A *quiver* is just a (finite) collection of *arrows* and its set of *vertices* is the union of the sets of *tails* and *heads* of the arrows. Fixing a field *k*, a *representation* of a quiver is a choice of (finite dimensional) vector spaces attached to the ends of each arrow and a linear transformation associated to each arrow. A *morphism* between two representations of the same quiver is a collection of linear maps between vector spaces attached to the same vertex that make the whole diagram commute. The *dimension vector* of a representation of a quiver *Q* is the function that assigns to each vertex of *Q* the dimension of the corresponding vector space.

The *trivial* representation is the one that assigns the zero vector space to each vertex of the quiver. Direct sums of representations, subrepresentations and quotients are defined in a natural fashion, and for a fixed quiver *Q* and field *k*, its finite dimensional representations and morphisms between them form an abelian category. A representation of a quiver *Q* is d*ecomposable* if it is a direct sum of non-trivial representations; otherwise it is called *indecomposable*.

By the Krull-Schmidt theorem every quiver representation can be expressed in a unique way, up to isomorphism, as a direct sum of indecomposable quiver representations. Thus, to classify all the representations of a given quiver it is enough to classify its indecomposable representations.

To do this, there are some cases to consider: either the quiver has finitely many indecomposable representations, in which case it is called a quiver of *finite type*, or has infinitely many indecomposable representations, which can further be classified into two subclasses, colorfully named *tame* or *wild* types, and the three classes are mutually disjoint.

Gabriel's theorem takes care of the first class: A connected quiver is of finite type if and only if the underlying non directed graph is a Dynkin diagram of type \(A_n, D_n\) or \(E_n\) for \(n=6,7,8\). The proof of Gabriel's theorem makes heavy use of the quadratic (Tits) form associated to a quiver \(Q\) (with no loops). It is defined like this: if \(h,t:Q_1\rightarrow Q_0\), are the head and tail functions, where \(Q_1\) and \(Q_0\) are the sets of arrows and vertices of \(Q\), and \(n=|Q_0|\), the associated integral quadratic form \(q:{\mathbb Z}^n\rightarrow {\mathbb Z}\) is \[q(x)=\sum_{i\in Q_0}x_i^2-\sum_{a\in Q_1}x_{t(a)}x_{h(a)}.\]

For a connected quiver \(Q\) the main result is: \(q\) is positive definite if and only if \(Q\) is of Dynkin of type \(A_n, D_n\) or \(E_n\). Now, to relate this result to the type of the quiver, recall that for a positive semi-definite quadratic form \(q:{\mathbb Z}^n\rightarrow {\mathbb Z}\), a non zero vector \(x\in{\mathbb Z}^n\) is a real root if \(q(x)=1\) and it is an imaginary root if \(q(x)=0\). Clearly the canonical vectors \(e_i\) are real roots. A root \(\alpha=\sum a_ie_i\) (real or imaginary) is positive if all \(a_i\geq 0\), and it is negative if all \(a_i\leq 0\). It can be proved that the set of all roots of \(q\) is a disjoint union of the set of positive roots and the set of negative roots. To finish the proof of Gabriel's theorem, the following stronger result is proved: The (isomorphism classes of) indecomposable representations are in one-to-one correspondence with the positive roots of the root system associated to the Dynkin diagram. The correspondence associates to each indecomposable representation of *Q* its dimension vector.

There is a corresponding theorem for quivers of tame type, due to Nazarova, Donovan and Freislich now using unions of Dynkin diagrams and their extended versions.

Dynkin diagrams tend to appear whenever there is some finite type structure underlying the classification problem at hand, for example in the classification of simple Lie algebras or finite Coxeter groups. Kac and Moody, in 1980, found a generalization of root systems with corresponding generalized Lie algebras for arbitrary quivers. One of their main results generalizes part of Gabriel's theorem: For an arbitrary quiver *Q* the dimension vectors of indecomposable representations of Q correspond to positive roots of the root system associated to the underlying diagram of *Q*, and thus, are independent of the orientation of the arrows of *Q*.

