38 research outputs found
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