[Paper Review] Semi-stable subcategories for Euclidean quivers

Colin Ingalls, Charles Paquette+1 more|Archipelago (Université du Québec à Montréal)|Dec 6, 2012

Algebraic structures and combinatorial models16 references4 citations

TL;DR

This paper investigates semi-stable subcategories in the representation category of Euclidean quivers, showing that while intersections of such subcategories are not always semi-stable, only finitely many exceptions exist. It characterizes these non-semi-stable intersections as thick subcategories contained in the regular part, containing all homogeneous tubes, and expressible as intersections of at most two semi-stable subcategories via regular orthogonal pairs in non-homogeneous tubes.

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.

Motivation & Objective

• To determine which abelian, extension-closed subcategories of the representation category of a Euclidean quiver arise as semi-stable subcategories for some stability condition.
• To understand the structure of intersections of semi-stable subcategories, particularly when such intersections fail to be semi-stable.
• To characterize the finite set of exceptions where the intersection of semi-stable subcategories is not itself semi-stable.
• To establish a connection between stability conditions and simplicial fans in Q^n, using real Schur roots and quasi-simple objects.
• To provide a canonical decomposition of dimension vectors via a fan structure that encodes ss-equivalence of stability conditions.

Proposed method

• Define a simplicial fan in Q^n associated with any acyclic quiver Q, using convex codimension-one subsets indexed by real Schur roots or quasi-simple objects.
• Use the Grothendieck group and Euler form to identify stability conditions with elements in a rational vector space, enabling geometric analysis.
• Characterize semi-stable subcategories via the canonical presentation of dimension vectors induced by the fan.
• Analyze intersections of semi-stable subcategories by restricting to the regular part of the representation category and examining tube decompositions.
• Apply the theory of regular orthogonal pairs (E_J, F) in tubes, where F ∩ OT = ∅, to describe non-semi-stable intersections.
• Use the structure of thick subcategories and quasi-length filtrations to prove that any such exceptional intersection is expressible as an intersection of at most two semi-stable subcategories.

Experimental results

Research questions

• RQ1 Which abelian and extension-closed subcategories of rep(Q) for a Euclidean quiver Q arise as θ-semi-stable subcategories for some stability condition θ?
• RQ2 Under what conditions is the intersection of two semi-stable subcategories not itself semi-stable?
• RQ3 How can the set of all such non-semi-stable intersections be characterized geometrically and combinatorially?
• RQ4 What is the role of the simplicial fan in encoding ss-equivalence of stability conditions and canonical dimension vector decompositions?
• RQ5 Can every thick subcategory of the regular part of rep(Q) that contains all homogeneous tubes and satisfies tube-wise conditions be realized as a finite intersection of semi-stable subcategories?

Key findings

• For a Euclidean quiver Q, an abelian and extension-closed subcategory B of rep(Q) is semi-stable if and only if it is either finitely generated or consists of all regular objects in a finitely generated subcategory equivalent to a Euclidean quiver.
• Intersections of semi-stable subcategories may fail to be semi-stable, but only finitely many such exceptions exist.
• All non-semi-stable intersections are contained in the regular part of rep(Q), contain all homogeneous tubes, and are characterized by regular orthogonal pairs (E_J, F) in non-homogeneous tubes with F ∩ OT = ∅.
• At least one of the index sets J_i in the tube decomposition must be empty for the intersection to fail to be semi-stable.
• Any such exceptional subcategory can be expressed as the intersection of at most two semi-stable subcategories.
• The simplicial fan associated with Q, built from real Schur roots and quasi-simple objects, encodes ss-equivalence of stability conditions and enables a canonical presentation of every dimension vector in Z^n.