Ringel, Claus Michael
Published in
Archiv der Mathematik

We are going to determine the self-injective cluster-tilted algebras. All are of finite representation type and special biserial. There are two different classes. The first class are the self-injective serial (or Nakayama) algebras with n ≥ 3 simple modules and Loewy length n–1. The second class of algebras has an even number 2m of simple modules; ...

Yang, ShiLin Liu, JianZhen
Published in
Science in China Series A: Mathematics

Let (Г, I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group ℤd. In this paper, we list all indecomposable representations of (θ, I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Пλ(Г, I). It is shown that those representations give...

Kosakowska, Justyna
Published in
Acta Mathematica Sinica, English Series

In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Li...

Ahmed, Munir Li, Fang
Published in
Acta Mathematica Sinica, English Series

In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these t...

Chen, Jianmin Lin, Yanan
Published in
Chinese Annals of Mathematics, Series B

The authors introduce an effective method to construct the rational function sheaf \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{K} $$\end{document} on an e...

Álvarez-Cónsul, Luis
Published in
Geometriae Dedicata

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can b...

Wei, Jiaqun
Published in
Advances in Mathematics

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radi...

Schiffler, Ralf
Published in
Advances in Mathematics

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conject...

Barot, Michael Fernández, Elsa Platzeck, María Inés Pratti, Nilda Isabel Trepode, Sonia
Published in
Advances in Mathematics

In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.

Aguadé, Jaume
Published in
Journal of Geometry

We introduce an invariant of sets of four lines in three-dimensional projective space which classifies these sets up to linear equivalence, and we relate this invariant to the representation theory of the infinite dihedral group.