Published in Algebras and Representation Theory
For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result...
Published in Algebras and Representation Theory
Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between...
Published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type A we construct a maximum antichain in the poset. Secondly, we consider the set of subrepresentations of a give...
Published in Algebras and Representation Theory
The ADR algebra RA of an Artin algebra A is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the Δ-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of RA-modules. As an application, we give a counterexample to a claim by Auslander–Platzeck–Todorov, concerning proje...
Published in Frontiers of Mathematics in China
Let Λ be a Koszul algebra, and let M be a graded Λ-bimodule. We prove that the trivial extension algebra of Λ by M is also a Koszul algebra whenever M is Koszul as a left Λ-module. Applications and examples are also provided.
Published in Mathematische Zeitschrift
Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting object...
Published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra. For each triangulated orbit category T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb...
Published in São Paulo Journal of Mathematical Sciences
In the seventies of the last century, M. Auslander introduced the notion of representation dimension of an algebra with the objective of having a measure of the complexity of its module category. In the last 20 years, there was a renewal interest in studying such a dimension. Our aim here is to survey the last developments concerning it.
Published in Algebras and Representation Theory
Every cluster-tilted algebra B is the relation extension C⋉ExtC2(DC,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C\ltimes \textup {Ext}^{2}_{C}(DC,C)$\end{document}...
In this note, we will explain symmetries of isomonodromic deformations as Weyl groups of some quivers and give classifications of isomondromic deformations of linear ordinary differential equations with at most unramified irregular singularities and 2 or 4 accessory parameters.
