On BD-algebra and CM-Auslander algebra for a gentle algebra and their representation types

TL;DR

This work shows that for a gentle algebra $A$, the BD-gentle algebra $B=\mathrm{BD}(A)$ and the CM-Auslander algebra $C=A^{\mathrm{CMA}}$ share the same representation-theoretic behavior: $A$, $B$, and $C$ are either all representation-finite or all representation-infinite, and likewise for representation-discrete and derived-discrete properties. The authors develop and leverage geometric marked-surface models to translate BD and CMA constructions into surface data, establishing homotopy-equivalences between the marked surfaces of $A$, $B$, and $C$. They provide explicit bound-quiver constructions for both BD and CMA algebras and show these constructions preserve gentleness, enabling a unified analysis of representation types across the three algebras. The results extend to derived categories, showing that derived-discreteness is also shared among $A$, $B$, and $C$, with proofs grounded in the AG/Bekkert–Merklen perspective on homotopy bands and forbidden threads. Overall, the paper offers a coherent, geometry-inspired framework for understanding how BD and CMA operations affect representation theory in the gentle algebra setting, with implications for stability under common algebraic transformations.

Abstract

Let $A$ be a gentle algebra, and $B$ and $C$ be its BD-gentle algebra and CM-Auslander algebra, respectively. In this paper, we show that the representation-finiteness of $A$, $B$ and $C$ coincide and the representation-discreteness of $A$, $B$ and $C$ coincide.

Figures (14)

• Figure 2.1: The polygon ${\mathsf{P}}_{{\color{red}\pmb{\circ}}}(\pmb{A}_m)$ corresponding to $\pmb{A}_m$
• Figure 2.2: An example for marked surface
• Figure 2.3: The ${\color{red}\pmb{\circ}}$-FFAS and ${\color{blue}\pmb{\bullet}}$-FFAS
• Figure 3.1: $\infty-$elementary polygon
• Figure 3.2: The marked surfaces of $H_i$ and $\mathrm{BD}(H_i)$

Theorems & Definitions (41)

• Theorem 1.1: Proposition \ref{['prop:repr type']} and Theorem \ref{['mainthm:repr-type']}
• Theorem 1.2: Proposition \ref{['prop:der-disc']} and Theorem \ref{['mainthm:der repr-type']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3: OPS2018BCS2021QZZ2022
• Example 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Theorem 2.7