arXiv subjects
Publications and source records attributed to Ray Maresca.
We show that isoclasses of iterated tilted algebras in type $A_n$ are in bijection with non-crossing spanning trees up to rotation on a convex n+1-gon. This is done by constructing a relationship between iterated tilted algebras up to isomorphism and exceptional sets up to isomorphic Hom-Ext quiver.
For a finite dimensional algebra $\Lambda$, we consider a torsion class $G$ in $mod$-$\Lambda$, which is not necessarily finitely generated. We construct a wall-and-chamber structure for $G$ where the chambers are the connected components of the complement of the union of walls. We also consider ``infinitesimal chambers". To each chamber we associate a ``pseudo-torsion class'' and a ``pseudo-torsionfree class'' and show that they are all distinct. We consider ``green paths'' in the stability space and associate to them Harder-Narasimhan stratifications of $G$. This paper is part of a series of papers whose goal is to study the ``ghosts'' which are remnants of the indecomposable $\Lambda$-modules which do not lie in $G$. In the special case when our torsion class is all of $mod$-$\Lambda$, we are in the classical well-known setting. All of our results apply to this classical setting. The ``pseudo-torsion classes'' are torsion classes. We point out that we do not take the closure of the set of walls. So, we get more chambers and our results are new even in this classical setting.
We study what we call the Hom-Ext quiver and characterize it as a type of `superquiver'. In type $\tilde{\mathbb{A}}$, the Hom-Ext quiver of an exceptional set is the tiling algebra of the corresponding geometric model. And, in that case, Hom-Ext quivers classify exceptional sets up to Dehn twist of the corresponding geometric model. We show that these Dehn twists are realized by twist functors and give autoequivalences of the derived category. We provide a generating set for the group of autoequivalences of the derived category in type $\tilde{\mathbb{A}}$, and show that the Hom-Ext quiver classifies exceptional sets up to the action of the subgroup of the automorphism group of the derived category generated by twist functors associated to exceptional cycles. We introduce superquivers, which are a generalization of Hom-Ext quivers. Exceptional sets over finite acyclic quivers are realized as representations of superquivers. Throughout, we list several questions and conjectures that make for, what we believe, exciting new research.
In this paper, we will place clusters in type $\tilde{\mathbb{A}}$ (equivalently triangluations of an annulus) into infinite families parametrized by winding numbers of certain arcs in the corresponding triangulation. We will count how many such families there are and provide an algebraic description of these families in terms of the corresponding cluster category.
In this note, we will illuminate some immediate consequences of work done by Reineke that may prove to be useful in the study of elliptic curves. In particular, we will construct an isomorphism between the category of smooth projective curves with a category of quiver grassmannians. We will use this to provide a 4-fold categorical equivalence between a category of quiver grassmannians, smooth projective curves, compact Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$. We finish with noting that the category of elliptic curves is isomorphic to a category of quiver grassmannians, whence providing an analytic group structure to a class of quiver grassmannians.
In this paper, we will present the author's interpretation and embellishment of five lectures on cluster theory given by Kiyoshi Igusa during the Spring semester of 2022 at Brandeis University. They are meant to be used as an introduction to cluster theory from a representation-theoretic point of view. It is assumed that the reader has some background in representations of quivers.
In this paper we first study clusters in type $\tilde{\mathbb{A}}$ by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type $\tilde{\mathbb{A}}$. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type $\mathbb{A}$ quivers with straight orientation and using this to count the number of families of exceptional sets of type $\tilde{\mathbb{A}}$ with straight orientation.
It is known that there are infinitely many exceptional sequences of quiver representations for Euclidean quivers. In this paper we study those of type $\tilde{\mathbb{A}}_n$ and classify them into finitely many parametrized families. We first give a bijection between exceptional collections and a combinatorial object known as strand diagrams. We will then realize these strand diagrams as chord diagrams and then arc diagrams on an annulus. Using arc diagrams, we will define parametrized families of exceptional collections and use arc diagrams to show that there are finitely many such families. We moreover show that these families of exceptional collections are in bijection with equivalence classes of small arc diagrams. Finally, we provide an algebraic explanation of parametrized families using the transjective component of the bounded derived category.