter 4, come the heavy artilleries of irreducible morphisms and almost-split sequences. Here, exposure to some basic homological algebra will suffice for a superficial understanding of the material, but a reader without a solid background will have some difficulty following everything. Chapter 5 is easier to follow; the ideas developed earlier are applied in this chapter to take the reader through a tour of Nakayama algebras, perhaps one of the best understood classes of algebras.

Chapter 6 is again theory-heavy; it introduces the very powerful techniques of tilting theory. This is, in its core, a generalization of Morita theory, in that one attempts to study module categories of algebras through the study of other algebras which are not really equivalent but somewhat simpler. Chapter 7 introduces yet another important tool of the theory of representations of associative algebras, integral quadratic forms. However this is not the single purpose of this chapter. It in fact is quite climactic, because it provides a full description of representation finite hereditary algebras via the use of all the earlier tools developed in the text. The beautiful theorem of Gabriel is also proved here. And so one can very comfortably stop reading at this point, with a satisfied mind. For those who are hungry for more, there are two more chapters, on tilted algebras and directive modules and postprojective components, respectively. These latter chapters are somewhat more technical, but also much closer to the forefront of current research in the field.

The book under review is the first volume of a three volume set, and the authors promise much more in the other volumes. However the first volume by itself gives its reader a lot to chew on, and is clearly valuable enough to stand on its own.

See also the reviews of volume 2 and volume 3.

Gizem Karaali is assistant professor of mathematics at Pomona College.

 0. Introduction; 1. Algebras and modules; 2. Quivers and algebras; 3. Representations and modules; 4. Auslander-Reiten theory; 5. Nakayama algebras and representation-finite group algebras; 6. Tilting theory; 7. Representation-finite hereditary algebras; 8. Tilted algebras; 9. Directing modules and postprojective components; A. Appendix. Categories functors and homology.