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arXiv · 1212.1424

Semi-stable subcategories for Euclidean quivers

Colin Ingalls·Charles Paquette·Hugh Thomas
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Abstract

In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in $\mathbb{Q}^n$ to any acyclic quiver $Q$, and this simplicial fan allows one to completely determine the canonical presentation of any element in $\mathbb{Z}^n$. This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-one subsets of $\mathbb{Q}^n$, each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

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BibTeXRIS

Colin Ingalls, Charles Paquette, Hugh Thomas. 2015-03-13. Semi-stable subcategories for Euclidean quivers. https://arxiv.org/abs/1212.1424

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