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Stable representations of quivers. (English) Zbl 1040.16011
Summary: Let \(Q\) be a finite quiver without oriented cycles and let \(kQ\) be the path algebra of \(Q\) over an algebraically closed field \(k\). We investigate stable finite-dimensional representations of \(Q\). That is for a fixed dimension vector \(d\) and a fixed weight \(\theta\) we consider \(\theta\)-stable representations of \(Q\) with dimension vector \(d\). If we wish to compare also representations with different dimension vectors, then it is more convenient to consider a slope \(\mu\) instead of a weight \(\theta\). In particular, we apply the results of Harder-Narasimhan on natural filtrations associated to any fixed slope \(\mu\) to the category of representations of \(Q\). Further we introduce the wall system for weights with respect to a fixed dimension vector \(d\) and consider several examples.

MSC:
16G20 Representations of quivers and partially ordered sets
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