Indecomposable representations for extended Dynkin quivers of type ${\widetilde{\mathbb E}}_8$

Dawid Kędzierski ¹   ; Hagen Meltzer ¹

¹ Institute of Mathematics Szczecin University Wielkopolska 15 70-451 Szczecin, Poland

Colloquium Mathematicum, Tome 124 (2011) no. 1, pp. 95-116

Résumé

We discuss the problem of classification of indecomposable representations for extended Dynkin quivers of type $\widetilde{\mathbb E}_8$, with a fixed orientation. We describe a method for an explicit determination of all indecomposable preprojective and preinjective representations for those quivers over an arbitrary field and for all indecomposable representations in case the field is algebraically closed. This method uses tilting theory and results about indecomposable modules for a canonical algebra of type $(5,3,2)$ obtained by Kussin and Meltzer and by Komoda and Meltzer. Using these techniques we calculate all series of preprojective indecomposable representations of rank $6$. The same method has been used by Kussin and Meltzer to determine indecomposable representations for extended Dynkin quivers of type $\widetilde{\mathbb D}_n$ and $\widetilde{\mathbb E}_6$. Moreover, our techniques can be applied to calculate indecomposable representations of extended Dynkin quivers of type $\widetilde{\mathbb E}_7$. The indecomposable representations for extended Dynkin quivers of type $\widetilde{\mathbb A}_n$ are known.

Détail

DOI : 10.4064/cm124-1-7

Keywords: discuss problem classification indecomposable representations extended dynkin quivers type widetilde mathbb fixed orientation describe method explicit determination indecomposable preprojective preinjective representations those quivers arbitrary field indecomposable representations field algebraically closed method uses tilting theory results about indecomposable modules canonical algebra type obtained kussin meltzer komoda meltzer using these techniques calculate series preprojective indecomposable representations rank method has kussin meltzer determine indecomposable representations extended dynkin quivers type widetilde mathbb widetilde mathbb moreover techniques applied calculate indecomposable representations extended dynkin quivers type widetilde mathbb indecomposable representations extended dynkin quivers type widetilde mathbb known