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Elements of the Representation Theory of Associative Algebras 2: Tubes and Concealed Algebras of Euclidean Type

Daniel Simson and Andrzej Skowroński
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Student Texts 71
[Reviewed by
Gizem Karaali
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This is the second volume of a trilogy, the promised sequel to the excellent Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory (which I reviewed for MAA Reviews), by the same authors together with Ibrahim Assem. In the first volume, the authors had promised us only a second volume. However, considering that this volume is 284 pages long without the bibliography and the index, and that what became the third volume (Elements of the Representation Theory of Associative Algebras 3: Representation-infinite Tilted Algebras ) is a whopping 456 pages, readers will certainly understand the need for making the set a trilogy.

Like the first volume, this book is published in the series of London Mathematical Society Student Texts . There is certainly no illusion, however, as to who the audience should be. The readership of this volume is definitely a proper subset of that of the first volume! Mathematicians with some graduate algebra under their belts could get much out of the previous book; those who would read this one will have to be a lot more dedicated and experienced.

The book begins with the introduction of the notion of stable tubes (Chapter X) whose geometry becomes a useful tool in the rest of the book. Then the focus moves on to concealed algebras of Euclidean type (Chapters XI and XII) which are tilted algebras of Euclidean type defined by a certain postprojective tilting module. (Euclidean in this context means that the quivers attached to the algebra are of Dynkin type A, D or E). In Chapter XIII, some of the constructions of the earlier chapters are applied to determine the complete list of indecomposable regular modules over a path algebra associated with any given canonically oriented Euclidean quiver. Then using elementary tilting theory (developed already in Volume I), the results are extended to algebras with arbitrary quivers of Euclidean type. Chapter XIV is on minimal representation-infinite algebras.

I must emphasize that the book is exquisitely written, in a way that certainly allows it to be used in the classroom; but there are not that many classrooms for it. The main thing that distinguishes this book from a standard research monograph is the existence of several end-of-the-chapter exercises for each chapter. 

On the other hand, the book is more realistically and perfectly suited for self-study by graduate students in representation theory, and mathematicians in other fields who have already mastered most of the material in the first volume will also find this book understandable. However, it should most definitely be emphasized that throughout this second volume, all the major tools developed in the first volume (theory of almost split sequences, tilting theory and integral quadratic forms) are extensively used, so a superficial read of the first volume will not suffice.

Gizem Karaali is assistant professor of mathematics at Pomona College.

Introduction; 10. Tubes; 11. Module categories over concealed algebras of Euclidean type; 12. Regular modules and tubes over concealed algebras of Euclidean type; 13. Indecomposable modules and tubes over hereditary algebras of Euclidean type; 14. Minimal representation-infinite algebras; Bibliography; Index; List of symbols.