Representations of quivers, while being interesting by themselves, are more than that: indeed, associated to a quiver *Q* we have its path algebra *kQ*, which is the *k*-vector space with a basis the set of all paths in the quiver and multiplication given by concatenation of paths. The path k-algebra *kQ *is finite-dimensional if and only if the quiver *Q* has no oriented cycles (paths with the same head and tail). The reason for which the study of finite-dimensional associative k-algebras is related to the representations of quivers is that for any finite-dimensional *k*-algebra *A*, the category of representations of *A* is equivalent to the category of representations of the algebra *kQ/*I, for some quiver *Q* and a two-sided ideal I of *kQ*. The notions of finite, tame or wild type can be extended to arbitrary finite dimensional *k*-algebras.

As the adjective wild suggests, the classification of indecomposable algebras of wild type is, for now, out of reach. In this direction, however, algebraic geometry has provided a new approach. To begin with, for a finite dimensional representation of a quiver *Q*, with dimension vector d, choosing bases for each vector space \(V_i\simeq k^{d_i}\), the linear maps of the representation correspond to matrices \(x_a\), for each arrow of *Q*, and therefore the representation corresponds to a vector in the representation space \[{\mathcal R}(Q,d)=\bigoplus_{a\in Q_1}\text{Hom}_k(k^{d_{t(a)}},k^{d_{h(a)}})\] and conversely. The linear group \(GL(d)=\prod_{i\in Q_0}GL(d_i)\) acts in this space by conjugation, and the orbits in the quotient \({\mathcal R}(Q,d)/GL(d)\) correspond to isomorphism classes of representations of *Q*. But there is a catch, the closed orbits of this quotient pick up only the semisimple representations of *Q*.

King had the idea to consider Mumford's GIT quotient, with adequate notions of stability and semi-stability in the category of finite dimensional representations of the path algebra *kQ*, over an algebraically closed field. Thus, this quotient parametrizes not all representations but only representations satisfying the stability conditions, and the main result is that this quotient is a coarse moduli space. When \(k={\mathbb C}\) there is an interpretation of this moduli space as a symplectic quotient. More recent developments include Nakajima’s geometric construction of the enveloping algebra of Kac-Moody Lie algebra. Nakajima’s construction of quiver varieties is rather involved and is the main topic of the book under review.

Kirillov’s book joins a relative long list of monographs and textbooks devoted to quiver representations. From Gabriel and Roiter's *Representations of Finite-Dimensional Algebras *(Springer, 1992), or Auslander, Reiten and Smalo's *Representation Theory of Artin Algebras* (Cambridge, 1995) or the three-volume *Elements of the Representation Theory of Associative Algebras*, by Assem, Simpson, Skowronski et al (Cambridge, 2006 and 2007) to Schiffler’s textbook Quiver Representations (Springer, 2014). What sets this book apart is that it covers more recent developments on the geometric aspects of quiver theory.

The book is divided in three parts. Part 1 covers in about 80 pages the representation theory of quivers of finite type. The exposition tends to be a little terse (for example, Schiffler's Quiver Representations covers essentially the same topics in the whole book) and the approach follows somehow to the classical paper by Bernstein, Gelfand and Pononarev, “Coxeter Functors and Gabriel’s Theorem” *Russian Mathematical Surveys* **28** (1973), 17-32.

Part 2 treats some aspects of the representation of quivers of infinite type up to the statement of Kac’s theorem, but without proofs. This part ends with a chapter on the McKay correspondence for finite subgroups of SU(2) and a geometric construction of representations of Euclidean quivers using these subgroups. Again, some important results are proved and some others are referred to the literature.

Part 3, the main contribution of this monograph, is devoted to geometric approach to the representation of quivers. The topics range from King’s construction of the moduli space of quiver representations and its interpretation as a symplectic quotient when the base field is the complex numbers to Nakajima’s quiver varieties for framed representations. Although the exposition in this last part is even terser, with many calls to the literature, the inclusion of selected examples illustrates the ideas and helps to clarify the underlying thread. For example, a whole chapter treats the quiver variety of the Jordan quiver and its relations to the Hilbert scheme of points on the plane, and the last chapter presents a construction of a geometric realization of the Kac-Moody Lie algebras due to Nakajima. An appendix summarizes basic facts about Cartan matrices and their quadratic and bilinear forms, root systems, Weyl groups, and Kac-Moody algebras.

With an adequate background in Lie theory and algebraic geometry, the book is accessible to an interested reader. Although the book has no proper sets of exercises, it engages the reader to fill in some arguments or to look for a result in the references. As such, the book can be used for a topics course on its subjects.

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