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A construction of simple-minded systems over domestic Brauer graph algebras I: the 2-domestic case

Zhen Zhang

TL;DR

This work provides a complete framework for constructing and classifying simple-minded systems in the stable module category of 2-domestic Brauer graph algebras. By explicitly building orthogonal systems on Euclidean components and on quasi-tubes, and then combining them, the authors produce sms that coincide with all sms in this setting, along with a new proof of the AR-conjecture for 2-domestic algebras. A key technical tool is the stable bi-perpendicular category, whose explicit rectangle/triangle descriptions for objects on Euclidean components and quasi-tubes enable precise control of Hom-spaces and extension-closure. The results show that any maximal orthogonal system containing Euclidean data extends to an sms with a functorially finite extension-closure, and that weakly sms with finite cardinality are sms, contributing to a robust understanding of stable representation theory for this tame, representation-infinite class.

Abstract

Let $A$ be a 2-domestic Brauer graph algebra. We present a construction for a family of objects on $A$-$\stmod$ to be a simple-minded system and our construction provides all simple-minded systems on $A$-$\stmod$. As a byproduct, we provide a new proof of AR-conjecture for 2-domestic Brauer graph algebras and we prove that a weakly simple-minded system with a finite cardinality is a simple-minded system on $A$-$\stmod$. We also prove that an orthogonal system $\mathcal{S}$ which contains at least one object for an Euclidean component extends to a simple-minded system on $A$-$\stmod$ and its extension closure $\mathcal{F}(\mathcal{S})$ is functorially finite.

A construction of simple-minded systems over domestic Brauer graph algebras I: the 2-domestic case

TL;DR

This work provides a complete framework for constructing and classifying simple-minded systems in the stable module category of 2-domestic Brauer graph algebras. By explicitly building orthogonal systems on Euclidean components and on quasi-tubes, and then combining them, the authors produce sms that coincide with all sms in this setting, along with a new proof of the AR-conjecture for 2-domestic algebras. A key technical tool is the stable bi-perpendicular category, whose explicit rectangle/triangle descriptions for objects on Euclidean components and quasi-tubes enable precise control of Hom-spaces and extension-closure. The results show that any maximal orthogonal system containing Euclidean data extends to an sms with a functorially finite extension-closure, and that weakly sms with finite cardinality are sms, contributing to a robust understanding of stable representation theory for this tame, representation-infinite class.

Abstract

Let be a 2-domestic Brauer graph algebra. We present a construction for a family of objects on - to be a simple-minded system and our construction provides all simple-minded systems on -. As a byproduct, we provide a new proof of AR-conjecture for 2-domestic Brauer graph algebras and we prove that a weakly simple-minded system with a finite cardinality is a simple-minded system on -. We also prove that an orthogonal system which contains at least one object for an Euclidean component extends to a simple-minded system on - and its extension closure is functorially finite.

Paper Structure

This paper contains 14 sections, 46 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Simple-minded system
  4. Brauer graph algebras
  5. The property of components over a 2-domestic Brauer graph algebra
  6. The stable bi-perpendicular category of a stable brick
  7. The intersection of the stable bi-perpendicular category of an object (0,a,b) and an Euclidean component
  8. The intersection of the stable bi-perpendicular category of $(0,a,b)$ and a quasi-tube
  9. The stable bi-perpendicular category of an object on a quasi-tube of rank not 1
  10. the construction of orthogonal systems
  11. The construction of orthogonal systems on Euclidean components
  12. The construction of orthogonal systems on quasi-tubes
  13. A maximal orthogonal systems on $A$-$\mathsf{\underline{mod}}$
  14. A sufficient and necessary condition for an orthogonal system to be a simple-minded system

Key Result

Theorem 2.5

$($CLZ$)$ Let $A$ be a self-injective algebra and $\mathcal{C}$ a quasi-tube of rank $n$. Then the number of elements in a simple-minded system of $A$ lying in $\mathcal{C}$ is strictly less than $n$. In particular, none of the indecomposable modules in a simple-minded system lie in the homogeneous

Theorems & Definitions (86)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Example 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 76 more