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Skew-Brauer graph algebras

Ana García Elsener, Victoria Guazzelli, Yadira Valdivieso

TL;DR

The paper introduces skew-Brauer graph algebras as symmetric, finite dimensional algebras built from skew-Brauer graphs and shows they arise as trivial extensions of skew-gentle algebras. It provides a detailed bridge between skew-gentle algebras and skew-Brauer graph algebras via sg-admissible presentations, and characterizes finite representation type in terms of skew-Brauer trees, linking them to trivial extensions of iterated tilted algebras of type D. A major contribution is the geometric interpretation of cuts and reflections through orbifold ×-dissections, enabling computation of trivial extensions from dissection data and revealing corresponding derived-equivalence phenomena. The results unify algebraic, combinatorial, and geometric perspectives, offering tools to study symmetric tame algebras and their representations. The work also discusses limitations and examples where derived equivalence is not preserved under geometric moves, highlighting nuanced interactions between algebraic structure and ambient geometry.

Abstract

In this work, we introduce a new class of algebras called skew-Brauer graph algebras, which generalize the well-known Brauer graph algebras. We establish that skew-Brauer graph algebras are symmetric and can be defined using a Brauer graph with additional information. We show that the class of trivial extensions of skew-gentle algebras coincides with a subclass of skew-Brauer graph algebras, where the associated skew-Brauer graph has multiplicity function identically equal to one, generalizing a result over gentle algebras. We also characterize skew-Brauer algebras of finite representation type. Finally, we provide a geometric interpretation of cut-sets and reflections of algebras using orbifold dissections.

Skew-Brauer graph algebras

TL;DR

The paper introduces skew-Brauer graph algebras as symmetric, finite dimensional algebras built from skew-Brauer graphs and shows they arise as trivial extensions of skew-gentle algebras. It provides a detailed bridge between skew-gentle algebras and skew-Brauer graph algebras via sg-admissible presentations, and characterizes finite representation type in terms of skew-Brauer trees, linking them to trivial extensions of iterated tilted algebras of type D. A major contribution is the geometric interpretation of cuts and reflections through orbifold ×-dissections, enabling computation of trivial extensions from dissection data and revealing corresponding derived-equivalence phenomena. The results unify algebraic, combinatorial, and geometric perspectives, offering tools to study symmetric tame algebras and their representations. The work also discusses limitations and examples where derived equivalence is not preserved under geometric moves, highlighting nuanced interactions between algebraic structure and ambient geometry.

Abstract

In this work, we introduce a new class of algebras called skew-Brauer graph algebras, which generalize the well-known Brauer graph algebras. We establish that skew-Brauer graph algebras are symmetric and can be defined using a Brauer graph with additional information. We show that the class of trivial extensions of skew-gentle algebras coincides with a subclass of skew-Brauer graph algebras, where the associated skew-Brauer graph has multiplicity function identically equal to one, generalizing a result over gentle algebras. We also characterize skew-Brauer algebras of finite representation type. Finally, we provide a geometric interpretation of cut-sets and reflections of algebras using orbifold dissections.
Paper Structure (16 sections, 25 theorems, 20 equations, 24 figures)

This paper contains 16 sections, 25 theorems, 20 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Gentle and skew-gentle algebras
  4. Trivial extensions
  5. Repetitive algebra and reflections
  6. Trivial extension of a skew-gentle algebra
  7. Skew-Brauer graph algebras and trivial extension of skew-gentle algebras
  8. Skew-Brauer algebras and a set of relations
  9. The trivial extension of a skew-gentle algebra
  10. Admissible cuts, the repetitive algebra and reflections
  11. Skew-Brauer graph algebras of finite representation type
  12. A geometric interpretation using dissections
  13. Computing trivial extensions of skew-gentle algebras using dissections
  14. Contracting and adding boundary segments.
  15. On derived equivalent skew-gentle algebras.
  16. ...and 1 more sections

Key Result

Theorem 2.4

FSTTV22 Let $A=KQ_A/I_A$ a finite dimensional algebra and let $T(A)=KQ_{T(A)}/I_{T(A)}$ be its trivial extension. Then, the quiver $Q_{T(A)}$ is as above and the ideal $I_{T(A)}$ is generated by the union of the following sets.

Figures (24)

  • Figure 1: Ordinary quiver of the trivial extension $T(A')$ of the auxiliary gentle algebra $A'$
  • Figure 2: Quiver of the trivial extension $T(A^{\text{sg}})$
  • Figure 3: Skew-Brauer graph with two special vertices.
  • Figure 4: Skew-Brauer quiver with two special vertices
  • Figure 5: Quivers of skew-Brauer graphs with underlying tree graph and 2 edges
  • ...and 19 more figures

Theorems & Definitions (79)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Example 3.5
  • ...and 69 more