[Paper Review] The elementary 3-Kronecker modules

Claus Michael Ringel|arXiv (Cornell University)|Dec 29, 2016

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper classifies the elementary 3-Kronecker modules—building blocks for all regular representations of the 3-Kronecker quiver—by showing they are precisely those with dimension vectors in the τ-orbits of (1,1), (2,1), (2,2), and (4,2). It proves that indecomposable modules of dimension (1,1) and (2,1) are always elementary, while those of dimension (2,2) or (4,2) are elementary iff isomorphic to X(α,β,γ) or Y(α,β,γ) for some basis α,β,γ of the arrow space A.

ABSTRACT

The 3-Kronecker quiver has two vertices, namely a sink and a source, and 3 arrows. A regular representation of a representation-infinite quiver such as the 3-Kronecker quiver is said to be elementary provided it is non-zero and not a proper extension of two regular representations. Of course, any regular representation has a filtration whose factors are elementary, thus the elementary representations may be considered as the building blocks for obtaining all the regular representations. We are going to determine the elementary $3$-Kronecker modules. It turns out that all the elementary modules are combinatorially defined.

Motivation & Objective

• To identify and classify all elementary 3-Kronecker modules, which serve as fundamental building blocks for regular representations of the 3-Kronecker quiver.
• To determine which indecomposable regular modules are elementary, based on their dimension vectors and structure.
• To provide a combinatorial characterization of elementary modules using basis-dependent constructions X(α,β,γ) and Y(α,β,γ).
• To clarify the role of the Auslander-Reiten translation τ in classifying these modules via τ-orbits.
• To demonstrate that filtrations of regular modules into elementary modules are non-unique, using explicit examples.

Proposed method

• Define the 3-Kronecker quiver Q = K(3) with two vertices and three arrows, and associate its path algebra Λ = [k A; 0 k], where A is a 3-dimensional k-vector space.
• Use the Auslander-Reiten translation τ to analyze the τ-orbits of dimension vectors, focusing on (1,1), (2,1), (2,2), and (4,2).
• Construct two families of modules: X(α,β,γ) and Y(α,β,γ), defined via specific linear maps on k² × k² with respect to a basis α,β,γ of A.
• Apply the definition of elementary modules: a regular module is elementary if it is not a proper extension of two non-zero regular modules.
• Verify that modules of dimension (1,1) and (2,1) are always elementary, while (2,2) and (4,2) are elementary iff isomorphic to X(α,β,γ) or Y(α,β,γ) for some basis.
• Use coefficient quivers and module filtrations to illustrate non-uniqueness of filtrations into elementary modules.

Experimental results

Research questions

• RQ1 Which indecomposable regular 3-Kronecker modules are elementary?
• RQ2 What are the dimension vectors of elementary 3-Kronecker modules, and how are they related via the Auslander-Reiten translation?
• RQ3 For which dimension vectors (2,2) and (4,2) are the corresponding indecomposable modules elementary?
• RQ4 Can the elementary modules be explicitly constructed using basis-dependent combinatorial data?
• RQ5 How do filtrations of regular modules into elementary modules behave, and what does this imply about their non-uniqueness?

Key findings

• The elementary 3-Kronecker modules are precisely those with dimension vectors in the τ-orbits of (1,1), (2,1), (2,2), and (4,2).
• All indecomposable modules of dimension vector (1,1) or (2,1) are elementary, regardless of the choice of basis.
• An indecomposable module of dimension vector (2,2) is elementary if and only if it is isomorphic to X(α,β,γ) for some basis α,β,γ of A.
• An indecomposable module of dimension vector (4,2) is elementary if and only if it is isomorphic to Y(α,β,γ) for some basis α,β,γ of A.
• The modules X(α,β,γ) and Y(α,β,γ) are explicitly constructed using linear maps defined by the basis elements α,β,γ.
• Examples show that regular modules can admit multiple non-isomorphic filtrations into elementary modules, demonstrating the non-uniqueness of such filtrations.