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.
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
Affiliations des auteurs :
Dawid Kędzierski
1
;
Hagen Meltzer
1
1
Institute of Mathematics Szczecin University Wielkopolska 15 70-451 Szczecin, Poland
Dawid Kędzierski; Hagen Meltzer. Indecomposable representations for extended Dynkin quivers of type
${\widetilde{\mathbb E}}_8$. Colloquium Mathematicum, Tome 124 (2011) no. 1, pp. 95-116. doi: 10.4064/cm124-1-7
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author = {Dawid K\k{e}dzierski and Hagen Meltzer},
title = {Indecomposable representations for extended {Dynkin} quivers of type
${\widetilde{\mathbb E}}_8$},
journal = {Colloquium Mathematicum},
pages = {95--116},
year = {2011},
volume = {124},
number = {1},
doi = {10.4064/cm124-1-7},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm124-1-7/}
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