Published in Forum Mathematicum
For an interval finite quiver Q, we introduce a class of flat representations. We classify the indecomposable projective objects in the category rep ( Q ) {\mathrm{rep}(Q)} of pointwise finite dimensional representations. We show that an object in rep ( Q ) {\mathrm{rep}(Q)} is projective if and only if it is a direct sum of countably generated...
Published in Acta Mathematica Sinica, English Series
An algebra A is called derived-unique provided that any algebra which is derived equivalent to A is necessarily Morita equivalent to A. We classify derived-unique gentle algebras with at most one cycle using the combinatorial invariant introduced by Avella-Alaminos and Geiss.
Published in Science China Mathematics
Bongartz (2013) and Ringel (2011) proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly studies this "no gaps" theorem as to cohomological length for the bounded derived category Db(A) of a gentle algebra A: if there is an inde...
Published in Japan Journal of Industrial and Applied Mathematics
The theory of persistence modules on the commutative ladders CLn(τ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CL_n(\tau )$$\end{document} provides an extension of ...
Published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
We generalize the notion of multi-Gieseker semistability for coherent sheaves, introduced by Greb, Ross, and Toma, to quiver sheaves for a quiver Q. We construct coarse moduli spaces for semistable quiver sheaves using a functorial method that realizes these as subschemes of moduli spaces of representations of a twisted quiver, depending on Q, with...
Published in Archiv der Mathematik
Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of modB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage...
Published in Mathematische Zeitschrift
We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the cardinality and the height of the poset. We explicitly describe spherical subcategories and their poset structure for derived categories of certain finite-dimensional algebras.
Published in Mathematische Annalen
Consider a finite-dimensional algebra A and any of its moduli spaces M(A,d)θss\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {M}}}(A,\mathbf {d})^{ss}_{\the...
Published in Algebras and Representation Theory
Let J(C) be the poset of order ideals of a cominuscule poset C where C comes from two of the three infinite families of cominuscule posets or the exceptional cases. We show that the Auslander-Reiten translation τ on the Grothendieck group of the bounded derived category for the incidence algebra of the poset J(C), which is called the Coxeter transf...
Published in Algebras and Representation Theory
In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A13\documentclass[12pt]{mi...
