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Quiver Representations

Ralf Schiffler
Publisher: 
Springer
Publication Date: 
2014
Number of Pages: 
230
Format: 
Hardcover
Series: 
CMS Books in Mathematics
Price: 
69.99
ISBN: 
9783319092034
Category: 
Textbook
[Reviewed by
Felipe Zaldivar
, on
12/30/2014
]

The representation theory of finite dimensional k-algebras, for k an algebraically closed field, had a remarkable boost in the 1970s with the introduction of diagrammatic methods by P. Gabriel. Diagrammatic methods allow the visualization of the modules associated to a given algebra as finite collections of k-vector spaces and linear maps between them. The underlying diagram is finite directed graph, and one has to like the choice of the word quiver for this finite collection of arrows. A representation of a quiver is a choice of one (finite-dimensional) vector space corresponding to each end of an arrow together with a linear transformation or matrix associated to each arrow. A morphism between two such representations of the same quiver is a collection of linear maps between corresponding vector spaces, i.e., those attached to the same vertex, that make the whole diagram commute.

All of these natural definitions beg for chalk or pencil to come alive. For a fixed quiver, finite dimensional representations and morphisms between them form a category, in fact an abelian category with enough injectives. It is then not surprising that for a given quiver one wants to classify all its representations.

The starting point is the Krull-Schmidt theorem (every quiver representation can be expressed in a unique way as a direct sum of indecomposable quiver representations). Therefore, it is enough to classify the indecomposable representations of a given quiver. This is achieved, in a very important case, by Gabriel’s theorem: A connected quiver has a finite number of isomorphism classes of indecomposable representations if and only if the underlying non directed graph is a Dynkin diagram of type An, Dn or En. This is another still unexplained appearance of these diagrams that seem to pop up any time there is some finite structure underlying the corresponding mathematical situation, from singularity theory to cluster algebras, Coxeter groups, root systems, and the classification of Lie algebras.

The book under review is an elementary introduction to the diagrammatic or quiver approach to the representation theory of finite-dimensional algebras. It is perhaps the first such textbook addressed to advanced undergraduates or beginning graduate students. Assuming some basic category theory and advanced linear algebra, the first three chapters give an easy to follow introduction to quiver representations up to the construction of Auslander-Reiten quivers and the knitting algorithm, a recursive method to construct one mesh after another. Chapter four is a quick and self-contained introduction of some topics from ring theory, algebras, and modules.

The second part of the book is a bit more demanding: every quiver Q gives rise to a k-algebra, the path algebra kQ of the quiver, whose underlying k-vector space has as basis the set of all paths in the quiver and whose multiplication is given by concatenation of paths. The starting point for this part is the equivalence of the category of (finite dimensional) representations of the quiver Q and the category of (finitely generated) kQ-modules. Moreover, every finite-dimensional k-algebra is a quotient of a path algebra kQ by an ideal satisfying certain admissibility condition. These bound quiver algebras are treated in chapter 5. The main theorem in this context gives an equivalence between the category of finitely generated kQ/I-modules and the category of representations of the quiver Q which satisfy the relations induced by the ideal I. The chapter ends with examples of Auslander-Reiten quivers for bound quiver algebras. Chapter six provides interesting examples of constructions of new algebras from other algebras, for example, tilted algebras or cluster-tilted algebras. The all-important Auslander-Reiten almost split sequences are introduced and studied in chapter seven. The last chapter gives a complete detailed proof of Gabriel’s theorem using the classification of positive definite integral quadratic forms associated to quivers.

This is a well-written and engaging book, full of examples and diagrams, as one would expect from this approach. Teaching a course from this book should be a pleasant experience. Sets of problems are provided at the end of every one of its chapters, and little notes point to the literature. For a motivated student, the book is well suited for self-study.


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

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