Semi-stable subcategories for Euclidean quivers

Abstract

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 Q n to any acyclic quiver Q, and this simplicial fan allows one to completely determine the canonical presentation of any element in 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 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.

An affine almost positive roots model

“…Since compatibility is the same in our setting and the representation-theoretic setting, our cluster expansions (Definition 6.1) are analogous to the canonical decompositions (or generic decompositions) of [23]. Thus, parts of Proposition 5.14 correspond to results proved in [39,Section 3] and [22,Section 6]. The canonical decomposition fan constructed in [22] coincides with Fan c (Φ) within the span of the positive roots.…”

“…In any case, in light of Corollary 1.7, it seems reasonable to call Φ c the "almost positive Schur roots". Comparison with [22,39] makes it clear that Λ c = Φ re c ∩ U c (Definition 3.1) is the set of dimension vectors of the regular representations. As usual, the deformed Coxeter element τ c corresponds to the Auslander-Reiten translation (or its inverse, depending on conventions).…”

Wide subcategories of finitely generated Λ-modules

“…Following M. Hovey, in [6], we say that the subcategory C ⊆ A is wide if it is closed under extensions, kernels and cokernels of morphisms in C. Wide categories have been extensively studied in the case of modules over commutative rings [6,11,17]. For the case of finitely generated modules over a hereditary algebra, we recommend the reader to see [7,8]. An easy observation is that any wide category C is closed under isomorphisms and furthermore it is an additive subcategory of A.…”

Generators versus projective generators in abelian categories

“…The converse is not true. However, if A is hereditary, thick is equivalent to being exact abelian and extension-closed; see [3], for instance. Hence, we get the following.…”