These notes give an introduction to the theory of representations of quivers, in its algebraic and geometric aspects. The main result is Gabriel’s theorem that characterizes quivers having only finitely many isomorphism classes of representations in any prescribed dimension. Résumé. Ces notes donnent une introduction à la théorie des représentations des carquois, sous ses aspects algébrique et ...

In recent years, the representation theory of quivers has enjoyed an enormous impact of new techniques both from algebraic geometry and from quantum group theory. Some examples of this are the realization of quantum groups in terms of the representation theory of quivers via C. M. Ringel’s Hall algebra approach [61, 22], G. Lusztig’s geometric interpretation of the Hall algebra approach [49, 50...

We show that the category of representations of the Euclidean group E(2) is equivalent to the category of representations of the preprojective algebra of the quiver of type A∞. Furthermore, we consider the moduli space of E(2)-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw o...

Let k be a field and A a finite dimensional (associative with 1) k-algebra. By modA we denote the category of finite dimensional left A-modules. In many important situations we may suppose that A is presented as a quiver with relations (Q, I) (e.g. if k is algebraically closed, then A is Morita equivalent to kQ/I). We recall that if A is presented by (Q, I), then Q is a finite quiver and I is a...

One of the main tools for the study of the category of finite dimensional modules over a basic algebra, over an algebraically closed field k is its presentation as quiver and relations. This theory is mainly due to P. Gabriel (see for example [GRo]). More precisely, it has been proved that for all finite dimensional and basic algebras over an algebraically closed field k, there exists a unique ...

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One can show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a quantized universal enveloping algebra.

We give an introduction to the theory of quiver representations, in its algebraic and geometric aspects. The main result is Gabriel’s theorem that characterizes quivers of finite representation type. Résumé. Nous donnons une introduction à la théorie des représentations des carquois, sous ses aspects algébrique et géométrique. Le résultat principal est le théorème de Gabriel qui caractérise les...

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials...