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Quasi-biserial algebras, special quasi-biserial algebras and symmetric fractional Brauer graph algebras

Bohan Xing

TL;DR

This work introduces quasi-biserial and special quasi-biserial algebras as natural generalizations of biserial and special biserial algebras and situates them within the Brauer-graph paradigm by proving that symmetric special quasi-biserial algebras are exactly sf Brauer graph algebras determined by labeled ribbon graphs with multiplicities. It establishes that every special quasi-biserial algebra is a quotient of a symmetric one, and that symmetric special quasi-biserial algebras admit a complete combinatorial realization as sf Brauer graph algebras. The paper further develops the derived-equivalence theory via Kauer moves (extended to labeled graphs), showing derived equivalences between sf Brauer graph algebras when their labeled ribbon graphs are connected by such moves and share multiplicity data. Collectively, these results extend the Brauer-graph framework to a broader class of algebras, providing a concrete combinatorial classification and a robust toolkit for studying representation type and derived categories in this setting.

Abstract

This paper develops the theory of quasi-biserial and special quasi-biserial algebras, relating them to symmetric fractional Brauer graph algebras. We prove that these algebras generalize key properties of classical biserial algebras, and establish that symmetric special quasi-biserial algebras are exactly those determined by combinatorial data (a labeled ribbon graph with a multiplicity function). Furthermore, we show that Kauer moves on labeled ribbon graphs induce derived equivalences between the corresponding symmetric special quasi-biserial algebras.

Quasi-biserial algebras, special quasi-biserial algebras and symmetric fractional Brauer graph algebras

TL;DR

This work introduces quasi-biserial and special quasi-biserial algebras as natural generalizations of biserial and special biserial algebras and situates them within the Brauer-graph paradigm by proving that symmetric special quasi-biserial algebras are exactly sf Brauer graph algebras determined by labeled ribbon graphs with multiplicities. It establishes that every special quasi-biserial algebra is a quotient of a symmetric one, and that symmetric special quasi-biserial algebras admit a complete combinatorial realization as sf Brauer graph algebras. The paper further develops the derived-equivalence theory via Kauer moves (extended to labeled graphs), showing derived equivalences between sf Brauer graph algebras when their labeled ribbon graphs are connected by such moves and share multiplicity data. Collectively, these results extend the Brauer-graph framework to a broader class of algebras, providing a concrete combinatorial classification and a robust toolkit for studying representation type and derived categories in this setting.

Abstract

This paper develops the theory of quasi-biserial and special quasi-biserial algebras, relating them to symmetric fractional Brauer graph algebras. We prove that these algebras generalize key properties of classical biserial algebras, and establish that symmetric special quasi-biserial algebras are exactly those determined by combinatorial data (a labeled ribbon graph with a multiplicity function). Furthermore, we show that Kauer moves on labeled ribbon graphs induce derived equivalences between the corresponding symmetric special quasi-biserial algebras.
Paper Structure (15 sections, 29 theorems, 27 equations, 3 figures)

This paper contains 15 sections, 29 theorems, 27 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Biserial and special biserial algebras
  3. Brauer graph algebras
  4. Brauer graphs
  5. Brauer graph algebras
  6. Multiserial and special multiserial algebras
  7. Quasi-biserial and special quasi-biserial algebras
  8. Quasi-biserial algebras
  9. Special quasi-biserial algebras
  10. Properties of special quasi-biserial algebras
  11. Symmetric fractional Brauer graph algebras
  12. Sf Brauer graphs and their associated algebras
  13. Representation-finite sf Brauer graph algebras
  14. Kauer moves
  15. Symmetric special quasi-biserial algebras

Key Result

Theorem 1

(see Theorem ssqba quotient) Every special quasi-biserial algebra $A=kQ/I$ is a quotient of a symmetric special quasi-biserial algebra.

Figures (3)

  • Figure 3: $\alpha_h$ in $Q_{\Gamma,L}$ and in $Q_{\Gamma\backslash L}$.
  • Figure 4: Kauer moves on ribbon graphs.
  • Figure 5: The tilting complex for the first Kauer move.

Theorems & Definitions (73)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Theorem 4
  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • ...and 63 more