Nakajima varieties and repetitive algebras

We realize certain graded Nakajima varieties of finite Dynkin type as orbit closures of repetitive algebras of Dynkin quivers. As an application, we obtain that the perverse sheaves introduced by Nakajima for describing irreducible characters of quantum loop algebras are isomorphic to the intersection cohomology sheaves of these orbit closures.Comment: 26 pages, v2 : Minor corrections. Final version to appear in PRIMS (Publications of the Research Institute for Mathematical Sciences, Kyoto

Cluster algebras via cluster categories with infinite-dimensional morphism spaces

We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.Comment: 32 pages, added referenc

Algèbres inclinées amassées aimables

Dans ce mémoire, les algèbres inclinées amassées aimables sont classifiées. Celles-ci sont précisément de type [anneau des adèles][indice inférieur n] et [anneau des adèles tilde][indice inférieur n]. La démonstration décrit explicitement le carquois lié de ces algèbres et fait usage de l'extension par relations définie par Assem, Brüstle et Schifiler. Il est également prouvé que celles-ci s'apparentent aux algèbres obtenues à partir de triangulations du disque et du cylindre